Research ArticleA Proposal to the Study of Contractions in Quasi-Metric Spaces

Hamed H Alsulami1 Erdal KarapJnar12

Farshid Khojasteh3 and Antonio-Francisco Roldaacuten-Loacutepez-de-Hierro4

1 Nonlinear Analysis and Applied Mathematics Research Group (NAAM) King Abdulaziz University Jeddah Saudi Arabia2Department of Mathematics Atilim University Incek 06836 Ankara Turkey3 Department of Mathematics Arak Branch Islamic Azad University Arak Iran4Department of Mathematics University of Jaen Campus las Lagunillas sn 23071 Jaen Spain

Correspondence should be addressed to Erdal Karapınar erdalkarapinaryahoocom

Received 28 March 2014 Accepted 10 July 2014 Published 6 August 2014

Academic Editor Janusz Brzdęk

Copyright copy 2014 Hamed H Alsulami et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate the existence and uniqueness of a fixed point of an operator via simultaneous functions in the setting of completequasi-metric spaces Our results generalize and improve several recent results in literature

1 Introduction and Preliminaries

One of the attractive research subjects in the fixed pointtheory is the investigation of the existence and uniquenessof (common) fixed point of various operators in the settingof quasi-metric space Very recently Jleli and Samet [1]and Samet et al [2] reported that 119866-metrics introduced byMustafa and Sims [3] can be deduced from quasi-metricsby taking 119902(119909 119910) = 119866(119909 119910 119910) Consequently the authors in[1 2] proved that several fixed point results in the setting of119866-metric spaces can be deduced from the corresponding the-orems in the context of quasi-metric spaces The importanceof these results follows from the simplicity of construction ofquasi-metric despite the notion of 119866-metric

In this paper we investigate the existence and uniquenessof a fixed point of operators via simultaneous functionsdefined by Khojasteh et al [4] in the setting of completequasi-metric spaces We also observed that several existingresults can be concluded from ourmain resultsWe also showthat some result in the context of 119866-metric spaces can bededuced from the corresponding theorems in the frameworkof quasi-metric spaces

For the sake of completeness we recollect basic notionsdefinitions and fundamental results Let 119860 119861 sube 119883 be twononempty subsets of a set119883 and let119879 119860 rarr 119861 be amapping

A point 119909 isin 119883 is called a fixed point of the mapping 119879 if 119879119909 =119909

Definition 1 Let 119883 be a nonempty set and let 119902 119883 times 119883 rarr

[0 +infin) be a given function which satisfies

(1) 119902(119909 119910) = 0 if and only if 119909 = 119910

(2) 119902(119909 119910) le 119902(119909 119911) + 119902(119911 119910) for any points 119909 119910 119911 isin 119883

Then 119902 is called a quasi-metric and the pair (119883 119902) is called aquasi-metric space

It is evident that any metric space is a quasi-metricspace but the converse is not true in general Now we recallconvergence and completeness on quasi-metric spaces

Definition 2 Let (119883 119902) be a quasi-metric space and let 119909119899

be a sequence in 119883 and 119909 isin 119883 The sequence 119909119899 converges

to 119909 if

lim119899rarrinfin

119902 (119909119899 119909) = lim

119899rarrinfin

119902 (119909 119909119899) = 0 (1)

Remark 3 Aconvergent sequence in a quasi-metric space hasa unique limit

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 269286 10 pageshttpdxdoiorg1011552014269286

2 Discrete Dynamics in Nature and Society

Remark 4 If 119909119899 converges to 119909 in a quasi-metric space

(119883 119902) then

lim119899rarrinfin

119902 (119909119899 119910) = 119902 (119909 119910) forall119910 isin 119883 (2)

In other words 119902 is a continuous mapping on its firstargument This property follows from 119902(119909

119899 119910) le 119902(119909

119899 119909) +

119902(119909 119910) and 119902(119909 119910) le 119902(119909 119909119899) + 119902(119909

119899 119910) Therefore

119902 (119909 119910) minus 119902 (119909 119909119899) le 119902 (119909

119899 119910) le 119902 (119909

119899 119909) + 119902 (119909 119910)

forall119899 isin N(3)

Definition 5 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

left-Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all 119899 ge 119898 gt 119873

Definition 6 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

right-Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all119898 ge 119899 gt 119873

Definition 7 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all119898 119899 gt 119873

Remark 8 A sequence 119909119899 in a quasi-metric space is Cauchy

if and only if it is left-Cauchy and right-Cauchy

Definition 9 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace We say that

(1) (119883 119902) is left-complete if each left-Cauchy sequence in119883 is convergent

(2) (119883 119902) is right-complete if each right-Cauchysequence in119883 is convergent

(3) (119883 119902) is complete if each Cauchy sequence in 119883 isconvergent

2 Simulation Functions

The notion of simulation function was introduced by Kho-jasteh et al in [4]

Definition 10 (see [4]) A simulation function is a mapping120577 [0infin) times [0infin) rarr R satisfying the following conditions

(1205771) 120577(0 0) = 0

(1205772) 120577(119905 119904) lt 119904 minus 119905 for all 119905 119904 gt 0

(1205773) if 119905

119899 119904119899 are sequences in (0infin) such that

lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899gt 0 then

lim sup119899rarrinfin

120577 (119905119899 119904119899) lt 0 (4)

LetZ be the family of all simulation functions 120577 [0infin)times

[0infin) rarr R

Before presenting our main fixed point results usingsimulation functions we show a wide range of examples tohighlight their potential applicability to the field of fixed pointtheory In the following results themapping 120577 is defined from[0infin) times [0infin) into R

Definition 11 (Khan et al [5]) An altering distance function isa continuous nondecreasing mapping 120601 [0infin) rarr [0infin)

such that 120601minus1(0) = 0

Example 12 Let 120601 and 120595 be two altering distance functionssuch that 120595(119905) lt 119905 le 120601(119905) for all 119905 gt 0 Then the mapping

1205771(119905 119904) = 120595 (119904) minus 120601 (119905) forall119905 119904 isin [0infin) (5)

is a simulation function

If in the previous example 120601(119905) = 119905 and 120595(119905) = 120582119905 forall 119905 ge 0 where 120582 isin [0 1) then we obtain the followingparticular case of simulation function

120577119861(119905 119904) = 120582119904 minus 119905 forall119905 119904 isin [0infin) (6)

Example 13 If 120593 [0infin) rarr [0infin) is a lower semicon-tinuous function such that 120593minus1(0) = 0 and we define 120577

119877

[0infin) times [0infin) rarr R by

120577119877(119905 119904) = 119904 minus 120593 (119904) minus 119905 forall119904 119905 isin [0infin) (7)

then 120577119877is a simulation function

If in the previous example 120593 is continuous we deducethe following case

Example 14 If 120593 [0infin) rarr [0infin) is a continuous functionsuch that 120593(119905) = 0 hArr 119905 = 0 and we define

120577119877(119905 119904) = 119904 minus 120593 (119904) minus 119905 forall119904 119905 isin [0infin) (8)

then 120577119877is a simulation function

Example 15 Let 119891 119892 [0infin) rarr (0infin) be two continuousfunctions with respect to each variable such that 119891(119905 119904) gt119892(119905 119904) for all 119905 119904 gt 0 and define

120577 (119905 119904) = 119904 minus119891 (119905 119904)

119892 (119905 119904)119905 forall119905 119904 isin [0infin) (9)

Then 120577 is a simulation function

Example 16 If 120593 [0infin) rarr [0 1) is a function such thatlim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 and we define

120577119879(119905 119904) = 119904120593 (119904) minus 119905 forall119904 119905 isin [0infin) (10)

then 120577119879is a simulation function

Example 17 If 120578 [0infin) rarr [0infin) is an upper semicontin-uous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and 120578(0) = 0

and we define

120577119861119882

(119905 119904) = 120578 (119904) minus 119905 forall119904 119905 isin [0infin) (11)

then 120577119861119882

is a simulation function

Discrete Dynamics in Nature and Society 3

Example 18 If 120601 [0infin) rarr [0infin) is a function such thatint120576

0

120601(119906)119889119906 exists and int1205760

120601(119906)119889119906 gt 120576 for each 120576 gt 0 and wedefine

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119889119906 forall119904 119905 isin [0infin) (12)

then 120577119870is a simulation function

Example 19 Let ℎ [0infin) times [0infin) rarr [0infin) be a functionsuch that ℎ(119905 119904) lt 1 for all 119905 119904 gt 0 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt

1 provided that 119905119899 and 119904

119899 sub (0 +infin) are two sequences

such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899gt 0 and we define

120577119880(119905 119904) = 119904ℎ (119905 119904) minus 119905 forall119904 119905 isin [0infin) (13)

and then 120577119880is a simulation function

The following results are more theoretical

Proposition 20 Let 120578 [0infin) times [0infin) rarr R be a functionsuch that 120578(0 0) = 0 and there exists 120577 isin Z verifying that120578(119905 119904) le 120577(119905 119904) for all 119904 119905 ge 0 Then 120578 isinZ

Proof For all 119905 119904 gt 0 120578(119905 119904) le 120577(119905 119904) lt 119904minus119905 If 119905119899 and 119904

119899 are

sequences in (0infin) such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt

0 then lim sup119899rarrinfin

120578(119905119899 119904119899) le lim sup

119899rarrinfin120577(119905119899 119904119899) lt 0

Proposition 21 Let 120578119894119894isinN sub Z Then the following state-

ments hold

(a) For each 119896 isin N the function 120578min(119896)

RtimesR rarr R definedby

120578min(119896)

(119905 119904) = min 1205781(119905 119904) 120578

2(119905 119904) 120578

119896(119905 119904) forall119905 119904 ge 0

(14)

is a simulation function (ie 120578min(119896)

isinZ for any 119896 isin N)(b) For each 119896 isin N the function 120578

(119896) RtimesR rarr R defined

by

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) forall119905 119904 ge 0 (15)

is a simulation function (ie 120578(119896)isin for any 119896 isin N)

Proof Since 120578min(119896)(119905 119904) le 120578

1(119905 119904) for all 119905 119904 gt 0 the conclusion

(a) is a direct consequence of Proposition 20 Next we provethe conclusion (b) Let 119896 isin N be given It is obvious that120578(119896)(119905 119904) lt 119904 minus 119905 for all 119904 119905 gt 0 because

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) lt

1

119896

119896

sum

119894=1

(119904 minus 119905) = 119904 minus 119905 (16)

Let 119905119899 119904119899 sub (0 +infin) be two sequences such that

lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt 0 For any 119899 isin N we have

lim sup119899rarrinfin

120578(119896)(119905119899 119904119899) =

1

119896

119896

sum

119894=1

lim sup119899rarrinfin

120578119894(119905119899 119904119899) lt 0 (17)

3 Main Results

In this section we use simulation functions to present a verygeneral kind of contractions on quasi-metric spaces and weprove related existence and uniqueness fixed point theorems

Definition 22 Let (119883 119902) be a quasi-metric space We will saythat a self-mapping 119879 119883 rarr 119883 is a Z-contraction if thereexists 120577 isinZ such that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910)) ge 0 forall119909 119910 isin 119883 (18)

For clarity we will use the termZ119902-contraction when we

want to highlight that 119879 is aZ-contraction on a quasi-metricspace involving the quasi-metric 119902 In such a case we will saythat 119879 is aZ

119902-contraction with respect to 120577

Next we observe some useful properties of Z119902-

contractions in the context of quasi-metric spaces

Remark 23 By axiom (1205773) it is clear that a simulation

function must verify 120577(119903 119903) lt 0 for all 119903 gt 0 Consequently if119879 is aZ

119902-contraction with respect to 120577 isinZ

119902 then

119902 (119879119909 119879119910) = 119902 (119909 119910) forall distinct 119909 119910 isin 119883 (19)

In other words if 119879 is aZ119902-contraction then it cannot be an

isometry

We will prove that if a Z119902-contraction has a fixed point

then it is unique

Lemma 24 If aZ119902-contraction in a quasi-metric space has a

fixed point then it is unique

Proof Let (119883 119902) be a quasi-metric space and let 119879 119883 rarr 119883

be aZ119902-contraction with respect to 120577 isinZ We are reasoning

by contradiction Suppose that there are two distinct fixedpoints 119906 V isin 119883 of the mapping 119879 Then 119902(119906 V) gt 0 By (18)we have

