Research ArticleApproximate Quadratic-Additive Mappings in FuzzyNormed Spaces

Ick-Soon Chang1 and Yang-Hi Lee2

1 Department of Mathematics Chungnam National University 79 Daehangno Yuseong-gu Daejeon 305-764 Republic of Korea2Department of Mathematics Education Gongju National University of Education Gongju 314-711 Republic of Korea

Correspondence should be addressed to Ick-Soon Chang ischangcnuackr

Received 11 March 2014 Revised 9 May 2014 Accepted 16 May 2014 Published 26 May 2014

Academic Editor Dorian Popa

Copyright copy 2014 I-S Chang and Y-H Lee 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 examine the generalized Hyers-Ulam stability of the following functional equation 2119891 (119909 + 119910 + 119911 + 119908) + 119891 (minus119909 minus 119910 + 119911 + 119908) +119891 (minus119909 + 119910 minus 119911 + 119908)+119891 (minus119909 + 119910 + 119911 minus 119908)+119891 (119909 minus 119910 minus 119911 + 119908)+119891 (119909 minus 119910 + 119911 minus 119908)+119891 (119909 + 119910 minus 119911 minus 119908)minus5119891 (119909)minus 3119891 (minus119909)minus 5119891 (119910)minus

3119891 (minus119910) minus 5119891 (119911) minus 3119891 (minus119911) minus 5119891 (119908) minus 3119891 (minus119908) = 0 in the fuzzy normed spaces with the fixed point method

1 Introduction

The problem of stability for functional equations originatedfrom questions of Ulam [1] concerning the stability of grouphomomorphisms Hyers [2] had answered affirmatively thequestion of Ulam for Banach spaces The theorem of Hyerswas generalized by Aoki [3] for additive mappings and byRassias [4] for linearmappings by considering an unboundedCauchy difference Thereafter many interesting results of thegeneralized Hyers-Ulam stability to a number of functionalequations and mappings have been investigated EspeciallyCadariu and Radu [5] observed that the existence of thesolution for a functional equation and the estimation of thedifference with the given mapping can be obtained from thefixed point alternative This method is called a fixed pointmethod Also they [6 7] applied this method to prove thestability theorems of the additive functional equation

Katsaras [8] defined a fuzzy norm on a linear space toconstruct a fuzzy structure on the space Since then somemathematicians have introduced several types of fuzzy normin different points of view In particular Bag and Samantafollowing Cheng andMordeson gave an idea of a fuzzy normin such a manner that the corresponding fuzzy metric is ofKramosil and Michalek type [9ndash11] In 2008 Mirmostafaeeand Moslehian [12 13] obtained a fuzzy stability for theadditive functional equation and for the quadratic functionalequation

On the other hand there are some papers where severalresults of stability for different functional equations areproved in probabilistic metric and random normed spaces(see eg [14ndash17]) after that the results were established infuzzy normed spaces or in non-Archimedean fuzzy normedspaces [18ndash21] In these papers except [20] the fixed pointmethod is used Moreover in some of them another type ofmetric is used (see eg [17])

In this paper we take into account the generalizedHyers-Ulam stability of the following quadratic-additive typefunctional equation

2119891 (119909 + 119910 + 119911 + 119908) + 119891 (minus119909 minus 119910 + 119911 + 119908)

+ 119891 (minus119909 + 119910 minus 119911 + 119908) + 119891 (minus119909 + 119910 + 119911 minus 119908)

+ 119891 (119909 minus 119910 minus 119911 + 119908) + 119891 (119909 minus 119910 + 119911 minus 119908)

+ 119891 (119909 + 119910 minus 119911 minus 119908) minus 5119891 (119909) minus 3119891 (minus119909)

minus 5119891 (119910) minus 3119891 (minus119910) minus 5119891 (119911)

minus 3119891 (minus119911) minus 5119891 (119908) minus 3119891 (minus119908) = 0

(1)

in the fuzzy normed spaces via the fixed point method Firstof all it is known that if a mapping 119891 satisfies the functionalequation (1) then 119891 is quadratic-additive mapping in [22]

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 494781 7 pageshttpdxdoiorg1011552014494781

CORE Metadata citation and similar papers at coreacuk

Provided by MUCC (Crossref)

2 Discrete Dynamics in Nature and Society

Thus the functional equation (1) may be called the quadratic-additive type functional equation and the general solution offunctional equation (1) may be called the quadratic-additivemapping However the stability problem for the functionalequation (11) in [22] is not investigated and so in this paperwe deal with the stability of this equation

2 Preliminaries

We first introduce one of the fundamental results of the fixedpoint theory For the proof we refer to [23] or [24]

Theorem 1 (the fixed point alternative) Assume that (119883 119889) isa complete generalized metric space andΛ 119883 rarr 119883 is a strictcontraction with the Lipschitz constant 119871 lt 1 If there existsa nonnegative integer 119899

0such that 119889(Λ1198990+1119909 Λ1198990119909) lt infin for

some 119909 isin 119883 then the following statements are true

(F1) The sequence Λ119899119909 converges to a fixed point 119909lowast of Λ(F2) 119909lowast is the unique fixed point of Λ in 119883lowast = 119910 isin 119883 |

119889(Λ1198990119909 119910) lt infin

(F3) If 119910 isin 119883lowast then

119889 (119910 119909lowast) le

1

1 minus 119871119889 (Λ119910 119910) (2)

We now introduce the definition of fuzzy normed spacesto establish a reasonable fuzzy stability for the quadratic andadditive functional equation (1) in the fuzzy normed spaces(cf [9])

Definition 2 Let119883 be a real linear space A function119873 119883times

R rarr [0 1] is said to be a fuzzy norm on 119883 if the followingconditions are true

(N1) 119873(119909 119905) = 0 for all 119909 isin 119883 and 119905 le 0(N2) 119909 = 0 if and only if119873(119909 119905) = 1 for all 119905 gt 0(N3) 119873(119888119909 119905) = 119873(119909 119905|119888|) for all 119909 isin 119883 and 119888 119905 isin R with

119888 = 0(N4) 119873(119909+119910 119904+ 119905) ge min119873(119909 119904)119873(119910 119905) for all 119909 119910 isin 119883

and 119904 119905 isin R(N5) 119873(119909 sdot) is a nondecreasing function on R and

lim119905rarrinfin

119873(119909 119905) = 1 for all 119909 isin 119883

The pair (119883119873) is called a fuzzy normed space Let (119883119873)be a fuzzy normed space A sequence 119909

119899 in 119883 is said to be

convergent if there exists an 119909 isin 119883 such that lim119899rarrinfin

119873(119909119899minus

119909 119905) = 1 for all 119905 gt 0 In this case 119909 is called the limit of thesequence 119909

119899 and we write 119873-lim

119899rarrinfin119909119899= 119909 A sequence

119909119899 in 119883 is called Cauchy if for each 120576 gt 0 and each 119905 gt 0

there exists an 1198990isin N such that 119873(119909

119899+119901minus 119909119899 119905) gt 1 minus 120576 for

all 119899 ge 1198990and all 119901 isin N It is known that every convergent

sequence in a fuzzy normed space is Cauchy If every Cauchysequence in 119883 converges in 119883 then the fuzzy norm is saidto be complete and the fuzzy normed space is called a fuzzyBanach space

In this paper we note that the triangular norm 119879 = minis used (see the definition of the fuzzy norm in the axiom

(1198734)) while in some recent papers properties of generalized

Hyers-Ulam stability by taking other triangular norms havebeen discussed (eg of Hadzic type in [25] )

3 Generalized Hyers-Ulam Stability of (1)Let (119883119873) and (1198841198731015840) be a fuzzy normed space and a fuzzyBanach space respectively For a given mapping 119891 119883 rarr 119884we use the abbreviation119863119891 (119909 119910 119911 119908) = 2119891 (119909 + 119910 + 119911 + 119908) + 119891 (minus119909 minus 119910 + 119911 + 119908)

+ 119891 (minus119909 + 119910 minus 119911 + 119908)