0 le 120577 (119902 (119879119906 119879V) 119902 (119906 V)) = 120577 (119902 (119906 V) 119902 (119906 V)) (20)

which is a contradiction due to Remark 23

Inspired by Browder and Petryshynrsquos paper [6] we willcharacterize the notions of asymptotically right-regularityand asymptotically left-regularity for a self-mapping 119879 in thecontext of quasi-metric space (119883 119902)

Definition 25 We will say that a self-mapping 119879 119883 rarr 119883

on a quasi-metric space (119883 119902) is

(i) asymptotically right-regular at a point 119909 isin 119883 iflim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0(ii) asymptotically left-regular at a point 119909 isin 119883 if

lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0(iii) asymptotically regular if it is both asymptotically

right-regular and asymptotically left-regular

Now we show that a Z119902-contraction is asymptotically

regular at every point of119883

4 Discrete Dynamics in Nature and Society

Lemma 26 Every Z119902-contraction on a quasi-metric space is

asymptotically regular

Proof Let 119909 be an arbitrary point of a quasi-metric space(119883 119902) and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect

to 120577 isin Z If there exists some 119901 isin N such that 119879119901119909 = 119879119901minus1119909then 119910 = 119879

119901minus1

119909 is a fixed point of 119879 that is 119879119910 = 119910Consequently we have that 119879119899119910 = 119910 for all 119899 isin N so

119902 (119879119899

119909 119879119899+1

119909) = 119902 (119879119899minus119901+1

119879119901minus1

119909 119879119899minus119901+2

119879119901minus1

119909)

= 119902 (119879119899minus119901+1

119910 119879119899minus119901+2

119910) = 119902 (119910 119910) = 0

(21)

for sufficient large 119899 isin N Thus we conclude that

lim119899rarrinfin

119902 (119879119899

119909 119879119899+1

119909) = 0 (22)

Similarly lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0 so 119879 is asymptoticallyregular at 119909 On the contrary suppose that 119879119899119909 = 119879

119899minus1

119909 forall 119899 isin N that is

119902 (119879119899

119909 119879119899minus1

119909) gt 0 forall119899 isin N (23)

On what follows from (18) and (1205772) we have that for all 119899 isin

N

0 le 120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909))

lt 119902 (119879119899

119909 119879119899minus1

119909) minus 119902 (119879119899+1

119909 119879119899

119909)

(24)

In particular

119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909) forall119899 isin N (25)

The above inequality yields that 119902(119879119899119909 119879119899minus1119909) is a mono-tonically decreasing sequence of nonnegative real numbersThus there exists 119903 isin [0infin) such that lim

119899rarrinfin119902(119879119899

119909

119879119899+1

119909) = 119903 ge 0 We will prove that 119903 = 0 Suppose on thecontrary that 119903 gt 0 Since 119879 is Z

119902-contraction with respect

to 120577 isinZ119902 by (120577

3) we have

0 le lim sup119899rarrinfin

120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909)) lt 0

(26)

which is a contradiction Thus 119903 = 0 and this proves thatlim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0 Hence 119879 is an asymptoticallyright-regular mapping at 119909 Similarly it can be demonstratedthat 119879 is asymptotically left-regular at 119909

Given a self-mapping119879 119883 rarr 119883 a sequence 119909119899 sube 119883 is

called a Picard sequence of 119879 (or generated by119879) if 119909119899= 119879119909119899minus1

for all 119899 isin N

Remark 27 In the proof of the previous result we have provedthat if 119879 119883 rarr 119883 is a Z

119902-contraction on a quasi-metric

space (119883 119902) and 119909119899= 119879119899minus1

1199091 is a Picard sequence of119879 then

either there exists 1198990isin N such that 119909

1198990

is a fixed point of 119879(ie 119909

1198990+1= 1198791199091198990

= 1199091198990

) or

0 lt 119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909)

0 lt 119902 (119879119899

119909 119879119899+1

119909) lt 119902 (119879119899minus1

119909 119879119899

119909)

forall119899 isin N

(27)

Now we show that every Picard sequence 119909119899 generated

by aZ119902-contraction is always bounded

Lemma 28 Let (119883 119902) be a quasi-metric space and let 119879

119883 rarr 119883 be a Z119902-contraction with respect to 120577 If 119909

119899 is a

Picard sequence generated by 119879 then 119902(119909119899 119909119898) 119899 119898 isin N is

bounded

Proof Let 1199090isin 119883 be arbitrary and let 119909

119899 be defined

iteratively by 119909119899+1

= 119879119909119899for all 119899 ge 0 If there exists some

119899 ge 0 and 119901 ge 1 such that 119909119899+119901

= 119909119899 then the set 119909

119899 119899 isin N

is finite so it is bounded Hence assume that 119909119899+119901

= 119909119899for

all 119899 ge 0 and 119901 ge 1 In this case by Remark 27 we have that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(28)

Notice that by Lemma 26

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (29)

In particular there exists 1198990isin N such that

119902 (119909119899+1 119909119899) lt 1 119902 (119909

119899 119909119899+1) lt 1 forall119899 ge 119899

0 (30)

We will prove that 119909119899 119899 isin N is bounded reasoning

by contradiction We distinguish between right and leftboundedness Suppose that the set

119863 = 119902 (119909119898 119909119899) 119898 gt 119899 (31)

is not bounded Then we can find 1198991

gt 1198990such that

119902(1199091198991

1199091198990

) gt 1 If 1198991is the smallest natural number greater

than 1198990 verifying this property then we can suppose that

119902 (119909119901 1199091198990

) le 1 forall119901 isin 1198990 1198990+ 1 119899

1minus 1 (32)

Again as119863 is not bounded there exists 1198992gt 1198991such that

119902 (1199091198992

1199091198991

) gt 1 119902 (119909119901 1199091198991

) le 1

forall119901 isin 1198991 1198991+ 1 119899

2minus 1

(33)

Repeating this process there exists a partial subsequence119909119899119896

of 119909119899 such that for all 119896 ge 1

119902 (119909119899119896+1

119909119899119896

) gt 1 119902 (119909119901 119909119899119896

) le 1

forall119901 isin 119899119896 119899119896+ 1 119899

119896+1minus 1

(34)

Discrete Dynamics in Nature and Society 5

Therefore by the triangular inequality we have that for all 119896

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1

119909119899119896+1minus1) + 119902 (119909

119899119896+1minus1 119909119899119896

)

le 119902 (119909119899119896+1

119909119899119896+1minus1) + 1

(35)

Letting 119896 rarr infin in (35) and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1

119909119899119896

) = 1 (36)

By (28) we have 119902(119909119899119896+1

119909119899119896

) le 119902(119909119899119896+1minus1 119909119899119896minus1) Therefore

using the triangular inequality we obtain

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1minus1 119909119899119896minus1)

le 119902 (119909119899119896+1minus1 119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

le 1 + 119902 (119909119899119896

119909119899119896minus1)

(37)

Letting 119896 rarr infin and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1minus1 119909119899119896minus1) = 1 (38)

Owing to the fact that 119879 is a Z119902-contraction with respect to

120577 isinZ119902 we deduce from (120577

3) that for all 119896

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119899119896+1minus1 119879119909119899119896minus1) 119902 (119909

119899119896+1minus1 119909119899119896minus1))

= lim sup119896rarrinfin

120577 (119902 (119909119899119896+1

119909119899119896

) 119902 (119909119899119896+1minus1 119909119899119896minus1)) lt 0

(39)

which is a contradiction This proves that 119863 = 119902(119909119898 119909119899)

119898 gt 119899 is bounded Similarly it can be proved that 1198631015840 =119902(119909119898 119909119899) 119898 lt 119899 is also bounded Therefore the set

119902(119909119898 119909119899) 119898 119899 isin N is bounded

In the next theorem we prove the existence of fixed pointof aZ

119902-contraction

Theorem29 EveryZ-contraction on a complete quasi-metricspace has a unique fixed point In fact every Picard sequenceconverges to its unique fixed point

Proof Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect to 120577 Take 119909

0isin

119883 and consider the Picard sequence 119909119899= 119879119899

1199090119899ge0

If 119909119899

contains a fixed point of119879 the proof is finished In other caseLemma 26 and Remark 27 guarantee that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(40)

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (41)

We are going to show that 119909119899 is a left Cauchy sequence For

this purpose taking into account that Lemma 28 guarantees

that 119902(119909119898 119909119899) 119898 119899 isin N is bounded we can consider the

sequence 119862119899 sub [0infin) given by

119862119899= sup (119902 (119909

119894 119909119895) 119894 ge 119895 ge 119899) forall119899 isin N (42)

It is clear that the sequence 119862119899 is a monotonically nonin-

creasing sequence of nonnegative real numbers Thereforeit is convergent that is there exists 119862 ge 0 such thatlim119899rarrinfin

119862119899= 119862 Let us show that 119862 = 0 reasoning by

contradiction If 119862 gt 0 then by definition of 119862119899 for every

119896 isin N there exists 119899119896 119898119896isin N such that119898

119896gt 119899119896ge 119896 and

119862119896minus1

119896lt 119902 (119909

119898119896

119909119899119896

) le 119862119896 (43)

Hence

lim119896rarrinfin

119902 (119909119898119896

119909119899119896

) = 119862 (44)

By using (40) and the triangular inequality we have for all 119896

119902 (119909119898119896

119909119899119896

) le 119902 (119909119898119896minus1 119909119899119896minus1)

le 119902 (119909119898119896minus1 119909119898119896

) + 119902 (119909119898119896

119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

(45)

Letting 119896 rarr infin in the above inequality and using (41) and(44) we derive that

lim119896rarrinfin

119902 (119909119898119896minus1 119909119899119896minus1) = 119862 (46)

Due to fact that 119879 is aZ119902-contraction with respect to 120577 isinZ

119902

and by using (1205773) (18) (44) and (46) we have

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119898119896

119879119909119899119896

) 119902 (119909119898119896

119909119899119896

))

= lim sup119896rarrinfin

120577 (119902 (119909119898119896minus1 119909119899119896minus1) 119902 (119909

119898119896

119909119899119896

)) lt 0

(47)

which is a contradiction This contradiction concludes that119862 = 0 and hence 119909

119899 is a left Cauchy sequence Similarly it

can be proved that 119909119899 is a right Cauchy sequenceTherefore

119909119899 is a Cauchy sequence Since (119883 119902) is a complete quasi-

metric space there exists 119906 isin 119883 such that lim119899rarrinfin

119909119899= 119906

We will show that the point 119906 is a fixed point of 119879reasoning by contradiction Suppose that 119879119906 = 119906 that is119902(119906 119879119906) gt 0 By Remark 4

lim119899rarrinfin

119902 (119879119909119899 119879119906) = lim

119899rarrinfin

119902 (119909119899+1 119879119906) = 119902 (119906 119879119906) gt 0

(48)

Therefore there is 1198990isin N such that

119902 (119879119909119899 119879119906) gt 0 forall119899 ge 119899

0 (49)

In particular 119879119909119899

= 119879119906 This also means that 119909119899

= 119906 for all119899 ge 119899

0 As 119902(119879119909

119899 119879119906) gt 0 and 119902(119909

119899 119906) gt 0 axiom (120577

2) and

property (18) imply that for all 119899 ge 1198990

0 le 120577 (119902 (119879119909119899 119879119906) 119902 (119909

119899 119906)) lt 119902 (119909

119899 119906) minus 119902 (119879119909

119899 119879119906)

(50)

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

Remark 4 If 119909119899 converges to 119909 in a quasi-metric space

(119883 119902) then

lim119899rarrinfin

119902 (119909119899 119910) = 119902 (119909 119910) forall119910 isin 119883 (2)

In other words 119902 is a continuous mapping on its firstargument This property follows from 119902(119909

119899 119910) le 119902(119909

119899 119909) +

119902(119909 119910) and 119902(119909 119910) le 119902(119909 119909119899) + 119902(119909

119899 119910) Therefore

119902 (119909 119910) minus 119902 (119909 119909119899) le 119902 (119909

119899 119910) le 119902 (119909

119899 119909) + 119902 (119909 119910)

forall119899 isin N(3)

Definition 5 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

left-Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all 119899 ge 119898 gt 119873

Definition 6 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

right-Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all119898 ge 119899 gt 119873

Definition 7 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace and let 119909

119899 be a sequence in 119883 We say that 119909

119899 is

Cauchy if for every 120576 gt 0 there exists a positive integer119873 = 119873(120576) such that 119902(119909

119899 119909119898) lt 120576 for all119898 119899 gt 119873

Remark 8 A sequence 119909119899 in a quasi-metric space is Cauchy

if and only if it is left-Cauchy and right-Cauchy

Definition 9 (see eg [1 2]) Let (119883 119902) be a quasi-metricspace We say that

(1) (119883 119902) is left-complete if each left-Cauchy sequence in119883 is convergent

(2) (119883 119902) is right-complete if each right-Cauchysequence in119883 is convergent

(3) (119883 119902) is complete if each Cauchy sequence in 119883 isconvergent

2 Simulation Functions

The notion of simulation function was introduced by Kho-jasteh et al in [4]

Definition 10 (see [4]) A simulation function is a mapping120577 [0infin) times [0infin) rarr R satisfying the following conditions

(1205771) 120577(0 0) = 0

(1205772) 120577(119905 119904) lt 119904 minus 119905 for all 119905 119904 gt 0

(1205773) if 119905

119899 119904119899 are sequences in (0infin) such that

lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899gt 0 then

lim sup119899rarrinfin

120577 (119905119899 119904119899) lt 0 (4)

LetZ be the family of all simulation functions 120577 [0infin)times

[0infin) rarr R

Before presenting our main fixed point results usingsimulation functions we show a wide range of examples tohighlight their potential applicability to the field of fixed pointtheory In the following results themapping 120577 is defined from[0infin) times [0infin) into R

Definition 11 (Khan et al [5]) An altering distance function isa continuous nondecreasing mapping 120601 [0infin) rarr [0infin)

such that 120601minus1(0) = 0

Example 12 Let 120601 and 120595 be two altering distance functionssuch that 120595(119905) lt 119905 le 120601(119905) for all 119905 gt 0 Then the mapping

1205771(119905 119904) = 120595 (119904) minus 120601 (119905) forall119905 119904 isin [0infin) (5)

is a simulation function

If in the previous example 120601(119905) = 119905 and 120595(119905) = 120582119905 forall 119905 ge 0 where 120582 isin [0 1) then we obtain the followingparticular case of simulation function

120577119861(119905 119904) = 120582119904 minus 119905 forall119905 119904 isin [0infin) (6)

Example 13 If 120593 [0infin) rarr [0infin) is a lower semicon-tinuous function such that 120593minus1(0) = 0 and we define 120577

119877

[0infin) times [0infin) rarr R by

120577119877(119905 119904) = 119904 minus 120593 (119904) minus 119905 forall119904 119905 isin [0infin) (7)

then 120577119877is a simulation function

If in the previous example 120593 is continuous we deducethe following case

Example 14 If 120593 [0infin) rarr [0infin) is a continuous functionsuch that 120593(119905) = 0 hArr 119905 = 0 and we define

120577119877(119905 119904) = 119904 minus 120593 (119904) minus 119905 forall119904 119905 isin [0infin) (8)

then 120577119877is a simulation function

Example 15 Let 119891 119892 [0infin) rarr (0infin) be two continuousfunctions with respect to each variable such that 119891(119905 119904) gt119892(119905 119904) for all 119905 119904 gt 0 and define

120577 (119905 119904) = 119904 minus119891 (119905 119904)

119892 (119905 119904)119905 forall119905 119904 isin [0infin) (9)

Then 120577 is a simulation function

Example 16 If 120593 [0infin) rarr [0 1) is a function such thatlim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 and we define

120577119879(119905 119904) = 119904120593 (119904) minus 119905 forall119904 119905 isin [0infin) (10)

then 120577119879is a simulation function

Example 17 If 120578 [0infin) rarr [0infin) is an upper semicontin-uous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and 120578(0) = 0

and we define

120577119861119882

(119905 119904) = 120578 (119904) minus 119905 forall119904 119905 isin [0infin) (11)

then 120577119861119882

is a simulation function

Discrete Dynamics in Nature and Society 3

Example 18 If 120601 [0infin) rarr [0infin) is a function such thatint120576

0

120601(119906)119889119906 exists and int1205760

120601(119906)119889119906 gt 120576 for each 120576 gt 0 and wedefine

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119889119906 forall119904 119905 isin [0infin) (12)

then 120577119870is a simulation function

Example 19 Let ℎ [0infin) times [0infin) rarr [0infin) be a functionsuch that ℎ(119905 119904) lt 1 for all 119905 119904 gt 0 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt

1 provided that 119905119899 and 119904

119899 sub (0 +infin) are two sequences

such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899gt 0 and we define

120577119880(119905 119904) = 119904ℎ (119905 119904) minus 119905 forall119904 119905 isin [0infin) (13)

and then 120577119880is a simulation function

The following results are more theoretical

Proposition 20 Let 120578 [0infin) times [0infin) rarr R be a functionsuch that 120578(0 0) = 0 and there exists 120577 isin Z verifying that120578(119905 119904) le 120577(119905 119904) for all 119904 119905 ge 0 Then 120578 isinZ

Proof For all 119905 119904 gt 0 120578(119905 119904) le 120577(119905 119904) lt 119904minus119905 If 119905119899 and 119904

119899 are

sequences in (0infin) such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt

0 then lim sup119899rarrinfin

120578(119905119899 119904119899) le lim sup

119899rarrinfin120577(119905119899 119904119899) lt 0

Proposition 21 Let 120578119894119894isinN sub Z Then the following state-

ments hold

(a) For each 119896 isin N the function 120578min(119896)

RtimesR rarr R definedby

120578min(119896)

(119905 119904) = min 1205781(119905 119904) 120578

2(119905 119904) 120578

119896(119905 119904) forall119905 119904 ge 0

(14)

is a simulation function (ie 120578min(119896)

isinZ for any 119896 isin N)(b) For each 119896 isin N the function 120578

(119896) RtimesR rarr R defined

by

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) forall119905 119904 ge 0 (15)

is a simulation function (ie 120578(119896)isin for any 119896 isin N)

Proof Since 120578min(119896)(119905 119904) le 120578

1(119905 119904) for all 119905 119904 gt 0 the conclusion

(a) is a direct consequence of Proposition 20 Next we provethe conclusion (b) Let 119896 isin N be given It is obvious that120578(119896)(119905 119904) lt 119904 minus 119905 for all 119904 119905 gt 0 because

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) lt

1

119896

119896

sum

119894=1

(119904 minus 119905) = 119904 minus 119905 (16)

Let 119905119899 119904119899 sub (0 +infin) be two sequences such that

lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt 0 For any 119899 isin N we have

lim sup119899rarrinfin

120578(119896)(119905119899 119904119899) =

1

119896

119896

sum

119894=1

lim sup119899rarrinfin

120578119894(119905119899 119904119899) lt 0 (17)

3 Main Results

In this section we use simulation functions to present a verygeneral kind of contractions on quasi-metric spaces and weprove related existence and uniqueness fixed point theorems

Definition 22 Let (119883 119902) be a quasi-metric space We will saythat a self-mapping 119879 119883 rarr 119883 is a Z-contraction if thereexists 120577 isinZ such that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910)) ge 0 forall119909 119910 isin 119883 (18)

For clarity we will use the termZ119902-contraction when we

want to highlight that 119879 is aZ-contraction on a quasi-metricspace involving the quasi-metric 119902 In such a case we will saythat 119879 is aZ

119902-contraction with respect to 120577

Next we observe some useful properties of Z119902-

contractions in the context of quasi-metric spaces

Remark 23 By axiom (1205773) it is clear that a simulation

function must verify 120577(119903 119903) lt 0 for all 119903 gt 0 Consequently if119879 is aZ

119902-contraction with respect to 120577 isinZ

119902 then

119902 (119879119909 119879119910) = 119902 (119909 119910) forall distinct 119909 119910 isin 119883 (19)

In other words if 119879 is aZ119902-contraction then it cannot be an

isometry

We will prove that if a Z119902-contraction has a fixed point

then it is unique

Lemma 24 If aZ119902-contraction in a quasi-metric space has a

fixed point then it is unique

Proof Let (119883 119902) be a quasi-metric space and let 119879 119883 rarr 119883

be aZ119902-contraction with respect to 120577 isinZ We are reasoning

by contradiction Suppose that there are two distinct fixedpoints 119906 V isin 119883 of the mapping 119879 Then 119902(119906 V) gt 0 By (18)we have

0 le 120577 (119902 (119879119906 119879V) 119902 (119906 V)) = 120577 (119902 (119906 V) 119902 (119906 V)) (20)

which is a contradiction due to Remark 23

Inspired by Browder and Petryshynrsquos paper [6] we willcharacterize the notions of asymptotically right-regularityand asymptotically left-regularity for a self-mapping 119879 in thecontext of quasi-metric space (119883 119902)

Definition 25 We will say that a self-mapping 119879 119883 rarr 119883

on a quasi-metric space (119883 119902) is

(i) asymptotically right-regular at a point 119909 isin 119883 iflim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0(ii) asymptotically left-regular at a point 119909 isin 119883 if

lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0(iii) asymptotically regular if it is both asymptotically

right-regular and asymptotically left-regular

Now we show that a Z119902-contraction is asymptotically

regular at every point of119883

4 Discrete Dynamics in Nature and Society

Lemma 26 Every Z119902-contraction on a quasi-metric space is

asymptotically regular

Proof Let 119909 be an arbitrary point of a quasi-metric space(119883 119902) and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect

to 120577 isin Z If there exists some 119901 isin N such that 119879119901119909 = 119879119901minus1119909then 119910 = 119879

119901minus1

119909 is a fixed point of 119879 that is 119879119910 = 119910Consequently we have that 119879119899119910 = 119910 for all 119899 isin N so

119902 (119879119899

119909 119879119899+1

119909) = 119902 (119879119899minus119901+1

119879119901minus1

119909 119879119899minus119901+2

119879119901minus1

119909)

= 119902 (119879119899minus119901+1

119910 119879119899minus119901+2

119910) = 119902 (119910 119910) = 0

(21)

for sufficient large 119899 isin N Thus we conclude that

lim119899rarrinfin

119902 (119879119899

119909 119879119899+1

119909) = 0 (22)

Similarly lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0 so 119879 is asymptoticallyregular at 119909 On the contrary suppose that 119879119899119909 = 119879

119899minus1

119909 forall 119899 isin N that is

119902 (119879119899

119909 119879119899minus1

119909) gt 0 forall119899 isin N (23)

On what follows from (18) and (1205772) we have that for all 119899 isin

N

0 le 120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909))

lt 119902 (119879119899

119909 119879119899minus1

119909) minus 119902 (119879119899+1

119909 119879119899

119909)

(24)

In particular

119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909) forall119899 isin N (25)

The above inequality yields that 119902(119879119899119909 119879119899minus1119909) is a mono-tonically decreasing sequence of nonnegative real numbersThus there exists 119903 isin [0infin) such that lim

119899rarrinfin119902(119879119899

119909

119879119899+1

119909) = 119903 ge 0 We will prove that 119903 = 0 Suppose on thecontrary that 119903 gt 0 Since 119879 is Z

119902-contraction with respect

to 120577 isinZ119902 by (120577

3) we have

0 le lim sup119899rarrinfin

120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909)) lt 0

(26)

which is a contradiction Thus 119903 = 0 and this proves thatlim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0 Hence 119879 is an asymptoticallyright-regular mapping at 119909 Similarly it can be demonstratedthat 119879 is asymptotically left-regular at 119909

Given a self-mapping119879 119883 rarr 119883 a sequence 119909119899 sube 119883 is

called a Picard sequence of 119879 (or generated by119879) if 119909119899= 119879119909119899minus1

for all 119899 isin N

Remark 27 In the proof of the previous result we have provedthat if 119879 119883 rarr 119883 is a Z

119902-contraction on a quasi-metric

space (119883 119902) and 119909119899= 119879119899minus1

1199091 is a Picard sequence of119879 then

either there exists 1198990isin N such that 119909

1198990

is a fixed point of 119879(ie 119909

1198990+1= 1198791199091198990

= 1199091198990

) or

0 lt 119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909)

0 lt 119902 (119879119899

119909 119879119899+1

119909) lt 119902 (119879119899minus1

119909 119879119899

119909)

forall119899 isin N

(27)