+ 119891 (minus119909 + 119910 + 119911 minus 119908)

+ 119891 (119909 minus 119910 minus 119911 + 119908) + 119891 (119909 minus 119910 + 119911 minus 119908)

+ 119891 (119909 + 119910 minus 119911 minus 119908) minus 5119891 (119909) minus 3119891 (minus119909)

minus 5119891 (119910) minus 3119891 (minus119910) minus 5119891 (119911)

minus 3119891 (minus119911) minus 5119891 (119908) minus 3119891 (minus119908)

(3)

for all 119909 119910 119911 119908 isin 119883In the following theorem we investigate the stability

problems of the functional equation (1) between fuzzynormed spaces

Theorem 3 Let (X 119873) and (11988511987310158401015840) be fuzzy normed spacesand let (1198841198731015840) be a fuzzy Banach space Assume that amapping120593 1198834rarr 119885 satisfies one of the following conditions

(i) 11987310158401015840(120572120593(119909 119910 119911 119908) 119905) le 11987310158401015840(120593(2119909 2119910 2119911 2119908) 119905) le

11987310158401015840(1205721015840120593(119909 119910 119911 119908) 119905) for some 1 le 1205721015840 le 120572 lt 2

(ii) 11987310158401015840(120572120593(119909 119910 119911 119908) 119905) le 11987310158401015840(120593(2119909 2119910 2119911 2119908) 119905) for

some 0 lt 120572 lt 1(iii) 11987310158401015840(120593(2119909 2119910 2119911 2119908) 119905) le 119873

10158401015840(120572120593(119909 119910 119911 119908) 119905) for

some 120572 gt 4for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 If a mapping 119891 119883 rarr 119884 with119891(0) = 0 satisfies

1198731015840(119863119891 (119909 119910 119911 119908) 119905) ge 119873

10158401015840(120593 (119909 119910 119911 119908) 119905) (4)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 then there exists a uniquequadratic-additive mapping 119865 119883 rarr 119884 such that

1198731015840(119865 (119909) minus 119891 (119909) 119905)

ge 119872(119909 2 (2 minus 120572) 119905) if 120593 satisfies (119894) or (119894119894)

119872 (119909 2 (120572 minus 4) 119905) if 120593 satisfies (119894119894119894)

(5)

for all 119909 isin 119883 and 119905 gt 0 where

119872(119909 119905)

= min 11987310158401015840 (120593 (119909 119909 0 0) 119905) 11987310158401015840 (120593 (minus119909 minus119909 0 0) 119905) (6)

Moreover if 11987310158401015840(120593(119909 119910 0 0) 119905) is continuous in 119909 119910 underthe condition (119894119894) then the mapping 119891 is a quadratic-additivemapping

Discrete Dynamics in Nature and Society 3

Proof We will take into account three different cases for theassumption of 120593

Case 1 Assume that 120593 satisfies the condition (i) We considerthe set of functions

119878 = 119892 119883 997888rarr 119884 | 119892 (0) = 0 (7)

and introduce a generalized metric on 119878 by

119889 (119892 ℎ) = inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905)

ge 119872 (119909 119905) forall119909 isin 119883

(8)

We first prove that 119889 is a generalized metric on 119878 If119889(119892 ℎ) = 0 that is

inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905) ge 119872 (119909 119905) forall119909 isin 119883

= 0

(9)

then we see that

1198731015840(119892 (119909) minus ℎ (119909) 119906119905) ge 119872 (119909 119905) (10)

for all 119909 isin 119883 and all 119906 gt 0 which means that

1198731015840(119892 (119909) minus ℎ (119909) 119905) ge 119872(119909

119905

119906) (11)

for all 119909 isin 119883 and all 119906 gt 0 It follows that

1198731015840(119892 (119909) minus ℎ (119909) 119905) ge lim

119906rarr0

119872(119909119905

119906) = 1 (12)

So we get 119892(119909) = ℎ(119909) for all 119909 isin 119883Conversely if 119892(119909) = ℎ(119909) for all 119909 isin 119883 then we have

1198731015840(119892 (119909) minus ℎ (119909) 119906119905) = 119873

1015840(0 119906119905) = 1 ge 119872 (119909 119905) (13)

for all 119906 and 119905 So we know that

119889 (119892 ℎ) = inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905)

ge 119872 (119909 119905) forall119909 isin 119883 = 0

(14)

Of course it is easily checked that 119889(119892 ℎ) = 119889(ℎ 119892) for all119892 ℎ isin 119878

Let 119906 V gt 0 such that 119889(119891 119892) lt 119906 and 119889(119892 ℎ) lt V Then

1198731015840(119891 (119909) minus 119892 (119909) 119906119905) ge 119872 (119909 119905)

1198731015840(119892 (119909) minus ℎ (119909) V119905) ge 119872 (119909 119905)

(15)

for all 119909 isin 119883 and all 119905 gt 0 Thus we find that

1198731015840(119891 (119909) minus ℎ (119909) (119906 + V) 119905)

ge min 1198731015840 (119891 (119909) minus 119892 (119909) 119906119905) 1198731015840 (119892 (119909) minus ℎ (119909) V119905)

ge 119872 (119909 119905)

(16)

This implies that 119906+V ge 119889(119891 ℎ) Hence we yield that 119889(119891 ℎ) le119889(119891 119892) + 119889(119892 ℎ) Therefore 119889 is a generalized metric on 119878

Now if we define a function 119869 119878 rarr 119878 by

119869119891 (119909) = (1

8) (119891 (2119909) + 119891 (minus2119909)) + (

1

4) (119891 (2119909) minus 119891 (minus2119909))

(17)

for all 119909 isin 119883 then we have

119869119899119891 (119909) = 2

minus2119899minus1(119891 (2119899119909) + 119891 (minus2

119899119909))

+2minus119899minus1

(119891 (2119899119909) minus 119891 (minus2

119899119909))

(18)

for all 119909 isin 119883 and all 119899 isin N cup 0For any 119891 119892 isin 119878 let 119906 isin [0infin] be an arbitrary constant

with 119889(119892 119891) le 119906 The definition of 119889 provides that for 0 lt

120572 lt 2

1198731015840(119869119892 (119909) minus 119869119891 (119909)

120572119906119905

2)

= 1198731015840((

3

8) (119892 (2119909) minus 119891 (2119909))

minus (1

8) (119892 (minus2119909) minus 119891 (minus2119909))

120572119906119905

2)

ge min 1198731015840 ((38) (119892 (2119909) minus 119891 (2119909))

3120572119906119905

8)

1198731015840((

1

8) (119892 (minus2119909) minus 119891 (minus2119909))

120572119906119905

8)

= min 1198731015840 (119892 (2119909) minus 119891 (2119909) 120572119906119905)

1198731015840(119892 (minus2119909) minus 119891 (minus2119909) 120572119906119905)

ge min 119872 (2119909 120572119905) 119872 (minus2119909 120572119905)

ge 119872 (119909 119905)

(19)

for all 119909 isin 119883 which implies that 119889(119869119891 119869119892) le (1205722)119889(119891 119892)Thus 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 1205722

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

4)

= 1198731015840(minus5119863119891 (119909 119909 0 0) + 3119863119891 (minus119909 minus119909 0 0)

32119905

4)

ge min1198731015840 (minus5119863119891 (119909 119909 0 0)

325119905

32)

1198731015840(3119863119891 (minus119909 minus119909 0 0)

323119905

32)

ge min 1198731015840 (119863119891 (119909 119909 0 0) 119905)

1198731015840(119863119891 (minus119909 minus119909 0 0) 119905)

4 Discrete Dynamics in Nature and Society

ge min 11987310158401015840 (120593 (119909 119909 0 0) 119905)

11987310158401015840(120593 (minus119909 minus119909 0 0) 119905)

= 119872 (119909 119905)

(20)

for all 119909 isin 119883 The above inequality and the definition of 119889show that 119889(119891 119869119891) le 14

According toTheorem 1 the sequence 119869119899119891 converges toa unique fixed point 119865 119883 rarr 119884 of 119869 in the set 119879 = 119892 isin 119878 |