Now we show that every Picard sequence 119909119899 generated

by aZ119902-contraction is always bounded

Lemma 28 Let (119883 119902) be a quasi-metric space and let 119879

119883 rarr 119883 be a Z119902-contraction with respect to 120577 If 119909

119899 is a

Picard sequence generated by 119879 then 119902(119909119899 119909119898) 119899 119898 isin N is

bounded

Proof Let 1199090isin 119883 be arbitrary and let 119909

119899 be defined

iteratively by 119909119899+1

= 119879119909119899for all 119899 ge 0 If there exists some

119899 ge 0 and 119901 ge 1 such that 119909119899+119901

= 119909119899 then the set 119909

119899 119899 isin N

is finite so it is bounded Hence assume that 119909119899+119901

= 119909119899for

all 119899 ge 0 and 119901 ge 1 In this case by Remark 27 we have that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(28)

Notice that by Lemma 26

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (29)

In particular there exists 1198990isin N such that

119902 (119909119899+1 119909119899) lt 1 119902 (119909

119899 119909119899+1) lt 1 forall119899 ge 119899

0 (30)

We will prove that 119909119899 119899 isin N is bounded reasoning

by contradiction We distinguish between right and leftboundedness Suppose that the set

119863 = 119902 (119909119898 119909119899) 119898 gt 119899 (31)

is not bounded Then we can find 1198991

gt 1198990such that

119902(1199091198991

1199091198990

) gt 1 If 1198991is the smallest natural number greater

than 1198990 verifying this property then we can suppose that

119902 (119909119901 1199091198990

) le 1 forall119901 isin 1198990 1198990+ 1 119899

1minus 1 (32)

Again as119863 is not bounded there exists 1198992gt 1198991such that

119902 (1199091198992

1199091198991

) gt 1 119902 (119909119901 1199091198991

) le 1

forall119901 isin 1198991 1198991+ 1 119899

2minus 1

(33)

Repeating this process there exists a partial subsequence119909119899119896

of 119909119899 such that for all 119896 ge 1

119902 (119909119899119896+1

119909119899119896

) gt 1 119902 (119909119901 119909119899119896

) le 1

forall119901 isin 119899119896 119899119896+ 1 119899

119896+1minus 1

(34)

Discrete Dynamics in Nature and Society 5

Therefore by the triangular inequality we have that for all 119896

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1

119909119899119896+1minus1) + 119902 (119909

119899119896+1minus1 119909119899119896

)

le 119902 (119909119899119896+1

119909119899119896+1minus1) + 1

(35)

Letting 119896 rarr infin in (35) and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1

119909119899119896

) = 1 (36)

By (28) we have 119902(119909119899119896+1

119909119899119896

) le 119902(119909119899119896+1minus1 119909119899119896minus1) Therefore

using the triangular inequality we obtain

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1minus1 119909119899119896minus1)

le 119902 (119909119899119896+1minus1 119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

le 1 + 119902 (119909119899119896

119909119899119896minus1)

(37)

Letting 119896 rarr infin and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1minus1 119909119899119896minus1) = 1 (38)

Owing to the fact that 119879 is a Z119902-contraction with respect to

120577 isinZ119902 we deduce from (120577

3) that for all 119896

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119899119896+1minus1 119879119909119899119896minus1) 119902 (119909

119899119896+1minus1 119909119899119896minus1))

= lim sup119896rarrinfin

120577 (119902 (119909119899119896+1

119909119899119896

) 119902 (119909119899119896+1minus1 119909119899119896minus1)) lt 0

(39)

which is a contradiction This proves that 119863 = 119902(119909119898 119909119899)

119898 gt 119899 is bounded Similarly it can be proved that 1198631015840 =119902(119909119898 119909119899) 119898 lt 119899 is also bounded Therefore the set

119902(119909119898 119909119899) 119898 119899 isin N is bounded

In the next theorem we prove the existence of fixed pointof aZ

119902-contraction

Theorem29 EveryZ-contraction on a complete quasi-metricspace has a unique fixed point In fact every Picard sequenceconverges to its unique fixed point

Proof Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect to 120577 Take 119909

0isin

119883 and consider the Picard sequence 119909119899= 119879119899

1199090119899ge0

If 119909119899

contains a fixed point of119879 the proof is finished In other caseLemma 26 and Remark 27 guarantee that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(40)

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (41)

We are going to show that 119909119899 is a left Cauchy sequence For

this purpose taking into account that Lemma 28 guarantees

that 119902(119909119898 119909119899) 119898 119899 isin N is bounded we can consider the

sequence 119862119899 sub [0infin) given by

119862119899= sup (119902 (119909

119894 119909119895) 119894 ge 119895 ge 119899) forall119899 isin N (42)

It is clear that the sequence 119862119899 is a monotonically nonin-

creasing sequence of nonnegative real numbers Thereforeit is convergent that is there exists 119862 ge 0 such thatlim119899rarrinfin

119862119899= 119862 Let us show that 119862 = 0 reasoning by

contradiction If 119862 gt 0 then by definition of 119862119899 for every

119896 isin N there exists 119899119896 119898119896isin N such that119898

119896gt 119899119896ge 119896 and

119862119896minus1

119896lt 119902 (119909

119898119896

119909119899119896

) le 119862119896 (43)

Hence

lim119896rarrinfin

119902 (119909119898119896

119909119899119896

) = 119862 (44)

By using (40) and the triangular inequality we have for all 119896

119902 (119909119898119896

119909119899119896

) le 119902 (119909119898119896minus1 119909119899119896minus1)

le 119902 (119909119898119896minus1 119909119898119896

) + 119902 (119909119898119896

119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

(45)

Letting 119896 rarr infin in the above inequality and using (41) and(44) we derive that

lim119896rarrinfin

119902 (119909119898119896minus1 119909119899119896minus1) = 119862 (46)

Due to fact that 119879 is aZ119902-contraction with respect to 120577 isinZ

119902

and by using (1205773) (18) (44) and (46) we have

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119898119896

119879119909119899119896

) 119902 (119909119898119896

119909119899119896

))

= lim sup119896rarrinfin

120577 (119902 (119909119898119896minus1 119909119899119896minus1) 119902 (119909

119898119896

119909119899119896

)) lt 0

(47)

which is a contradiction This contradiction concludes that119862 = 0 and hence 119909

119899 is a left Cauchy sequence Similarly it

can be proved that 119909119899 is a right Cauchy sequenceTherefore

119909119899 is a Cauchy sequence Since (119883 119902) is a complete quasi-

metric space there exists 119906 isin 119883 such that lim119899rarrinfin

119909119899= 119906

We will show that the point 119906 is a fixed point of 119879reasoning by contradiction Suppose that 119879119906 = 119906 that is119902(119906 119879119906) gt 0 By Remark 4

lim119899rarrinfin

119902 (119879119909119899 119879119906) = lim

119899rarrinfin

119902 (119909119899+1 119879119906) = 119902 (119906 119879119906) gt 0

(48)

Therefore there is 1198990isin N such that

119902 (119879119909119899 119879119906) gt 0 forall119899 ge 119899

0 (49)

In particular 119879119909119899

= 119879119906 This also means that 119909119899

= 119906 for all119899 ge 119899

0 As 119902(119879119909

119899 119879119906) gt 0 and 119902(119909

119899 119906) gt 0 axiom (120577

2) and

property (18) imply that for all 119899 ge 1198990

0 le 120577 (119902 (119879119909119899 119879119906) 119902 (119909

119899 119906)) lt 119902 (119909

119899 119906) minus 119902 (119879119909

119899 119879119906)

(50)

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

Example 18 If 120601 [0infin) rarr [0infin) is a function such thatint120576

0

120601(119906)119889119906 exists and int1205760

120601(119906)119889119906 gt 120576 for each 120576 gt 0 and wedefine

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119889119906 forall119904 119905 isin [0infin) (12)

then 120577119870is a simulation function

Example 19 Let ℎ [0infin) times [0infin) rarr [0infin) be a functionsuch that ℎ(119905 119904) lt 1 for all 119905 119904 gt 0 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt

1 provided that 119905119899 and 119904

119899 sub (0 +infin) are two sequences

such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899gt 0 and we define

120577119880(119905 119904) = 119904ℎ (119905 119904) minus 119905 forall119904 119905 isin [0infin) (13)

and then 120577119880is a simulation function

The following results are more theoretical

Proposition 20 Let 120578 [0infin) times [0infin) rarr R be a functionsuch that 120578(0 0) = 0 and there exists 120577 isin Z verifying that120578(119905 119904) le 120577(119905 119904) for all 119904 119905 ge 0 Then 120578 isinZ

Proof For all 119905 119904 gt 0 120578(119905 119904) le 120577(119905 119904) lt 119904minus119905 If 119905119899 and 119904

119899 are

sequences in (0infin) such that lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt

0 then lim sup119899rarrinfin

120578(119905119899 119904119899) le lim sup

119899rarrinfin120577(119905119899 119904119899) lt 0

Proposition 21 Let 120578119894119894isinN sub Z Then the following state-

ments hold

(a) For each 119896 isin N the function 120578min(119896)

RtimesR rarr R definedby

120578min(119896)

(119905 119904) = min 1205781(119905 119904) 120578

2(119905 119904) 120578

119896(119905 119904) forall119905 119904 ge 0

(14)

is a simulation function (ie 120578min(119896)

isinZ for any 119896 isin N)(b) For each 119896 isin N the function 120578

(119896) RtimesR rarr R defined

by

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) forall119905 119904 ge 0 (15)

is a simulation function (ie 120578(119896)isin for any 119896 isin N)

Proof Since 120578min(119896)(119905 119904) le 120578

1(119905 119904) for all 119905 119904 gt 0 the conclusion

(a) is a direct consequence of Proposition 20 Next we provethe conclusion (b) Let 119896 isin N be given It is obvious that120578(119896)(119905 119904) lt 119904 minus 119905 for all 119904 119905 gt 0 because

120578(119896)(119905 119904) =

1

119896

119896

sum

119894=1

120578119894(119905 119904) lt

1

119896

119896

sum

119894=1

(119904 minus 119905) = 119904 minus 119905 (16)

Let 119905119899 119904119899 sub (0 +infin) be two sequences such that

lim119899rarrinfin

119905119899= lim119899rarrinfin

119904119899= 120575 gt 0 For any 119899 isin N we have

lim sup119899rarrinfin

120578(119896)(119905119899 119904119899) =

1

119896

119896

sum

119894=1

lim sup119899rarrinfin

120578119894(119905119899 119904119899) lt 0 (17)

3 Main Results

In this section we use simulation functions to present a verygeneral kind of contractions on quasi-metric spaces and weprove related existence and uniqueness fixed point theorems

Definition 22 Let (119883 119902) be a quasi-metric space We will saythat a self-mapping 119879 119883 rarr 119883 is a Z-contraction if thereexists 120577 isinZ such that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910)) ge 0 forall119909 119910 isin 119883 (18)

For clarity we will use the termZ119902-contraction when we

want to highlight that 119879 is aZ-contraction on a quasi-metricspace involving the quasi-metric 119902 In such a case we will saythat 119879 is aZ

119902-contraction with respect to 120577

Next we observe some useful properties of Z119902-

contractions in the context of quasi-metric spaces

Remark 23 By axiom (1205773) it is clear that a simulation

function must verify 120577(119903 119903) lt 0 for all 119903 gt 0 Consequently if119879 is aZ

119902-contraction with respect to 120577 isinZ

119902 then

119902 (119879119909 119879119910) = 119902 (119909 119910) forall distinct 119909 119910 isin 119883 (19)

In other words if 119879 is aZ119902-contraction then it cannot be an

isometry

We will prove that if a Z119902-contraction has a fixed point

then it is unique

Lemma 24 If aZ119902-contraction in a quasi-metric space has a

fixed point then it is unique

Proof Let (119883 119902) be a quasi-metric space and let 119879 119883 rarr 119883

be aZ119902-contraction with respect to 120577 isinZ We are reasoning

by contradiction Suppose that there are two distinct fixedpoints 119906 V isin 119883 of the mapping 119879 Then 119902(119906 V) gt 0 By (18)we have

0 le 120577 (119902 (119879119906 119879V) 119902 (119906 V)) = 120577 (119902 (119906 V) 119902 (119906 V)) (20)

which is a contradiction due to Remark 23

Inspired by Browder and Petryshynrsquos paper [6] we willcharacterize the notions of asymptotically right-regularityand asymptotically left-regularity for a self-mapping 119879 in thecontext of quasi-metric space (119883 119902)

Definition 25 We will say that a self-mapping 119879 119883 rarr 119883

on a quasi-metric space (119883 119902) is

(i) asymptotically right-regular at a point 119909 isin 119883 iflim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0(ii) asymptotically left-regular at a point 119909 isin 119883 if

lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0(iii) asymptotically regular if it is both asymptotically

right-regular and asymptotically left-regular

Now we show that a Z119902-contraction is asymptotically

regular at every point of119883

4 Discrete Dynamics in Nature and Society

Lemma 26 Every Z119902-contraction on a quasi-metric space is

asymptotically regular

Proof Let 119909 be an arbitrary point of a quasi-metric space(119883 119902) and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect

to 120577 isin Z If there exists some 119901 isin N such that 119879119901119909 = 119879119901minus1119909then 119910 = 119879

119901minus1

119909 is a fixed point of 119879 that is 119879119910 = 119910Consequently we have that 119879119899119910 = 119910 for all 119899 isin N so

119902 (119879119899

119909 119879119899+1

119909) = 119902 (119879119899minus119901+1

119879119901minus1

119909 119879119899minus119901+2

119879119901minus1

119909)

= 119902 (119879119899minus119901+1

119910 119879119899minus119901+2

119910) = 119902 (119910 119910) = 0

(21)

for sufficient large 119899 isin N Thus we conclude that

lim119899rarrinfin

119902 (119879119899

119909 119879119899+1

119909) = 0 (22)

Similarly lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0 so 119879 is asymptoticallyregular at 119909 On the contrary suppose that 119879119899119909 = 119879

119899minus1

119909 forall 119899 isin N that is

119902 (119879119899

119909 119879119899minus1

119909) gt 0 forall119899 isin N (23)

On what follows from (18) and (1205772) we have that for all 119899 isin

N

0 le 120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909))

lt 119902 (119879119899

119909 119879119899minus1

119909) minus 119902 (119879119899+1

119909 119879119899

119909)

(24)

In particular

119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909) forall119899 isin N (25)

The above inequality yields that 119902(119879119899119909 119879119899minus1119909) is a mono-tonically decreasing sequence of nonnegative real numbersThus there exists 119903 isin [0infin) such that lim

119899rarrinfin119902(119879119899

119909

119879119899+1

119909) = 119903 ge 0 We will prove that 119903 = 0 Suppose on thecontrary that 119903 gt 0 Since 119879 is Z

119902-contraction with respect

to 120577 isinZ119902 by (120577

3) we have

0 le lim sup119899rarrinfin

120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909)) lt 0

(26)

which is a contradiction Thus 119903 = 0 and this proves thatlim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0 Hence 119879 is an asymptoticallyright-regular mapping at 119909 Similarly it can be demonstratedthat 119879 is asymptotically left-regular at 119909

Given a self-mapping119879 119883 rarr 119883 a sequence 119909119899 sube 119883 is

called a Picard sequence of 119879 (or generated by119879) if 119909119899= 119879119909119899minus1

for all 119899 isin N

Remark 27 In the proof of the previous result we have provedthat if 119879 119883 rarr 119883 is a Z

119902-contraction on a quasi-metric

space (119883 119902) and 119909119899= 119879119899minus1

1199091 is a Picard sequence of119879 then

either there exists 1198990isin N such that 119909

1198990

is a fixed point of 119879(ie 119909

1198990+1= 1198791199091198990

= 1199091198990

) or

0 lt 119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909)

0 lt 119902 (119879119899

119909 119879119899+1

119909) lt 119902 (119879119899minus1

119909 119879119899

119909)

forall119899 isin N

(27)

Now we show that every Picard sequence 119909119899 generated

by aZ119902-contraction is always bounded

Lemma 28 Let (119883 119902) be a quasi-metric space and let 119879

119883 rarr 119883 be a Z119902-contraction with respect to 120577 If 119909

119899 is a

Picard sequence generated by 119879 then 119902(119909119899 119909119898) 119899 119898 isin N is

bounded

Proof Let 1199090isin 119883 be arbitrary and let 119909

119899 be defined

iteratively by 119909119899+1

= 119879119909119899for all 119899 ge 0 If there exists some

119899 ge 0 and 119901 ge 1 such that 119909119899+119901

= 119909119899 then the set 119909

119899 119899 isin N

is finite so it is bounded Hence assume that 119909119899+119901

= 119909119899for

all 119899 ge 0 and 119901 ge 1 In this case by Remark 27 we have that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(28)

Notice that by Lemma 26

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (29)

In particular there exists 1198990isin N such that

119902 (119909119899+1 119909119899) lt 1 119902 (119909

119899 119909119899+1) lt 1 forall119899 ge 119899

0 (30)

We will prove that 119909119899 119899 isin N is bounded reasoning

by contradiction We distinguish between right and leftboundedness Suppose that the set

119863 = 119902 (119909119898 119909119899) 119898 gt 119899 (31)

is not bounded Then we can find 1198991

gt 1198990such that

119902(1199091198991

1199091198990

) gt 1 If 1198991is the smallest natural number greater

than 1198990 verifying this property then we can suppose that

119902 (119909119901 1199091198990

) le 1 forall119901 isin 1198990 1198990+ 1 119899

1minus 1 (32)

Again as119863 is not bounded there exists 1198992gt 1198991such that

119902 (1199091198992

1199091198991

) gt 1 119902 (119909119901 1199091198991

) le 1

forall119901 isin 1198991 1198991+ 1 119899

2minus 1

(33)

Repeating this process there exists a partial subsequence119909119899119896

of 119909119899 such that for all 119896 ge 1

119902 (119909119899119896+1

119909119899119896

) gt 1 119902 (119909119901 119909119899119896

) le 1

forall119901 isin 119899119896 119899119896+ 1 119899

119896+1minus 1

(34)

Discrete Dynamics in Nature and Society 5

Therefore by the triangular inequality we have that for all 119896

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1

119909119899119896+1minus1) + 119902 (119909

119899119896+1minus1 119909119899119896

)

le 119902 (119909119899119896+1

119909119899119896+1minus1) + 1

(35)

Letting 119896 rarr infin in (35) and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1

119909119899119896

) = 1 (36)

By (28) we have 119902(119909119899119896+1

119909119899119896

) le 119902(119909119899119896+1minus1 119909119899119896minus1) Therefore

using the triangular inequality we obtain

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1minus1 119909119899119896minus1)

le 119902 (119909119899119896+1minus1 119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

le 1 + 119902 (119909119899119896

119909119899119896minus1)

(37)

Letting 119896 rarr infin and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1minus1 119909119899119896minus1) = 1 (38)

Owing to the fact that 119879 is a Z119902-contraction with respect to

120577 isinZ119902 we deduce from (120577

3) that for all 119896

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119899119896+1minus1 119879119909119899119896minus1) 119902 (119909

119899119896+1minus1 119909119899119896minus1))

= lim sup119896rarrinfin

120577 (119902 (119909119899119896+1

119909119899119896

) 119902 (119909119899119896+1minus1 119909119899119896minus1)) lt 0

(39)

which is a contradiction This proves that 119863 = 119902(119909119898 119909119899)

119898 gt 119899 is bounded Similarly it can be proved that 1198631015840 =119902(119909119898 119909119899) 119898 lt 119899 is also bounded Therefore the set

119902(119909119898 119909119899) 119898 119899 isin N is bounded

In the next theorem we prove the existence of fixed pointof aZ

119902-contraction

Theorem29 EveryZ-contraction on a complete quasi-metricspace has a unique fixed point In fact every Picard sequenceconverges to its unique fixed point

Proof Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect to 120577 Take 119909

0isin

119883 and consider the Picard sequence 119909119899= 119879119899

1199090119899ge0

If 119909119899

contains a fixed point of119879 the proof is finished In other caseLemma 26 and Remark 27 guarantee that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(40)

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (41)

We are going to show that 119909119899 is a left Cauchy sequence For

this purpose taking into account that Lemma 28 guarantees

that 119902(119909119898 119909119899) 119898 119899 isin N is bounded we can consider the

sequence 119862119899 sub [0infin) given by

119862119899= sup (119902 (119909

119894 119909119895) 119894 ge 119895 ge 119899) forall119899 isin N (42)

It is clear that the sequence 119862119899 is a monotonically nonin-

creasing sequence of nonnegative real numbers Thereforeit is convergent that is there exists 119862 ge 0 such thatlim119899rarrinfin

119862119899= 119862 Let us show that 119862 = 0 reasoning by

contradiction If 119862 gt 0 then by definition of 119862119899 for every

119896 isin N there exists 119899119896 119898119896isin N such that119898

119896gt 119899119896ge 119896 and

119862119896minus1

119896lt 119902 (119909

119898119896

119909119899119896

) le 119862119896 (43)

Hence

lim119896rarrinfin

119902 (119909119898119896

119909119899119896

) = 119862 (44)

By using (40) and the triangular inequality we have for all 119896

119902 (119909119898119896

119909119899119896

) le 119902 (119909119898119896minus1 119909119899119896minus1)

le 119902 (119909119898119896minus1 119909119898119896

) + 119902 (119909119898119896

119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

(45)

Letting 119896 rarr infin in the above inequality and using (41) and(44) we derive that

lim119896rarrinfin

119902 (119909119898119896minus1 119909119899119896minus1) = 119862 (46)

Due to fact that 119879 is aZ119902-contraction with respect to 120577 isinZ

119902

and by using (1205773) (18) (44) and (46) we have

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119898119896

119879119909119899119896

) 119902 (119909119898119896

119909119899119896

))

= lim sup119896rarrinfin

120577 (119902 (119909119898119896minus1 119909119899119896minus1) 119902 (119909

119898119896

119909119899119896

)) lt 0

(47)

which is a contradiction This contradiction concludes that119862 = 0 and hence 119909

119899 is a left Cauchy sequence Similarly it

can be proved that 119909119899 is a right Cauchy sequenceTherefore

119909119899 is a Cauchy sequence Since (119883 119902) is a complete quasi-

metric space there exists 119906 isin 119883 such that lim119899rarrinfin

119909119899= 119906

We will show that the point 119906 is a fixed point of 119879reasoning by contradiction Suppose that 119879119906 = 119906 that is119902(119906 119879119906) gt 0 By Remark 4

lim119899rarrinfin

119902 (119879119909119899 119879119906) = lim

119899rarrinfin

119902 (119909119899+1 119879119906) = 119902 (119906 119879119906) gt 0

(48)

Therefore there is 1198990isin N such that

119902 (119879119909119899 119879119906) gt 0 forall119899 ge 119899

0 (49)

In particular 119879119909119899

= 119879119906 This also means that 119909119899

= 119906 for all119899 ge 119899

0 As 119902(119879119909

119899 119879119906) gt 0 and 119902(119909

119899 119906) gt 0 axiom (120577

2) and

property (18) imply that for all 119899 ge 1198990

0 le 120577 (119902 (119879119909119899 119879119906) 119902 (119909

119899 119906)) lt 119902 (119909

119899 119906) minus 119902 (119879119909

119899 119879119906)

(50)

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

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Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Lemma 26 Every Z119902-contraction on a quasi-metric space is

asymptotically regular

Proof Let 119909 be an arbitrary point of a quasi-metric space(119883 119902) and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect

to 120577 isin Z If there exists some 119901 isin N such that 119879119901119909 = 119879119901minus1119909then 119910 = 119879

119901minus1

119909 is a fixed point of 119879 that is 119879119910 = 119910Consequently we have that 119879119899119910 = 119910 for all 119899 isin N so

119902 (119879119899

119909 119879119899+1

119909) = 119902 (119879119899minus119901+1

119879119901minus1

119909 119879119899minus119901+2

119879119901minus1

119909)

= 119902 (119879119899minus119901+1

119910 119879119899minus119901+2

119910) = 119902 (119910 119910) = 0

(21)

for sufficient large 119899 isin N Thus we conclude that

lim119899rarrinfin

119902 (119879119899

119909 119879119899+1

119909) = 0 (22)

Similarly lim119899rarrinfin

119902(119879119899+1

119909 119879119899

119909) = 0 so 119879 is asymptoticallyregular at 119909 On the contrary suppose that 119879119899119909 = 119879