119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2minus2119899minus1

(119891 (2119899119909) + 119891 (minus2

119899119909))

+2minus119899minus1

(119891 (2119899119909) minus 119891 (minus2

119899119909)))

(21)

for all 119909 isin 119883 We observe that

119889 (119891 119865) le1

1 minus 1205722119889 (119891 119869119891) le

1

2 (2 minus 120572) (22)

This guarantees that inequality (5) holdsNext we are in the position to prove that 119865 is quadratic-

additive mapping Now we figure out the relation

1198731015840(119863119865 (119909 119910 119911 119908) 119905)

ge min 1198731015840 (2 (119865 minus 119869119899119891) (119909 + 119910 + 119911 + 119908) 11990530)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 minus 119908)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119909)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119909)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119910)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119910)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119911)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119911)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119908)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119908)

119905

30)

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

(23)

for all 119909 119910 119911 119908 isin 119883 and all 119899 isin N The first fifteen termson the right-hand side of the above inequality tend to 1 as119899 rarr infin by the definition of 119865 Moreover we find that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (2minus2119899minus1119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 1199058)

1198731015840(2minus2119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (2119899119909 2119899119910 2119899119911 2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

ge min 1198731015840 (119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 22119899minus2119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 22119899minus2

119905)

1198731015840(119863119891 (2

119899119909 2119899119910 2119899119911 2119899119908) 2119899minus2

119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 2119899minus2

119905)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 22119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

2119899minus2120572minus119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

119899minus2120572minus119899119905)

(24)

which tends to 1 as 119899 rarr infin since 2120572 gt 1 Thereforeinequality (23) gives that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (25)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 So we deduce that119863119865(119909 119910 119911 119908) = 0 for all 119909 119910 119911 119908 isin 119883

In order to show the uniqueness of 119865 we assume that 1198651015840 119883 rarr 119884 is another quadratic-additivemapping satisfying (5)and then we yield that

1198691198651015840(119909) =

51198631198651015840(119909 119909 0 0) minus 3119863119865

1015840(minus119909 minus119909 0 0)

32+ 1198651015840(119909)

= 1198651015840(119909)

(26)

Discrete Dynamics in Nature and Society 5

for all 119909 isin 119883That is1198651015840 is another fixed point of 119869 Since119865 is aunique fixed point of 119869 in the set 119879 we conclude that 119865 = 119865

1015840

Case 2 Assume that 120593 satisfies the condition (ii)The proof ofthis case can be carried out similarly as the proof of Case 1In particular assume that11987310158401015840(120593(119909 119910 0 0) 119905) is continuous in119909 119910 If 119872 119886 119887 119888 119889 are any fixed nonzero integers then wehave

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 120572

minus119899119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 119872119905)

= 11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905)

(27)

for all 119909 119910 isin 119883 and 119905 gt 0 Since 119899 is arbitrary we have

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119872rarrinfin

11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905) = 1

(28)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 From these and the followingequality

(119865 minus 119891) (119909) =3

8(119863119865 minus 119863119891) ((2

119899+ 1) 119909 minus2

119899119909 0 0)

minus1

8(119863119865 minus 119863119891) (minus (2

119899+ 1) 119909 2

119899119909 0 0)

minus1

2(119865 minus 119891) ((2

119899+1+ 1) 119909)

minus1

2(119865 minus 119891) (minus (2

119899+1+ 1) 119909)

+3

2(119865 minus 119891) ((2

119899+ 1) 119909)

+1

2(119865 minus 119891) (minus (2

119899+ 1) 119909)

+3

2(119865 minus 119891) (2

119899119909) +

1

2(119865 minus 119891) (2

119899119909)

(29)

we get the inequality

1198731015840((119865 minus 119891) (119909)

11119905

2)

ge lim119899rarrinfin

min 11987310158401015840 (38120593 ((2119899+ 1) 119909 minus2

119899119909 0 0)

3119905

8)

11987310158401015840(1

8120593 (minus (2

119899+ 1) 119909 2

119899119909 0 0)

119905

8)

119872((2119899+1

+ 1) 119909 2 (2 minus 120572) 119905)

119872 ((2119899+ 1) 119909 2 (2 minus 120572) 119905)

119872 (2119899119909 2 (2 minus 120572) 119905) = 1

(30)

for all 119909 isin 119883 Due to the previous inequality and the fact that119891(0) = 0 = 119865(0) we obtain that 119891 equiv 119865

Case 3 Assume that 120593 satisfies the condition (iii) Let the set(119878 119889) be as in the proof of Case 1 Now we take into accountthe function 119869 119878 rarr 119878 defined by

119869119892 (119909) = 119892 (119909

2) minus 119892(minus

119909

2) + 2 (119892 (

119909

2) + 119892(minus

119909

2)) (31)

for all 119892 isin 119878 and 119909 isin 119883 Note that

119869119899119892 (119909) = 2

119899minus1(119892 (2minus119899119909) minus 119892 (minus2

minus119899119909))

+22119899minus1

(119892 (2minus119899119909) + 119892 (minus2

minus119899119909))

(32)

and 1198690119892(119909) = 119892(119909) for all 119909 isin 119883 Let 119891 119892 isin 119878 and let 119906 isin

[0infin] be an arbitrary constant with 119889(119892 119891) le 119906 From thedefinition of 119889 we have

1198731015840(119869119892 (119909) minus 119869119891 (119909)

4119906119905

120572)

= 1198731015840(3 (119892 (

119909

2) minus 119891(

119909

2))

+(119892(minus119909

2) minus 119891(minus

119909

2))

4119906119905

120572)

ge min 1198731015840 (3 (119892 (1199092) minus 119891(

119909

2))

3119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 1198731015840 (119892(1199092) minus 119891(

119909

2)

119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 119872(119909

2119905

120572) 119872(minus

119909

2119905

120572)

= 119872 (119909 119905)

(33)

for all 119909 isin 119883 which means that 119889(119869119891 119869119892) le (4120572)119889(119891 119892)Hence 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 0 lt 4120572 lt 1

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

2120572)

= 1198731015840(3119863119891 (1199092 1199092 0 0)

8

minus119863119891 (minus1199092 minus1199092 0 0)

8119905

2120572)

ge min 11987310158401015840 (3120593 (1199092

119909

2 0 0)

3119905

120572)

11987310158401015840(120593(minus

119909

2 minus

119909

2 0 0)

119905

120572)

ge 119872 (119909 119905)

(34)

6 Discrete Dynamics in Nature and Society

for all 119909 isin 119883 It implies that 119889(119891 119869119891) le 12120572 by the definitionof 119889 Therefore according to Theorem 1 the sequence 119869119899119891converges to a unique fixed point 119865 119883 rarr 119884 of 119869 in the set119879 = 119892 isin 119878 | 119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2119899minus1

(119891 (2minus119899119909) minus 119891 (minus2

minus119899119909))

+22119899minus1

(119891 (2minus119899119909) + 119891 (minus2

minus119899119909)))

(35)

for all 119909 isin 119883 Since

119889 (119891 119865) le1

1 minus 4120572119889 (119891 119869119891) le

1

2 (120572 minus 4) (36)

inequality (5) holdsNext we will show that 119865 is quadratic-additive mapping

As in the previous case we have inequality (23) for all119909 119910 119911 119908 isin 119883 and all 119899 isin N The first terms on the right-handside of inequality (23) tend to 1 as 119899 rarr infin by the definitionof 119865 Now consider that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (22119899minus1119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908) 1199058)

1198731015840(22119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

1198731015840(2119899minus1

119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908)

119905

8)

1198731015840( minus 2119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 2minus2119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus2119899minus2120572119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

minus119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus119899minus2120572119899119905)

(37)

which tends to 1 as 119899 rarr infin for all 119909 119910 119911 119908 isin 119883 Thereforeit follows from (23) that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (38)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 That is 119863119865(119909 119910 119911 119908) = 0 forall 119909 119910 119911 119908 isin 119883

In particular instead of the assumption ofTheorem 3 that(119883119873) is a fuzzy normed space it is enough to consider that119883is a linear spaceMoreover we can useTheorem3 to get a clas-sical result in the framework of normed spaces Let (119883 sdot )be a normed linear space Then we can define a fuzzy norm119873119883on119883 by following

119873119883 (119909 119905) =

0 119905 le 119909

1 119905 gt 119909 (39)

where 119909 isin 119883 and 119905 isin R and see [12]

Theorem 4 Let 119883 and 119884 be a normed space and a completenormed space respectively If a mapping 119891 119883 rarr 119884 satisfies

1003817100381710038171003817119863119891 (119909 119910 119911 119908)1003817100381710038171003817 le 119909

119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (40)for all 119909 119910 119911 119908 isin 119883 and a fixed 119901 isin (0 1) cup (2infin) then thereexists a unique quadratic-additive mapping 119865 119883 rarr 119884 suchthat

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le

119909119901

2 minus 2119901

if 0 lt 119901 lt 1

119909119901

2119901minus 4

if 119901 gt 2(41)

for all 119909 isin 119883

Proof Let119873119884be a fuzzy norm on 119884 Then we get

119873119884(119863119891 (119909 119910 119911 119908) 119905) =

0 if 119905 le 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

1 if 119905 gt 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

(42)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R and

119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

= 0 if 119905 le 119909119901 + 1003817100381710038171003817119910

1003817100381710038171003817

119901+ 119911119901+ 119908

119901

1 if 119905 gt 119909119901 + 10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901

(43)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R Therefore it follows from(40) that

119873119884(119863119891 (119909 119910 119911 119908) 119905)

ge 119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

(44)

If we define a mapping 120593 1198834 rarr R by

120593 (119909 119910 119911 119908) = 119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (45)

then we see that 119891 and 120593 satisfy the conditions of Theorem 3with 120572 = 120572

1015840= 2119901 for 0 lt 119901 lt 1 and 120572 = 2

119901 for 119901 gt 2 So wefeel that (41) holds for all 119909 isin 119883

Theorem 5 Let 119883 and 119884 be a normed space and a completenormed space respectively Assume that 120601 119883 rarr R is amapping defined by

120601 (119909) = 119909119901 if 119909 = 0

0 if 119909 = 0(46)

for all 119909 isin 119883 and a fixed 119901 lt 0 Suppose that 119891 119883 rarr 119884 is amapping such that

1003817100381710038171003817119863119891 (119909 119910 119911 119908)

1003817100381710038171003817le 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (47)

for all 119909 119910 119911 119908 isin 119883 Then the mapping 119891 is a quadratic-additive mapping

Proof If we define a mapping 120593 1198834 rarr R by120593 (119909 119910 119911 119908) = 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (48)

then119891 and 120593 are fulfilled in the conditions ofTheorem 3 with120572 lt 2

119901lt 1 Based on the fact that 119873R(120593(119909 119910 0 0) 119905) is

continuous in 119909 119910 under the condition (ii) we arrive at thedesired conclusion

Discrete Dynamics in Nature and Society 7

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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 3

Proof We will take into account three different cases for theassumption of 120593

Case 1 Assume that 120593 satisfies the condition (i) We considerthe set of functions

119878 = 119892 119883 997888rarr 119884 | 119892 (0) = 0 (7)

and introduce a generalized metric on 119878 by

119889 (119892 ℎ) = inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905)

ge 119872 (119909 119905) forall119909 isin 119883

(8)

We first prove that 119889 is a generalized metric on 119878 If119889(119892 ℎ) = 0 that is

inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905) ge 119872 (119909 119905) forall119909 isin 119883

= 0

(9)

then we see that

1198731015840(119892 (119909) minus ℎ (119909) 119906119905) ge 119872 (119909 119905) (10)

for all 119909 isin 119883 and all 119906 gt 0 which means that

1198731015840(119892 (119909) minus ℎ (119909) 119905) ge 119872(119909

119905

119906) (11)

for all 119909 isin 119883 and all 119906 gt 0 It follows that

1198731015840(119892 (119909) minus ℎ (119909) 119905) ge lim

119906rarr0

119872(119909119905

119906) = 1 (12)

So we get 119892(119909) = ℎ(119909) for all 119909 isin 119883Conversely if 119892(119909) = ℎ(119909) for all 119909 isin 119883 then we have

1198731015840(119892 (119909) minus ℎ (119909) 119906119905) = 119873

1015840(0 119906119905) = 1 ge 119872 (119909 119905) (13)

for all 119906 and 119905 So we know that

119889 (119892 ℎ) = inf 119906 isin R+| 1198731015840(119892 (119909) minus ℎ (119909) 119906119905)

ge 119872 (119909 119905) forall119909 isin 119883 = 0

(14)

Of course it is easily checked that 119889(119892 ℎ) = 119889(ℎ 119892) for all119892 ℎ isin 119878

Let 119906 V gt 0 such that 119889(119891 119892) lt 119906 and 119889(119892 ℎ) lt V Then

1198731015840(119891 (119909) minus 119892 (119909) 119906119905) ge 119872 (119909 119905)

1198731015840(119892 (119909) minus ℎ (119909) V119905) ge 119872 (119909 119905)

(15)

for all 119909 isin 119883 and all 119905 gt 0 Thus we find that

1198731015840(119891 (119909) minus ℎ (119909) (119906 + V) 119905)

ge min 1198731015840 (119891 (119909) minus 119892 (119909) 119906119905) 1198731015840 (119892 (119909) minus ℎ (119909) V119905)

ge 119872 (119909 119905)

(16)

This implies that 119906+V ge 119889(119891 ℎ) Hence we yield that 119889(119891 ℎ) le119889(119891 119892) + 119889(119892 ℎ) Therefore 119889 is a generalized metric on 119878

Now if we define a function 119869 119878 rarr 119878 by

119869119891 (119909) = (1

8) (119891 (2119909) + 119891 (minus2119909)) + (

1

4) (119891 (2119909) minus 119891 (minus2119909))

(17)

for all 119909 isin 119883 then we have

119869119899119891 (119909) = 2

minus2119899minus1(119891 (2119899119909) + 119891 (minus2

119899119909))

+2minus119899minus1

(119891 (2119899119909) minus 119891 (minus2

119899119909))

(18)

for all 119909 isin 119883 and all 119899 isin N cup 0For any 119891 119892 isin 119878 let 119906 isin [0infin] be an arbitrary constant

with 119889(119892 119891) le 119906 The definition of 119889 provides that for 0 lt

120572 lt 2

1198731015840(119869119892 (119909) minus 119869119891 (119909)

120572119906119905

2)

= 1198731015840((

3

8) (119892 (2119909) minus 119891 (2119909))

minus (1

8) (119892 (minus2119909) minus 119891 (minus2119909))

120572119906119905

2)

ge min 1198731015840 ((38) (119892 (2119909) minus 119891 (2119909))

3120572119906119905

8)

1198731015840((

1

8) (119892 (minus2119909) minus 119891 (minus2119909))

120572119906119905

8)

= min 1198731015840 (119892 (2119909) minus 119891 (2119909) 120572119906119905)

1198731015840(119892 (minus2119909) minus 119891 (minus2119909) 120572119906119905)

ge min 119872 (2119909 120572119905) 119872 (minus2119909 120572119905)

ge 119872 (119909 119905)

(19)

for all 119909 isin 119883 which implies that 119889(119869119891 119869119892) le (1205722)119889(119891 119892)Thus 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 1205722

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

4)

= 1198731015840(minus5119863119891 (119909 119909 0 0) + 3119863119891 (minus119909 minus119909 0 0)

32119905

4)

ge min1198731015840 (minus5119863119891 (119909 119909 0 0)

325119905

32)

1198731015840(3119863119891 (minus119909 minus119909 0 0)

323119905

32)

ge min 1198731015840 (119863119891 (119909 119909 0 0) 119905)

1198731015840(119863119891 (minus119909 minus119909 0 0) 119905)