119899minus1

119909 forall 119899 isin N that is

119902 (119879119899

119909 119879119899minus1

119909) gt 0 forall119899 isin N (23)

On what follows from (18) and (1205772) we have that for all 119899 isin

N

0 le 120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909))

lt 119902 (119879119899

119909 119879119899minus1

119909) minus 119902 (119879119899+1

119909 119879119899

119909)

(24)

In particular

119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909) forall119899 isin N (25)

The above inequality yields that 119902(119879119899119909 119879119899minus1119909) is a mono-tonically decreasing sequence of nonnegative real numbersThus there exists 119903 isin [0infin) such that lim

119899rarrinfin119902(119879119899

119909

119879119899+1

119909) = 119903 ge 0 We will prove that 119903 = 0 Suppose on thecontrary that 119903 gt 0 Since 119879 is Z

119902-contraction with respect

to 120577 isinZ119902 by (120577

3) we have

0 le lim sup119899rarrinfin

120577 (119902 (119879119899+1

119909 119879119899

119909) 119902 (119879119899

119909 119879119899minus1

119909)) lt 0

(26)

which is a contradiction Thus 119903 = 0 and this proves thatlim119899rarrinfin

119902(119879119899

119909 119879119899+1

119909) = 0 Hence 119879 is an asymptoticallyright-regular mapping at 119909 Similarly it can be demonstratedthat 119879 is asymptotically left-regular at 119909

Given a self-mapping119879 119883 rarr 119883 a sequence 119909119899 sube 119883 is

called a Picard sequence of 119879 (or generated by119879) if 119909119899= 119879119909119899minus1

for all 119899 isin N

Remark 27 In the proof of the previous result we have provedthat if 119879 119883 rarr 119883 is a Z

119902-contraction on a quasi-metric

space (119883 119902) and 119909119899= 119879119899minus1

1199091 is a Picard sequence of119879 then

either there exists 1198990isin N such that 119909

1198990

is a fixed point of 119879(ie 119909

1198990+1= 1198791199091198990

= 1199091198990

) or

0 lt 119902 (119879119899+1

119909 119879119899

119909) lt 119902 (119879119899

119909 119879119899minus1

119909)

0 lt 119902 (119879119899

119909 119879119899+1

119909) lt 119902 (119879119899minus1

119909 119879119899

119909)

forall119899 isin N

(27)

Now we show that every Picard sequence 119909119899 generated

by aZ119902-contraction is always bounded

Lemma 28 Let (119883 119902) be a quasi-metric space and let 119879

119883 rarr 119883 be a Z119902-contraction with respect to 120577 If 119909

119899 is a

Picard sequence generated by 119879 then 119902(119909119899 119909119898) 119899 119898 isin N is

bounded

Proof Let 1199090isin 119883 be arbitrary and let 119909

119899 be defined

iteratively by 119909119899+1

= 119879119909119899for all 119899 ge 0 If there exists some

119899 ge 0 and 119901 ge 1 such that 119909119899+119901

= 119909119899 then the set 119909

119899 119899 isin N

is finite so it is bounded Hence assume that 119909119899+119901

= 119909119899for

all 119899 ge 0 and 119901 ge 1 In this case by Remark 27 we have that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(28)

Notice that by Lemma 26

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (29)

In particular there exists 1198990isin N such that

119902 (119909119899+1 119909119899) lt 1 119902 (119909

119899 119909119899+1) lt 1 forall119899 ge 119899

0 (30)

We will prove that 119909119899 119899 isin N is bounded reasoning

by contradiction We distinguish between right and leftboundedness Suppose that the set

119863 = 119902 (119909119898 119909119899) 119898 gt 119899 (31)

is not bounded Then we can find 1198991

gt 1198990such that

119902(1199091198991

1199091198990

) gt 1 If 1198991is the smallest natural number greater

than 1198990 verifying this property then we can suppose that

119902 (119909119901 1199091198990

) le 1 forall119901 isin 1198990 1198990+ 1 119899

1minus 1 (32)

Again as119863 is not bounded there exists 1198992gt 1198991such that

119902 (1199091198992

1199091198991

) gt 1 119902 (119909119901 1199091198991

) le 1

forall119901 isin 1198991 1198991+ 1 119899

2minus 1

(33)

Repeating this process there exists a partial subsequence119909119899119896

of 119909119899 such that for all 119896 ge 1

119902 (119909119899119896+1

119909119899119896

) gt 1 119902 (119909119901 119909119899119896

) le 1

forall119901 isin 119899119896 119899119896+ 1 119899

119896+1minus 1

(34)

Discrete Dynamics in Nature and Society 5

Therefore by the triangular inequality we have that for all 119896

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1

119909119899119896+1minus1) + 119902 (119909

119899119896+1minus1 119909119899119896

)

le 119902 (119909119899119896+1

119909119899119896+1minus1) + 1

(35)

Letting 119896 rarr infin in (35) and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1

119909119899119896

) = 1 (36)

By (28) we have 119902(119909119899119896+1

119909119899119896

) le 119902(119909119899119896+1minus1 119909119899119896minus1) Therefore

using the triangular inequality we obtain

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1minus1 119909119899119896minus1)

le 119902 (119909119899119896+1minus1 119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

le 1 + 119902 (119909119899119896

119909119899119896minus1)

(37)

Letting 119896 rarr infin and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1minus1 119909119899119896minus1) = 1 (38)

Owing to the fact that 119879 is a Z119902-contraction with respect to

120577 isinZ119902 we deduce from (120577

3) that for all 119896

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119899119896+1minus1 119879119909119899119896minus1) 119902 (119909

119899119896+1minus1 119909119899119896minus1))

= lim sup119896rarrinfin

120577 (119902 (119909119899119896+1

119909119899119896

) 119902 (119909119899119896+1minus1 119909119899119896minus1)) lt 0

(39)

which is a contradiction This proves that 119863 = 119902(119909119898 119909119899)

119898 gt 119899 is bounded Similarly it can be proved that 1198631015840 =119902(119909119898 119909119899) 119898 lt 119899 is also bounded Therefore the set

119902(119909119898 119909119899) 119898 119899 isin N is bounded

In the next theorem we prove the existence of fixed pointof aZ

119902-contraction

Theorem29 EveryZ-contraction on a complete quasi-metricspace has a unique fixed point In fact every Picard sequenceconverges to its unique fixed point

Proof Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect to 120577 Take 119909

0isin

119883 and consider the Picard sequence 119909119899= 119879119899

1199090119899ge0

If 119909119899

contains a fixed point of119879 the proof is finished In other caseLemma 26 and Remark 27 guarantee that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(40)

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (41)

We are going to show that 119909119899 is a left Cauchy sequence For

this purpose taking into account that Lemma 28 guarantees

that 119902(119909119898 119909119899) 119898 119899 isin N is bounded we can consider the

sequence 119862119899 sub [0infin) given by

119862119899= sup (119902 (119909

119894 119909119895) 119894 ge 119895 ge 119899) forall119899 isin N (42)

It is clear that the sequence 119862119899 is a monotonically nonin-

creasing sequence of nonnegative real numbers Thereforeit is convergent that is there exists 119862 ge 0 such thatlim119899rarrinfin

119862119899= 119862 Let us show that 119862 = 0 reasoning by

contradiction If 119862 gt 0 then by definition of 119862119899 for every

119896 isin N there exists 119899119896 119898119896isin N such that119898

119896gt 119899119896ge 119896 and

119862119896minus1

119896lt 119902 (119909

119898119896

119909119899119896

) le 119862119896 (43)

Hence

lim119896rarrinfin

119902 (119909119898119896

119909119899119896

) = 119862 (44)

By using (40) and the triangular inequality we have for all 119896

119902 (119909119898119896

119909119899119896

) le 119902 (119909119898119896minus1 119909119899119896minus1)

le 119902 (119909119898119896minus1 119909119898119896

) + 119902 (119909119898119896

119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

(45)

Letting 119896 rarr infin in the above inequality and using (41) and(44) we derive that

lim119896rarrinfin

119902 (119909119898119896minus1 119909119899119896minus1) = 119862 (46)

Due to fact that 119879 is aZ119902-contraction with respect to 120577 isinZ

119902

and by using (1205773) (18) (44) and (46) we have

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119898119896

119879119909119899119896

) 119902 (119909119898119896

119909119899119896

))

= lim sup119896rarrinfin

120577 (119902 (119909119898119896minus1 119909119899119896minus1) 119902 (119909

119898119896

119909119899119896

)) lt 0

(47)

which is a contradiction This contradiction concludes that119862 = 0 and hence 119909

119899 is a left Cauchy sequence Similarly it

can be proved that 119909119899 is a right Cauchy sequenceTherefore

119909119899 is a Cauchy sequence Since (119883 119902) is a complete quasi-

metric space there exists 119906 isin 119883 such that lim119899rarrinfin

119909119899= 119906

We will show that the point 119906 is a fixed point of 119879reasoning by contradiction Suppose that 119879119906 = 119906 that is119902(119906 119879119906) gt 0 By Remark 4

lim119899rarrinfin

119902 (119879119909119899 119879119906) = lim

119899rarrinfin

119902 (119909119899+1 119879119906) = 119902 (119906 119879119906) gt 0

(48)

Therefore there is 1198990isin N such that

119902 (119879119909119899 119879119906) gt 0 forall119899 ge 119899

0 (49)

In particular 119879119909119899

= 119879119906 This also means that 119909119899

= 119906 for all119899 ge 119899

0 As 119902(119879119909

119899 119879119906) gt 0 and 119902(119909

119899 119906) gt 0 axiom (120577

2) and

property (18) imply that for all 119899 ge 1198990

0 le 120577 (119902 (119879119909119899 119879119906) 119902 (119909

119899 119906)) lt 119902 (119909

119899 119906) minus 119902 (119879119909

119899 119879119906)

(50)

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Therefore by the triangular inequality we have that for all 119896

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1

119909119899119896+1minus1) + 119902 (119909

119899119896+1minus1 119909119899119896

)

le 119902 (119909119899119896+1

119909119899119896+1minus1) + 1

(35)

Letting 119896 rarr infin in (35) and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1

119909119899119896

) = 1 (36)

By (28) we have 119902(119909119899119896+1

119909119899119896

) le 119902(119909119899119896+1minus1 119909119899119896minus1) Therefore

using the triangular inequality we obtain

1 lt 119902 (119909119899119896+1

119909119899119896

) le 119902 (119909119899119896+1minus1 119909119899119896minus1)

le 119902 (119909119899119896+1minus1 119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

le 1 + 119902 (119909119899119896

119909119899119896minus1)

(37)

Letting 119896 rarr infin and using (29) we obtain

lim119896rarrinfin

119902 (119909119899119896+1minus1 119909119899119896minus1) = 1 (38)

Owing to the fact that 119879 is a Z119902-contraction with respect to

120577 isinZ119902 we deduce from (120577

3) that for all 119896

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119899119896+1minus1 119879119909119899119896minus1) 119902 (119909

119899119896+1minus1 119909119899119896minus1))

= lim sup119896rarrinfin

120577 (119902 (119909119899119896+1

119909119899119896

) 119902 (119909119899119896+1minus1 119909119899119896minus1)) lt 0

(39)

which is a contradiction This proves that 119863 = 119902(119909119898 119909119899)

119898 gt 119899 is bounded Similarly it can be proved that 1198631015840 =119902(119909119898 119909119899) 119898 lt 119899 is also bounded Therefore the set

119902(119909119898 119909119899) 119898 119899 isin N is bounded

In the next theorem we prove the existence of fixed pointof aZ

119902-contraction

Theorem29 EveryZ-contraction on a complete quasi-metricspace has a unique fixed point In fact every Picard sequenceconverges to its unique fixed point

Proof Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a Z

119902-contraction with respect to 120577 Take 119909

0isin

119883 and consider the Picard sequence 119909119899= 119879119899

1199090119899ge0

If 119909119899

contains a fixed point of119879 the proof is finished In other caseLemma 26 and Remark 27 guarantee that

0 lt 119902 (119909119899+1 119909119899) lt 119902 (119909

119899 119909119899minus1)

0 lt 119902 (119909119899 119909119899+1) lt 119902 (119909

119899minus1 119909119899)

forall119899 isin N

(40)

lim119899rarrinfin

119902 (119909119899+1 119909119899) = lim119899rarrinfin

119902 (119909119899 119909119899+1) = 0 (41)