4 Discrete Dynamics in Nature and Society

ge min 11987310158401015840 (120593 (119909 119909 0 0) 119905)

11987310158401015840(120593 (minus119909 minus119909 0 0) 119905)

= 119872 (119909 119905)

(20)

for all 119909 isin 119883 The above inequality and the definition of 119889show that 119889(119891 119869119891) le 14

According toTheorem 1 the sequence 119869119899119891 converges toa unique fixed point 119865 119883 rarr 119884 of 119869 in the set 119879 = 119892 isin 119878 |

119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2minus2119899minus1

(119891 (2119899119909) + 119891 (minus2

119899119909))

+2minus119899minus1

(119891 (2119899119909) minus 119891 (minus2

119899119909)))

(21)

for all 119909 isin 119883 We observe that

119889 (119891 119865) le1

1 minus 1205722119889 (119891 119869119891) le

1

2 (2 minus 120572) (22)

This guarantees that inequality (5) holdsNext we are in the position to prove that 119865 is quadratic-

additive mapping Now we figure out the relation

1198731015840(119863119865 (119909 119910 119911 119908) 119905)

ge min 1198731015840 (2 (119865 minus 119869119899119891) (119909 + 119910 + 119911 + 119908) 11990530)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 minus 119908)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119909)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119909)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119910)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119910)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119911)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119911)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119908)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119908)

119905

30)

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

(23)

for all 119909 119910 119911 119908 isin 119883 and all 119899 isin N The first fifteen termson the right-hand side of the above inequality tend to 1 as119899 rarr infin by the definition of 119865 Moreover we find that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (2minus2119899minus1119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 1199058)

1198731015840(2minus2119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (2119899119909 2119899119910 2119899119911 2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

ge min 1198731015840 (119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 22119899minus2119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 22119899minus2

119905)

1198731015840(119863119891 (2

119899119909 2119899119910 2119899119911 2119899119908) 2119899minus2

119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 2119899minus2

119905)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 22119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

2119899minus2120572minus119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

119899minus2120572minus119899119905)

(24)

which tends to 1 as 119899 rarr infin since 2120572 gt 1 Thereforeinequality (23) gives that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (25)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 So we deduce that119863119865(119909 119910 119911 119908) = 0 for all 119909 119910 119911 119908 isin 119883

In order to show the uniqueness of 119865 we assume that 1198651015840 119883 rarr 119884 is another quadratic-additivemapping satisfying (5)and then we yield that

1198691198651015840(119909) =

51198631198651015840(119909 119909 0 0) minus 3119863119865

1015840(minus119909 minus119909 0 0)

32+ 1198651015840(119909)

= 1198651015840(119909)

(26)

Discrete Dynamics in Nature and Society 5

for all 119909 isin 119883That is1198651015840 is another fixed point of 119869 Since119865 is aunique fixed point of 119869 in the set 119879 we conclude that 119865 = 119865

1015840

Case 2 Assume that 120593 satisfies the condition (ii)The proof ofthis case can be carried out similarly as the proof of Case 1In particular assume that11987310158401015840(120593(119909 119910 0 0) 119905) is continuous in119909 119910 If 119872 119886 119887 119888 119889 are any fixed nonzero integers then wehave

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 120572

minus119899119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 119872119905)

= 11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905)

(27)

for all 119909 119910 isin 119883 and 119905 gt 0 Since 119899 is arbitrary we have

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119872rarrinfin

11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905) = 1

(28)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 From these and the followingequality

(119865 minus 119891) (119909) =3

8(119863119865 minus 119863119891) ((2

119899+ 1) 119909 minus2

119899119909 0 0)

minus1

8(119863119865 minus 119863119891) (minus (2

119899+ 1) 119909 2

119899119909 0 0)

minus1

2(119865 minus 119891) ((2

119899+1+ 1) 119909)

minus1

2(119865 minus 119891) (minus (2

119899+1+ 1) 119909)

+3

2(119865 minus 119891) ((2

119899+ 1) 119909)

+1

2(119865 minus 119891) (minus (2

119899+ 1) 119909)

+3

2(119865 minus 119891) (2

119899119909) +

1

2(119865 minus 119891) (2

119899119909)

(29)

we get the inequality

1198731015840((119865 minus 119891) (119909)

11119905

2)

ge lim119899rarrinfin

min 11987310158401015840 (38120593 ((2119899+ 1) 119909 minus2

119899119909 0 0)

3119905

8)

11987310158401015840(1

8120593 (minus (2

119899+ 1) 119909 2

119899119909 0 0)

119905

8)

119872((2119899+1

+ 1) 119909 2 (2 minus 120572) 119905)

119872 ((2119899+ 1) 119909 2 (2 minus 120572) 119905)

119872 (2119899119909 2 (2 minus 120572) 119905) = 1

(30)

for all 119909 isin 119883 Due to the previous inequality and the fact that119891(0) = 0 = 119865(0) we obtain that 119891 equiv 119865

Case 3 Assume that 120593 satisfies the condition (iii) Let the set(119878 119889) be as in the proof of Case 1 Now we take into accountthe function 119869 119878 rarr 119878 defined by

119869119892 (119909) = 119892 (119909

2) minus 119892(minus

119909

2) + 2 (119892 (

119909

2) + 119892(minus

119909

2)) (31)

for all 119892 isin 119878 and 119909 isin 119883 Note that

119869119899119892 (119909) = 2

119899minus1(119892 (2minus119899119909) minus 119892 (minus2

minus119899119909))

+22119899minus1

(119892 (2minus119899119909) + 119892 (minus2

minus119899119909))

(32)

and 1198690119892(119909) = 119892(119909) for all 119909 isin 119883 Let 119891 119892 isin 119878 and let 119906 isin

[0infin] be an arbitrary constant with 119889(119892 119891) le 119906 From thedefinition of 119889 we have

1198731015840(119869119892 (119909) minus 119869119891 (119909)

4119906119905

120572)

= 1198731015840(3 (119892 (

119909

2) minus 119891(

119909

2))

+(119892(minus119909

2) minus 119891(minus

119909

2))

4119906119905

120572)

ge min 1198731015840 (3 (119892 (1199092) minus 119891(

119909

2))

3119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 1198731015840 (119892(1199092) minus 119891(

119909

2)

119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 119872(119909

2119905

120572) 119872(minus

119909

2119905

120572)

= 119872 (119909 119905)

(33)

for all 119909 isin 119883 which means that 119889(119869119891 119869119892) le (4120572)119889(119891 119892)Hence 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 0 lt 4120572 lt 1

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

2120572)

= 1198731015840(3119863119891 (1199092 1199092 0 0)

8

minus119863119891 (minus1199092 minus1199092 0 0)

8119905

2120572)

ge min 11987310158401015840 (3120593 (1199092

119909

2 0 0)

3119905

120572)

11987310158401015840(120593(minus

119909

2 minus

119909

2 0 0)

119905

120572)

ge 119872 (119909 119905)

(34)

6 Discrete Dynamics in Nature and Society

for all 119909 isin 119883 It implies that 119889(119891 119869119891) le 12120572 by the definitionof 119889 Therefore according to Theorem 1 the sequence 119869119899119891converges to a unique fixed point 119865 119883 rarr 119884 of 119869 in the set119879 = 119892 isin 119878 | 119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2119899minus1

(119891 (2minus119899119909) minus 119891 (minus2

minus119899119909))

+22119899minus1

(119891 (2minus119899119909) + 119891 (minus2

minus119899119909)))

(35)

for all 119909 isin 119883 Since

119889 (119891 119865) le1

1 minus 4120572119889 (119891 119869119891) le

1

2 (120572 minus 4) (36)

inequality (5) holdsNext we will show that 119865 is quadratic-additive mapping

As in the previous case we have inequality (23) for all119909 119910 119911 119908 isin 119883 and all 119899 isin N The first terms on the right-handside of inequality (23) tend to 1 as 119899 rarr infin by the definitionof 119865 Now consider that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (22119899minus1119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908) 1199058)

1198731015840(22119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

1198731015840(2119899minus1

119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908)

119905

8)

1198731015840( minus 2119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 2minus2119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus2119899minus2120572119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

minus119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus119899minus2120572119899119905)

(37)

which tends to 1 as 119899 rarr infin for all 119909 119910 119911 119908 isin 119883 Thereforeit follows from (23) that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (38)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 That is 119863119865(119909 119910 119911 119908) = 0 forall 119909 119910 119911 119908 isin 119883

In particular instead of the assumption ofTheorem 3 that(119883119873) is a fuzzy normed space it is enough to consider that119883is a linear spaceMoreover we can useTheorem3 to get a clas-sical result in the framework of normed spaces Let (119883 sdot )be a normed linear space Then we can define a fuzzy norm119873119883on119883 by following

119873119883 (119909 119905) =

0 119905 le 119909

1 119905 gt 119909 (39)

where 119909 isin 119883 and 119905 isin R and see [12]

Theorem 4 Let 119883 and 119884 be a normed space and a completenormed space respectively If a mapping 119891 119883 rarr 119884 satisfies

1003817100381710038171003817119863119891 (119909 119910 119911 119908)1003817100381710038171003817 le 119909

119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (40)for all 119909 119910 119911 119908 isin 119883 and a fixed 119901 isin (0 1) cup (2infin) then thereexists a unique quadratic-additive mapping 119865 119883 rarr 119884 suchthat

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le

119909119901

2 minus 2119901

if 0 lt 119901 lt 1

119909119901

2119901minus 4

if 119901 gt 2(41)

for all 119909 isin 119883

Proof Let119873119884be a fuzzy norm on 119884 Then we get

119873119884(119863119891 (119909 119910 119911 119908) 119905) =

0 if 119905 le 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

1 if 119905 gt 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

(42)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R and

119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

= 0 if 119905 le 119909119901 + 1003817100381710038171003817119910

1003817100381710038171003817

119901+ 119911119901+ 119908

119901

1 if 119905 gt 119909119901 + 10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901

(43)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R Therefore it follows from(40) that

119873119884(119863119891 (119909 119910 119911 119908) 119905)

ge 119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

(44)

If we define a mapping 120593 1198834 rarr R by

120593 (119909 119910 119911 119908) = 119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (45)

then we see that 119891 and 120593 satisfy the conditions of Theorem 3with 120572 = 120572

1015840= 2119901 for 0 lt 119901 lt 1 and 120572 = 2

119901 for 119901 gt 2 So wefeel that (41) holds for all 119909 isin 119883

Theorem 5 Let 119883 and 119884 be a normed space and a completenormed space respectively Assume that 120601 119883 rarr R is amapping defined by

120601 (119909) = 119909119901 if 119909 = 0

0 if 119909 = 0(46)

for all 119909 isin 119883 and a fixed 119901 lt 0 Suppose that 119891 119883 rarr 119884 is amapping such that

1003817100381710038171003817119863119891 (119909 119910 119911 119908)

1003817100381710038171003817le 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (47)

for all 119909 119910 119911 119908 isin 119883 Then the mapping 119891 is a quadratic-additive mapping

Proof If we define a mapping 120593 1198834 rarr R by120593 (119909 119910 119911 119908) = 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (48)

then119891 and 120593 are fulfilled in the conditions ofTheorem 3 with120572 lt 2

119901lt 1 Based on the fact that 119873R(120593(119909 119910 0 0) 119905) is

continuous in 119909 119910 under the condition (ii) we arrive at thedesired conclusion

Discrete Dynamics in Nature and Society 7

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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

4 Discrete Dynamics in Nature and Society

ge min 11987310158401015840 (120593 (119909 119909 0 0) 119905)

11987310158401015840(120593 (minus119909 minus119909 0 0) 119905)

= 119872 (119909 119905)

(20)

for all 119909 isin 119883 The above inequality and the definition of 119889show that 119889(119891 119869119891) le 14

According toTheorem 1 the sequence 119869119899119891 converges toa unique fixed point 119865 119883 rarr 119884 of 119869 in the set 119879 = 119892 isin 119878 |

119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2minus2119899minus1

(119891 (2119899119909) + 119891 (minus2

119899119909))

+2minus119899minus1

(119891 (2119899119909) minus 119891 (minus2

119899119909)))

(21)

for all 119909 isin 119883 We observe that

119889 (119891 119865) le1

1 minus 1205722119889 (119891 119869119891) le

1

2 (2 minus 120572) (22)

This guarantees that inequality (5) holdsNext we are in the position to prove that 119865 is quadratic-

additive mapping Now we figure out the relation

1198731015840(119863119865 (119909 119910 119911 119908) 119905)

ge min 1198731015840 (2 (119865 minus 119869119899119891) (119909 + 119910 + 119911 + 119908) 11990530)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 minus 119911 + 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 minus 119910 + 119911 minus 119908)

119905

30)

1198731015840((119865 minus 119869

119899119891) (119909 + 119910 minus 119911 minus 119908)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119909)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119909)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119910)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119910)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119911)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119911)

119905

30)

1198731015840(5 (119869119899119891 minus 119865) (119908)

119905

30)

1198731015840(3 (119869119899119891 minus 119865) (minus119908)

119905

30)

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

(23)

for all 119909 119910 119911 119908 isin 119883 and all 119899 isin N The first fifteen termson the right-hand side of the above inequality tend to 1 as119899 rarr infin by the definition of 119865 Moreover we find that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (2minus2119899minus1119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 1199058)

1198731015840(2minus2119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (2119899119909 2119899119910 2119899119911 2119899119908)

119905

8)

1198731015840(2minus119899minus1

119863119891 (minus2119899119909 minus2119899119910 minus2119899119911 minus2119899119908)

119905

8)

ge min 1198731015840 (119863119891 (2119899119909 2119899119910 2119899119911 2119899119908) 22119899minus2119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 22119899minus2

119905)

1198731015840(119863119891 (2

119899119909 2119899119910 2119899119911 2119899119908) 2119899minus2

119905)

1198731015840(119863119891 (minus2

119899119909 minus2119899119910 minus2119899119911 minus2119899119908) 2119899minus2

119905)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 22119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

2119899minus2120572minus119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

119899minus2120572minus119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

119899minus2120572minus119899119905)

(24)

which tends to 1 as 119899 rarr infin since 2120572 gt 1 Thereforeinequality (23) gives that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (25)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 So we deduce that119863119865(119909 119910 119911 119908) = 0 for all 119909 119910 119911 119908 isin 119883

In order to show the uniqueness of 119865 we assume that 1198651015840 119883 rarr 119884 is another quadratic-additivemapping satisfying (5)and then we yield that

1198691198651015840(119909) =

51198631198651015840(119909 119909 0 0) minus 3119863119865

1015840(minus119909 minus119909 0 0)

32+ 1198651015840(119909)

= 1198651015840(119909)

(26)

Discrete Dynamics in Nature and Society 5

for all 119909 isin 119883That is1198651015840 is another fixed point of 119869 Since119865 is aunique fixed point of 119869 in the set 119879 we conclude that 119865 = 119865

1015840

Case 2 Assume that 120593 satisfies the condition (ii)The proof ofthis case can be carried out similarly as the proof of Case 1In particular assume that11987310158401015840(120593(119909 119910 0 0) 119905) is continuous in119909 119910 If 119872 119886 119887 119888 119889 are any fixed nonzero integers then wehave

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 120572

minus119899119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 119872119905)

= 11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905)

(27)

for all 119909 119910 isin 119883 and 119905 gt 0 Since 119899 is arbitrary we have

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119872rarrinfin

11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905) = 1

(28)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 From these and the followingequality

(119865 minus 119891) (119909) =3

8(119863119865 minus 119863119891) ((2

119899+ 1) 119909 minus2

119899119909 0 0)

minus1

8(119863119865 minus 119863119891) (minus (2

119899+ 1) 119909 2

119899119909 0 0)

minus1

2(119865 minus 119891) ((2

119899+1+ 1) 119909)

minus1

2(119865 minus 119891) (minus (2

119899+1+ 1) 119909)