We are going to show that 119909119899 is a left Cauchy sequence For

this purpose taking into account that Lemma 28 guarantees

that 119902(119909119898 119909119899) 119898 119899 isin N is bounded we can consider the

sequence 119862119899 sub [0infin) given by

119862119899= sup (119902 (119909

119894 119909119895) 119894 ge 119895 ge 119899) forall119899 isin N (42)

It is clear that the sequence 119862119899 is a monotonically nonin-

creasing sequence of nonnegative real numbers Thereforeit is convergent that is there exists 119862 ge 0 such thatlim119899rarrinfin

119862119899= 119862 Let us show that 119862 = 0 reasoning by

contradiction If 119862 gt 0 then by definition of 119862119899 for every

119896 isin N there exists 119899119896 119898119896isin N such that119898

119896gt 119899119896ge 119896 and

119862119896minus1

119896lt 119902 (119909

119898119896

119909119899119896

) le 119862119896 (43)

Hence

lim119896rarrinfin

119902 (119909119898119896

119909119899119896

) = 119862 (44)

By using (40) and the triangular inequality we have for all 119896

119902 (119909119898119896

119909119899119896

) le 119902 (119909119898119896minus1 119909119899119896minus1)

le 119902 (119909119898119896minus1 119909119898119896

) + 119902 (119909119898119896

119909119899119896

) + 119902 (119909119899119896

119909119899119896minus1)

(45)

Letting 119896 rarr infin in the above inequality and using (41) and(44) we derive that

lim119896rarrinfin

119902 (119909119898119896minus1 119909119899119896minus1) = 119862 (46)

Due to fact that 119879 is aZ119902-contraction with respect to 120577 isinZ

119902

and by using (1205773) (18) (44) and (46) we have

0 le lim sup119896rarrinfin

120577 (119902 (119879119909119898119896

119879119909119899119896

) 119902 (119909119898119896

119909119899119896

))

= lim sup119896rarrinfin

120577 (119902 (119909119898119896minus1 119909119899119896minus1) 119902 (119909

119898119896

119909119899119896

)) lt 0

(47)

which is a contradiction This contradiction concludes that119862 = 0 and hence 119909

119899 is a left Cauchy sequence Similarly it

can be proved that 119909119899 is a right Cauchy sequenceTherefore

119909119899 is a Cauchy sequence Since (119883 119902) is a complete quasi-

metric space there exists 119906 isin 119883 such that lim119899rarrinfin

119909119899= 119906

We will show that the point 119906 is a fixed point of 119879reasoning by contradiction Suppose that 119879119906 = 119906 that is119902(119906 119879119906) gt 0 By Remark 4

lim119899rarrinfin

119902 (119879119909119899 119879119906) = lim

119899rarrinfin

119902 (119909119899+1 119879119906) = 119902 (119906 119879119906) gt 0

(48)

Therefore there is 1198990isin N such that

119902 (119879119909119899 119879119906) gt 0 forall119899 ge 119899

0 (49)

In particular 119879119909119899

= 119879119906 This also means that 119909119899

= 119906 for all119899 ge 119899

0 As 119902(119879119909

119899 119879119906) gt 0 and 119902(119909

119899 119906) gt 0 axiom (120577

2) and

property (18) imply that for all 119899 ge 1198990

0 le 120577 (119902 (119879119909119899 119879119906) 119902 (119909

119899 119906)) lt 119902 (119909

119899 119906) minus 119902 (119879119909

119899 119879119906)

(50)

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

In particular 0 le 119902(119879119909119899 119879119906) le 119902(119909

119899 119906) for all 119899 ge 119899

0 which

means that

lim119899rarrinfin

119902 (119909119899+1 119879119906) = lim

119899rarrinfin

119902 (119879119909119899 119879119906) = 0 (51)

Similarly it can be proved that lim119899rarrinfin

119902(119879119906 119909119899+1) = 0

Therefore 119909119899 converges at the same time to 119906 and to 119879119906

By the unicity of the limit 119906 = 119879119906 which contradicts119879119906 = 119906As a consequence 119906 is a fixed point of 119879 Notice that theuniqueness of the fixed point follows from Lemma 24

Next we show a variety of cases in whichTheorem 29 canbe applied Firstly we mention the analog of the celebratedBanach contraction principle [7] in quasi-metric spaces

Corollary 30 (see eg [1]) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping such that

119902 (119879119909 119879119910) le 120582119902 (119909 119910) forall119909 119910 isin 119883 (52)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119861isin Z

where 120577119861is defined by 120577

119861(119905 119904) = 120582119904minus 119905 for all 119904 119905 isin [0infin) (see

(6))

The following example shows that the above theorem isa proper generalization of the analog of Banach contractionprinciple

Example 31 Let 120572 120573 119896 isin (0 1) be such that 120572 le 119896 Let 119883 =

[0 1] and 119902 119883 times 119883 rarr [0infin) be a function defined by

119902 (119909 119910) = 119909 minus 119910 if 119909 ge 119910120573 (119910 minus 119909) if 119909 lt 119910

(53)

Then (119883 119902) is a complete quasi-metric space (but it is not ametric space) Consider the mapping 119879 119883 rarr 119883 defined as119879119909 = 120572119909 for all 119909 isin 119883 It is clear that it is a Z

119902-contraction

with respect to 120577 isinZ where

120577 (119905 119904) = 119896119904 minus 119905 forall119905 119904 isin [0infin) (54)

Indeed if 119909 ge 119910 then 119879119909 ge 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120572 (119909 minus 119910) 119909 minus 119910)

= 119896 (119909 minus 119910) minus 120572 (119909 minus 119910) = (119896 minus 120572) (119909 minus 119910) ge 0

(55)

If 119909 lt 119910 then 119879119909 lt 119879119910 Hence we get that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

= 120577 (119902 (120572119909 120572119910) 119902 (119909 119910)) = 120577 (120573 (120572119910 minus 120572119909) 120573 (119910 minus 119909))

= 119896120573 (119910 minus 119909) minus 120573 (120572119910 minus 120572119909) = 120573 (119896 minus 120572) (119910 minus 119909) ge 0

(56)

Notice that all conditions in Theorem 29 are satisfied and 119879has a unique fixed point which is 119909 = 0

In the following corollaries we obtain some knownand some new results in fixed point theory via simulationfunctions

Corollary 32 (Rhoades type) Let (119883 119902) be a complete quasi-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119902 (119879119909 119879119910) le 119902 (119909 119910) minus 120593 (119902 (119909 119910)) forall119909 119910 isin 119883 (57)

where 120593 [0infin) rarr [0infin) is a lower semicontinuousfunction and 120593minus1(0) = 0 Then 119879 has a unique fixed pointin 119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119877isin Z

where 120577119877is defined by 120577

119877(119905 119904) = 119904minus120593(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 13)

Remark 33 Note that Rhoades assumed in [8] that thefunction 120593 was continuous and nondecreasing and it verifiedlim119905rarrinfin

120593(119905) = infin In Corollary 32 we replace these condi-tions by the lower semicontinuity of 120593 which is a weakercondition Therefore our result is stronger than Rhoadesrsquooriginal version

Corollary 34 Let (119883 119902) be a complete quasi-metric space andlet 119879 119883 rarr 119883 be a mapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120593 (119902 (119909 119910)) 119902 (119909 119910) (58)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is a function suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119879isin Z

where 120577119879is defined by 120577

119879(119905 119904) = 119904 120593(119904) minus 119905 for all 119904 119905 isin [0infin)

(see Example 16)

Corollary 35 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping Suppose that for every 119909 119910 isin 119883

119902 (119879119909 119879119910) le 120578 (119902 (119909 119910)) (59)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Proof The result follows from Theorem 29 taking intoaccount that 119879 is aZ

119902-contraction with respect to 120577

119861119882isin Z

where 120577119861119882

is defined by 120577119861119882(119905 119904) = 120578(119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 17)

Corollary 36 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

int

119902(119879119909119879119910)

0

120601 (119905) 119889119905 le 119902 (119909 119910) forall119909 119910 isin 119883 (60)

where 120601 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119889119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in119883

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119870isin Z

where 120577119870is defined by

120577119870(119905 119904) = 119904 minus int

119905

0

120601 (119906) 119902119906 forall119904 119905 isin [0infin) (61)

(see Example 18)

Corollary 37 Let (119883 119902) be a complete quasi-metric space andlet119879 119883 rarr 119883 be amapping satisfying the following condition

119902 (119879119909 119879119910) le ℎ (119902 (119879119909 119879119910) 119902 (119909 119910)) 119902 (119909 119910) forall119909 119910 isin 119883

(62)

where ℎ [0infin) times [0infin) rarr [0infin) is a function such thatℎ(119905 119904) lt 1 and lim sup

119899rarrinfinℎ(119905119899 119904119899) lt 1 provided that 119905

119899

and 119904119899 sub (0 +infin) are two sequences such that lim

119899rarrinfin119905119899=

lim119899rarrinfin

119904119899 Then 119879 has a unique fixed point in119883

Proof The result follows from Theorem 29 taking intoaccount that 119879 is a Z

119902-contraction with respect to 120577

119880isin Z

where 120577119880is defined by 120577

119880(119905 119904) = 119904ℎ(119905 119904)minus119905 for all 119904 119905 isin [0infin)

(see Example 19)

Example 38 The following example is inspired by Remark 3in Boyd andWong [9] Let119883 = [0 1] cup 2 3 4 and let usdefine

119902 (119909 119910) =

0 if 119909 = 119910119909 minus 119910 if 119909 119910 isin [0 1] 119909 gt 119910119909 + 119910 otherwise

(63)

It is apparent that (119883 119902) is a complete quasi-metric space butit is not a metric space (for instance 119902(1 2) = 119902(2 1)) Let usconsider the mappings 119879 119883 rarr 119883 120578 [0infin) rarr R and120577 [0infin) times [0infin) rarr R defined by

119879119909 =

119909 minus1

21199092

if 119909 isin [0 1] 119909 minus 1 if 119909 isin 2 3 4

120578 (119905) =

119905 minus1

41199052

if 119905 isin [0 2]

119905 minus1

2 if 119905 gt 2

120577 (119905 119904) = 120578 (119904) minus 119905 forall119905 119904 ge 0

(64)

Although 120578 is not an upper semicontinuous mapping it iseasy to show that 120577 is a simulation function (if 119905

119899 rarr 120575 gt 0

and 119904119899 rarr 120575 then lim sup

119899rarrinfin120577(119905119899 119904119899) le max(minus12057524

minus12) lt 0) Furthermore it can be proved that

120577 (119902 (119879119909 119879119910) 119902 (119909 119910))

=

0 if 119909 = 119910 or 119909 119910 = 0 2

(119909 + 3119910) (119909 minus 119910)

4 if 119909 119910 isin [0 1] 119909 gt 119910

(119909 minus 119910)2

4 if 119909 119910 isin [0 1] 119909 lt 119910

1 + 119910 (4 minus 119910)

2 if 119909 = 2 0 lt 119910 le 1

1 + 1199092

2 if 119910 = 2 0 lt 119909 le 1

1 + 1199102

2 if 119909 isin 3 4 5 119910 isin [0 1]

1 + 1199092

2 if 119910 isin 3 4 5 119909 isin [0 1]

3

2 if 119909 119910 isin 2 3 4 119909 = 119910

(65)

Therefore 119879 is a Z119902-contraction with respect to 120577 Using

Theorem 29 119879 has a unique fixed point which is 119909 = 0As Boyd and Wong pointed out in [9] as

lim119899rarrinfin

119902 (119879119899 0)

119902 (119899 0)= lim119899rarrinfin

119879119899

119899= lim119899rarrinfin

119899 minus 1

119899= 1 (66)

there can be no decreasing function 120593 with 120593(119905) lt 1 for 119905 gt 0and for which (58) holds Furthermore since

lim119909rarr0

+

119902 (119879119909 0)

119902 (119909 0)= lim119909rarr0

+

119879119909

119909= lim119909rarr0

+

119909 minus 1199092

2

119909

= lim119909rarr0

+

(1 minus119909

2) = 1

(67)

there is no increasing function 120593 with 120593(119905) lt 1 for 119905 gt 0 andfor which (58) holds