+3

2(119865 minus 119891) ((2

119899+ 1) 119909)

+1

2(119865 minus 119891) (minus (2

119899+ 1) 119909)

+3

2(119865 minus 119891) (2

119899119909) +

1

2(119865 minus 119891) (2

119899119909)

(29)

we get the inequality

1198731015840((119865 minus 119891) (119909)

11119905

2)

ge lim119899rarrinfin

min 11987310158401015840 (38120593 ((2119899+ 1) 119909 minus2

119899119909 0 0)

3119905

8)

11987310158401015840(1

8120593 (minus (2

119899+ 1) 119909 2

119899119909 0 0)

119905

8)

119872((2119899+1

+ 1) 119909 2 (2 minus 120572) 119905)

119872 ((2119899+ 1) 119909 2 (2 minus 120572) 119905)

119872 (2119899119909 2 (2 minus 120572) 119905) = 1

(30)

for all 119909 isin 119883 Due to the previous inequality and the fact that119891(0) = 0 = 119865(0) we obtain that 119891 equiv 119865

Case 3 Assume that 120593 satisfies the condition (iii) Let the set(119878 119889) be as in the proof of Case 1 Now we take into accountthe function 119869 119878 rarr 119878 defined by

119869119892 (119909) = 119892 (119909

2) minus 119892(minus

119909

2) + 2 (119892 (

119909

2) + 119892(minus

119909

2)) (31)

for all 119892 isin 119878 and 119909 isin 119883 Note that

119869119899119892 (119909) = 2

119899minus1(119892 (2minus119899119909) minus 119892 (minus2

minus119899119909))

+22119899minus1

(119892 (2minus119899119909) + 119892 (minus2

minus119899119909))

(32)

and 1198690119892(119909) = 119892(119909) for all 119909 isin 119883 Let 119891 119892 isin 119878 and let 119906 isin

[0infin] be an arbitrary constant with 119889(119892 119891) le 119906 From thedefinition of 119889 we have

1198731015840(119869119892 (119909) minus 119869119891 (119909)

4119906119905

120572)

= 1198731015840(3 (119892 (

119909

2) minus 119891(

119909

2))

+(119892(minus119909

2) minus 119891(minus

119909

2))

4119906119905

120572)

ge min 1198731015840 (3 (119892 (1199092) minus 119891(

119909

2))

3119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 1198731015840 (119892(1199092) minus 119891(

119909

2)

119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 119872(119909

2119905

120572) 119872(minus

119909

2119905

120572)

= 119872 (119909 119905)

(33)

for all 119909 isin 119883 which means that 119889(119869119891 119869119892) le (4120572)119889(119891 119892)Hence 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 0 lt 4120572 lt 1

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

2120572)

= 1198731015840(3119863119891 (1199092 1199092 0 0)

8

minus119863119891 (minus1199092 minus1199092 0 0)

8119905

2120572)

ge min 11987310158401015840 (3120593 (1199092

119909

2 0 0)

3119905

120572)

11987310158401015840(120593(minus

119909

2 minus

119909

2 0 0)

119905

120572)

ge 119872 (119909 119905)

(34)

6 Discrete Dynamics in Nature and Society

for all 119909 isin 119883 It implies that 119889(119891 119869119891) le 12120572 by the definitionof 119889 Therefore according to Theorem 1 the sequence 119869119899119891converges to a unique fixed point 119865 119883 rarr 119884 of 119869 in the set119879 = 119892 isin 119878 | 119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2119899minus1

(119891 (2minus119899119909) minus 119891 (minus2

minus119899119909))

+22119899minus1

(119891 (2minus119899119909) + 119891 (minus2

minus119899119909)))

(35)

for all 119909 isin 119883 Since

119889 (119891 119865) le1

1 minus 4120572119889 (119891 119869119891) le

1

2 (120572 minus 4) (36)

inequality (5) holdsNext we will show that 119865 is quadratic-additive mapping

As in the previous case we have inequality (23) for all119909 119910 119911 119908 isin 119883 and all 119899 isin N The first terms on the right-handside of inequality (23) tend to 1 as 119899 rarr infin by the definitionof 119865 Now consider that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (22119899minus1119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908) 1199058)

1198731015840(22119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

1198731015840(2119899minus1

119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908)

119905

8)

1198731015840( minus 2119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 2minus2119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus2119899minus2120572119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

minus119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus119899minus2120572119899119905)

(37)

which tends to 1 as 119899 rarr infin for all 119909 119910 119911 119908 isin 119883 Thereforeit follows from (23) that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (38)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 That is 119863119865(119909 119910 119911 119908) = 0 forall 119909 119910 119911 119908 isin 119883

In particular instead of the assumption ofTheorem 3 that(119883119873) is a fuzzy normed space it is enough to consider that119883is a linear spaceMoreover we can useTheorem3 to get a clas-sical result in the framework of normed spaces Let (119883 sdot )be a normed linear space Then we can define a fuzzy norm119873119883on119883 by following

119873119883 (119909 119905) =

0 119905 le 119909

1 119905 gt 119909 (39)

where 119909 isin 119883 and 119905 isin R and see [12]

Theorem 4 Let 119883 and 119884 be a normed space and a completenormed space respectively If a mapping 119891 119883 rarr 119884 satisfies

1003817100381710038171003817119863119891 (119909 119910 119911 119908)1003817100381710038171003817 le 119909

119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (40)for all 119909 119910 119911 119908 isin 119883 and a fixed 119901 isin (0 1) cup (2infin) then thereexists a unique quadratic-additive mapping 119865 119883 rarr 119884 suchthat

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le

119909119901

2 minus 2119901

if 0 lt 119901 lt 1

119909119901

2119901minus 4

if 119901 gt 2(41)

for all 119909 isin 119883

Proof Let119873119884be a fuzzy norm on 119884 Then we get

119873119884(119863119891 (119909 119910 119911 119908) 119905) =

0 if 119905 le 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

1 if 119905 gt 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

(42)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R and

119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

= 0 if 119905 le 119909119901 + 1003817100381710038171003817119910

1003817100381710038171003817

119901+ 119911119901+ 119908

119901

1 if 119905 gt 119909119901 + 10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901

(43)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R Therefore it follows from(40) that

119873119884(119863119891 (119909 119910 119911 119908) 119905)

ge 119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

(44)

If we define a mapping 120593 1198834 rarr R by

120593 (119909 119910 119911 119908) = 119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (45)

then we see that 119891 and 120593 satisfy the conditions of Theorem 3with 120572 = 120572

1015840= 2119901 for 0 lt 119901 lt 1 and 120572 = 2

119901 for 119901 gt 2 So wefeel that (41) holds for all 119909 isin 119883

Theorem 5 Let 119883 and 119884 be a normed space and a completenormed space respectively Assume that 120601 119883 rarr R is amapping defined by

120601 (119909) = 119909119901 if 119909 = 0

0 if 119909 = 0(46)

for all 119909 isin 119883 and a fixed 119901 lt 0 Suppose that 119891 119883 rarr 119884 is amapping such that

1003817100381710038171003817119863119891 (119909 119910 119911 119908)

1003817100381710038171003817le 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (47)

for all 119909 119910 119911 119908 isin 119883 Then the mapping 119891 is a quadratic-additive mapping

Proof If we define a mapping 120593 1198834 rarr R by120593 (119909 119910 119911 119908) = 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (48)

then119891 and 120593 are fulfilled in the conditions ofTheorem 3 with120572 lt 2

119901lt 1 Based on the fact that 119873R(120593(119909 119910 0 0) 119905) is

continuous in 119909 119910 under the condition (ii) we arrive at thedesired conclusion

Discrete Dynamics in Nature and Society 7

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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 5

for all 119909 isin 119883That is1198651015840 is another fixed point of 119869 Since119865 is aunique fixed point of 119869 in the set 119879 we conclude that 119865 = 119865