Example 39 Let 119883 = [0infin) (it is also possible to consider119883 = [0 119860] where 119860 gt 0) and let us define

119902 (119909 119910) =

119909 minus 119910 if 119909 ge 119910119910 minus 119909

2 if 119909 lt 119910

(68)

It is clear that (119883 119902) is a complete quasi-metric space but itis not a metric space since 119902(1 2) = 119902(2 1) Let us define 119879 119883 rarr 119883 and 120577 [0infin) times [0infin) rarr R by

119879119909 = log (119909 + 1) forall119909 isin 119883

120577 (119905 119904) = log (119904 + 1) minus 119905 forall119905 119904 ge 0

(69)

Then 120577 isin Z and 119879 is a Z119902-contraction with respect to 120577

Therefore 119879 has a unique fixed point which is 119909 = 0

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

4 Consequences Fixed Point Results inthe Context of 119866-Metric Spaces

In this section we show the applicability of our main resultsto the framework of 119866-metric spaces and we indicate thatsome existing fixed point results in that setting can be easilyderived from our main theorems First we recall some basicdefinitions and fundamental results on this topic which canbe found in the literature

Definition 40 (Mustafa and Sims [3]) A generalized metric(or a 119866-metric) on a nonempty set 119883 is a mapping 119866 119883 times

119883 times 119883 rarr [0infin) satisfying the following properties for all119909 119910 119911 119886 isin 119883

(1198661) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911

(1198662) 0 lt 119866(119909 119909 119910) for all 119909 119910 isin 119883 with 119909 = 119910

(1198663) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119910 = 119911

(1198664) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetryin all three variables)

(1198665) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) (rectangle inequal-ity)

In such a case the pair (119883 119866) is called a 119866-metric space

The following result gives some examples of well-known119866-metrics

Lemma 41 If (119883 119889) is a metric space and we define119866max 119866sum 119883 times 119883 times 119883 rarr [0 +infin) for all 119909 119910 119911 isin 119883by

119866max (119909 119910 119911) = max 119889 (119909 119910) 119889 (119910 119911) 119889 (119911 119909)

119866sum (119909 119910 119911) = 119889 (119909 119910) + 119889 (119910 119911) + 119889 (119911 119909)

(70)

then 119866max and 119866sum are 119866-metrics on 119883

Example 42 Let119883 = [0infin) The function 119866 119883times119883times119883 rarr

[0 +infin) defined by

119866 (119909 119910 119911) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119911

1003816100381610038161003816 + |119911 minus 119909| (71)

for all 119909 119910 119911 isin 119883 is a 119866-metric on119883

Conversely a 119866-metric always induces quasi-metrics andalso metrics

Lemma 43 Let (119883 119866) be a 119866-metric space and let us define119902119866 1199021015840

119866 119889

m119866 119889

s119866 119883 times 119883 rarr [0infin) for all 119909 119910 isin 119883 by

119902119866(119909 119910) = 119866 (119909 119909 119910) 119902

1015840

119866(119909 119910) = 119866 (119909 119910 119910)

119889m119866(119909 119910) = max 119866 (119909 119909 119910) 119866 (119909 119910 119910)

119889s119866(119909 119910) = 119866 (119909 119909 119910) + 119866 (119909 119910 119910)

(72)

Then 119902119866and 1199021015840

119866are quasi-metrics on 119883 and 119889m

119866and 119889 s

119866are

metrics on119883

The notions of convergence Cauchy sequence and com-pleteness in a 119866-metric space are as follows

Definition 44 Let (119883 119866) be a 119866-metric space and let 119909119899 be

a sequence of points of 119883 We say that 119909119899 is 119866-convergent

to 119909 isin 119883 iflim119899119898rarrinfin

119866 (119909 119909119899 119909119898) = 0 (73)

that is for any 120576 gt 0 there exists 119873 isin N such that119866(119909 119909

119899 119909119898) lt 120576 for all 119899119898 ge 119873 We call 119909 the limit of the

sequence and write 119909119899 rarr 119909 or lim

119899rarrinfin119909119899= 119909

Proposition 45 If (119883 119866) is a 119866-metric space then thefollowing statements are equivalent

(1) 119909119899 is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

Definition 46 Let (119883 119866) be a 119866-metric space A sequence119909119899 is called a 119866-Cauchy sequence if for any 120576 gt 0 there

exists 119873 isin N such that 119866(119909119899 119909119898 119909119897) lt 120576 for all 119898 119899 119897 ge 119873

that is 119866(119909119899 119909119898 119909119897) rarr 0 as 119899119898 119897 rarr +infin

Proposition 47 Let (119883 119866) be a 119866-metric space Then thefollowing are equivalent

(1) the sequence 119909119899 is 119866-Cauchy

(2) for any 120576 gt 0 there exists 119873 isin N such that119866(119909119899 119909119898 119909119898) lt 120576 for all119898 119899 ge 119873

Definition 48 A 119866-metric space (119883 119866) is called 119866-completeif every 119866-Cauchy sequence is 119866-convergent in (119883 119866)

Formore details on119866-metric space we refer for exampleto [3 10 11]

Lemma 49 (Agarwal et al [12]) Let (119883 119866) be a 119866-metricspace and let us consider the quasi-metrics 119902

119866and 1199021015840

119866as in

Lemma 43 Then the following statements hold(1) 119902119866(119909 119910) le 2119902

1015840

119866(119909 119910) le 4119902

119866(119909 119910) for all 119909 119910 isin 119883

(2) In (119883 119902119866) and in (119883 119902

1015840

119866) a sequence is right-

convergent (resp left-convergent) if and only if it isconvergent In such a case its right-limit its left-limitand its limit coincide

(3) In (119883 119902119866) and in (119883 1199021015840

119866) a sequence is right-Cauchy

(resp left-Cauchy) if and only if it is Cauchy(4) In (119883 119902

119866) and in (119883 119902

1015840

119866) every right-convergent

(resp left-convergent) sequence has a unique right-limit (resp left-limit)

(5) If 119909119899 sube 119883 and 119909 isin 119883 then 119909

119899119866

997888rarr 119909 hArr 119909119899119902119866

997888997888rarr

119909 hArr 1199091198991199021015840

119866

997888997888rarr 119909(6) If 119909

119899 sube 119883 then 119909

119899 is 119866-Cauchy hArr 119909

119899 is 119902119866-

CauchyhArr 119909119899 is 1199021015840119866-Cauchy

(7) (119883 119866) is completehArr (119883 119902119866) is completehArr (119883 119902

1015840

119866) is

complete

We present the following version of Theorem 29 in thecontext of119866-metric spaces using the quasi-metric 119902

119866defined

in Lemma 43

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Corollary 50 Let (119883 119866) be a complete 119866-metric space andlet 119879 119883 rarr 119883 be a mapping such that there exists 120577 isin Zverifying

120577 (119866 (119879119909 119879119910 119879119910) 119866 (119909 119910 119910)) ge 0 forall119909 119910 isin 119883 (74)

Then 119879 has a unique fixed point in 119883 Furthermore everyPicard sequence generated by 119879 converges to the unique fixedpoint of 119879

Proof Since (119883 119866) is complete then item 7 of Lemma 49guarantees that (119883 119902

119866) is a complete quasi-metric space and

119879 is aZ119902119866

-contraction in (119883 119902119866) with respect to 120577

The following results are consequence of Corollaries 30ndash37 applied to the quasi-metric 119902

119866(119909 119910) = 119866(119909 119909 119910) for all

119909 119910 isin 119883 (generated by a 119866-metric)

Corollary 51 Let (119883 119866) be a complete 119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

119866 (119879119909 119879119910 119879119910) le 120582119866 (119909 119910 119910) forall119909 119910 isin 119883 (75)

where 120582 isin [0 1) Then 119879 has a unique fixed point in119883

Corollary 52 (see eg [13]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping satisfying thefollowing condition

119866 (119879119909 119879119910 119879119910) le 119866 (119909 119910 119910) minus 120593 (119866 (119909 119910 119910)) forall119909 119910 isin 119883

(76)

where 120593 [0infin) rarr [0infin) is lower semicontinuous functionand 120593minus1(0) = 0 Then 119879 has a unique fixed point in119883

Corollary 53 (see eg [14]) Let (119883 119866) be a complete 119866-metric space and let 119879 119883 rarr 119883 be a mapping Suppose thatfor every 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120593 (119866 (119909 119910 119910)) 119866 (119909 119910 119910) (77)

for all 119909 119910 isin 119883 where 120593 [0 +infin) rarr [0 1) is amapping suchthat lim sup

119905rarr 119903+120593(119905) lt 1 for all 119903 gt 0 Then 119879 has a unique

fixed point

Corollary 54 (cf [15]) Let (119883 119866) be a complete 119866-metricspace and let 119879 119883 rarr 119883 be a mapping Suppose that forevery 119909 119910 isin 119883

119866 (119879119909 119879119910 119879119910) le 120578 (119866 (119909 119910 119910)) (78)

for all 119909 119910 isin 119883 where 120578 [0 +infin) rarr [0 +infin) is an uppersemicontinuous mapping such that 120578(119905) lt 119905 for all 119905 gt 0 and120578(0) = 0 Then 119879 has a unique fixed point

Corollary 55 Let (119883 119866) be a complete119866-metric space and let119879 119883 rarr 119883 be a mapping satisfying the following condition

int

119866(119879119909119879119910119879119910)

0

120601 (119905) 119902119905 le 119866 (119909 119910 119910) forall119909 119910 isin 119883 (79)

where 120593 [0infin) rarr [0infin) is a function such that int1205980

120601(119905)119902119905

exists and int1205980

120601(119905)119902119905 gt 120598 for each 120598 gt 0 Then 119879 has a uniquefixed point in 119883

Finally we point out that obviously if we replace119866(119879119909 119879119910 119879119910) and 119866(119909 119910 119910) in Corollaries 50ndash55 by theexpressions 119866(119879119909 119879119910 119879119911) and 119866(119909 119910 119911) respectively thenthe conclusion is still valid (because the contractive condi-tions are stronger)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

This research was supported by Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah SaudiArabia The authors thank the anonymous referees for theirremarkable comments suggestions and ideas that helped toimprove this paper

References

[1] M Jleli and B Samet ldquoRemarks on G-metric spaces and fixedpoint theoremsrdquo Fixed Point Theory and Applications vol 2012article 210 2012

[2] B Samet C Vetro and F Vetro ldquoRemarks on119866-metric spacesrdquoInternational Journal of Analysis vol 2013 Article ID 917158 6pages 2013

[3] Z Mustafa and B Sims ldquoA new approach to generalized metricspacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash297 2006

[4] F Khojasteh S Shukla and S Radenovic ldquoA new approachto the study of fixed point theorems via simulation functionsrdquoFilomat In press

[5] M S Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984

[6] F E Browder and W V Petryshyn ldquoThe solution by iterationof nonlinear functional equations in Banach spacesrdquo Bulletin ofthe American Mathematical Society vol 72 pp 571ndash575 1966

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application auxequations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications vol 47 pp2683ndash2693 2001

[9] D W Boyd and J S W Wong ldquoOn nonlinear contractionsrdquoProceedings of the American Mathematical Society vol 20 no2 pp 458ndash464 1969

[10] Z Mustafa and B Sims ldquoFixed point theorems for contractivemappings in complete119866-metric spacesrdquo Fixed PointTheory andApplications vol 2009 Article ID 917175 10 pages 2009

[11] Z Mustafa A new structure for generalized metric spaces withapplications to fixed point theory [PhD thesis] The Universityof Newcastle Callaghan Australia 2005

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

[12] R Agarwal E Karapınar and A F Roldan-Lopez-de-HierroldquoFixed point theorems in quasi-metric spaces and applicationsto coupledtripled fixed points on Glowast-metric spacesrdquo Journal ofNonlinear and Convex Analysis In press

[13] C TAage and JN Salunke ldquoFixed points forweak contractionsin 119866-metric spacesrdquo Applied Mathematics E-Notes vol 12 pp23ndash28 2012

[14] R P Agarwal and E Karapınar ldquoRemarks on some coupledfixed point theorems in 119866-metric spacesrdquo Fixed Point Theoryand Applications vol 2013 article 2 33 pages 2013

[15] W Shatanawi ldquoFixed point theory for contractive mappingssatisfying 120601-maps in 119866-metric spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 181650 9 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of