1015840

Case 2 Assume that 120593 satisfies the condition (ii)The proof ofthis case can be carried out similarly as the proof of Case 1In particular assume that11987310158401015840(120593(119909 119910 0 0) 119905) is continuous in119909 119910 If 119872 119886 119887 119888 119889 are any fixed nonzero integers then wehave

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 120572

minus119899119905)

ge lim119899rarrinfin

11987310158401015840(120593 ((119886 + 2

minus119899119887) 119909 (119888 + 2

minus119899119889) 119910 0 0) 119872119905)

= 11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905)

(27)

for all 119909 119910 isin 119883 and 119905 gt 0 Since 119899 is arbitrary we have

lim119899rarrinfin

11987310158401015840(120593 ((2

119899119886 + 119887) 119909 (2

119899119888 + 119889) 119910 0 0) 119905)

ge lim119872rarrinfin

11987310158401015840(120593 (119886119909 119888119910 0 0) 119872119905) = 1

(28)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 From these and the followingequality

(119865 minus 119891) (119909) =3

8(119863119865 minus 119863119891) ((2

119899+ 1) 119909 minus2

119899119909 0 0)

minus1

8(119863119865 minus 119863119891) (minus (2

119899+ 1) 119909 2

119899119909 0 0)

minus1

2(119865 minus 119891) ((2

119899+1+ 1) 119909)

minus1

2(119865 minus 119891) (minus (2

119899+1+ 1) 119909)

+3

2(119865 minus 119891) ((2

119899+ 1) 119909)

+1

2(119865 minus 119891) (minus (2

119899+ 1) 119909)

+3

2(119865 minus 119891) (2

119899119909) +

1

2(119865 minus 119891) (2

119899119909)

(29)

we get the inequality

1198731015840((119865 minus 119891) (119909)

11119905

2)

ge lim119899rarrinfin

min 11987310158401015840 (38120593 ((2119899+ 1) 119909 minus2

119899119909 0 0)

3119905

8)

11987310158401015840(1

8120593 (minus (2

119899+ 1) 119909 2

119899119909 0 0)

119905

8)

119872((2119899+1

+ 1) 119909 2 (2 minus 120572) 119905)

119872 ((2119899+ 1) 119909 2 (2 minus 120572) 119905)

119872 (2119899119909 2 (2 minus 120572) 119905) = 1

(30)

for all 119909 isin 119883 Due to the previous inequality and the fact that119891(0) = 0 = 119865(0) we obtain that 119891 equiv 119865

Case 3 Assume that 120593 satisfies the condition (iii) Let the set(119878 119889) be as in the proof of Case 1 Now we take into accountthe function 119869 119878 rarr 119878 defined by

119869119892 (119909) = 119892 (119909

2) minus 119892(minus

119909

2) + 2 (119892 (

119909

2) + 119892(minus

119909

2)) (31)

for all 119892 isin 119878 and 119909 isin 119883 Note that

119869119899119892 (119909) = 2

119899minus1(119892 (2minus119899119909) minus 119892 (minus2

minus119899119909))

+22119899minus1

(119892 (2minus119899119909) + 119892 (minus2

minus119899119909))

(32)

and 1198690119892(119909) = 119892(119909) for all 119909 isin 119883 Let 119891 119892 isin 119878 and let 119906 isin

[0infin] be an arbitrary constant with 119889(119892 119891) le 119906 From thedefinition of 119889 we have

1198731015840(119869119892 (119909) minus 119869119891 (119909)

4119906119905

120572)

= 1198731015840(3 (119892 (

119909

2) minus 119891(

119909

2))

+(119892(minus119909

2) minus 119891(minus

119909

2))

4119906119905

120572)

ge min 1198731015840 (3 (119892 (1199092) minus 119891(

119909

2))

3119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 1198731015840 (119892(1199092) minus 119891(

119909

2)

119906119905

120572)

1198731015840(119892(minus

119909

2) minus 119891(minus

119909

2)

119906119905

120572)

ge min 119872(119909

2119905

120572) 119872(minus

119909

2119905

120572)

= 119872 (119909 119905)

(33)

for all 119909 isin 119883 which means that 119889(119869119891 119869119892) le (4120572)119889(119891 119892)Hence 119869 is a strictly contractive self-mapping of 119878 with theLipschitz constant 0 lt 4120572 lt 1

Moreover by (4) we see that

1198731015840(119891 (119909) minus 119869119891 (119909)

119905

2120572)

= 1198731015840(3119863119891 (1199092 1199092 0 0)

8

minus119863119891 (minus1199092 minus1199092 0 0)

8119905

2120572)

ge min 11987310158401015840 (3120593 (1199092

119909

2 0 0)

3119905

120572)

11987310158401015840(120593(minus

119909

2 minus

119909

2 0 0)

119905

120572)

ge 119872 (119909 119905)

(34)

6 Discrete Dynamics in Nature and Society

for all 119909 isin 119883 It implies that 119889(119891 119869119891) le 12120572 by the definitionof 119889 Therefore according to Theorem 1 the sequence 119869119899119891converges to a unique fixed point 119865 119883 rarr 119884 of 119869 in the set119879 = 119892 isin 119878 | 119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2119899minus1

(119891 (2minus119899119909) minus 119891 (minus2

minus119899119909))

+22119899minus1

(119891 (2minus119899119909) + 119891 (minus2

minus119899119909)))

(35)

for all 119909 isin 119883 Since

119889 (119891 119865) le1

1 minus 4120572119889 (119891 119869119891) le

1

2 (120572 minus 4) (36)

inequality (5) holdsNext we will show that 119865 is quadratic-additive mapping

As in the previous case we have inequality (23) for all119909 119910 119911 119908 isin 119883 and all 119899 isin N The first terms on the right-handside of inequality (23) tend to 1 as 119899 rarr infin by the definitionof 119865 Now consider that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (22119899minus1119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908) 1199058)

1198731015840(22119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

1198731015840(2119899minus1

119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908)

119905

8)

1198731015840( minus 2119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 2minus2119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus2119899minus2120572119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

minus119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus119899minus2120572119899119905)

(37)

which tends to 1 as 119899 rarr infin for all 119909 119910 119911 119908 isin 119883 Thereforeit follows from (23) that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (38)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 That is 119863119865(119909 119910 119911 119908) = 0 forall 119909 119910 119911 119908 isin 119883

In particular instead of the assumption ofTheorem 3 that(119883119873) is a fuzzy normed space it is enough to consider that119883is a linear spaceMoreover we can useTheorem3 to get a clas-sical result in the framework of normed spaces Let (119883 sdot )be a normed linear space Then we can define a fuzzy norm119873119883on119883 by following

119873119883 (119909 119905) =

0 119905 le 119909

1 119905 gt 119909 (39)

where 119909 isin 119883 and 119905 isin R and see [12]

Theorem 4 Let 119883 and 119884 be a normed space and a completenormed space respectively If a mapping 119891 119883 rarr 119884 satisfies

1003817100381710038171003817119863119891 (119909 119910 119911 119908)1003817100381710038171003817 le 119909

119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (40)for all 119909 119910 119911 119908 isin 119883 and a fixed 119901 isin (0 1) cup (2infin) then thereexists a unique quadratic-additive mapping 119865 119883 rarr 119884 suchthat

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le

119909119901

2 minus 2119901

if 0 lt 119901 lt 1

119909119901

2119901minus 4

if 119901 gt 2(41)

for all 119909 isin 119883

Proof Let119873119884be a fuzzy norm on 119884 Then we get

119873119884(119863119891 (119909 119910 119911 119908) 119905) =

0 if 119905 le 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

1 if 119905 gt 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

(42)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R and

119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

= 0 if 119905 le 119909119901 + 1003817100381710038171003817119910

1003817100381710038171003817

119901+ 119911119901+ 119908

119901

1 if 119905 gt 119909119901 + 10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901

(43)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R Therefore it follows from(40) that

119873119884(119863119891 (119909 119910 119911 119908) 119905)

ge 119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

(44)

If we define a mapping 120593 1198834 rarr R by

120593 (119909 119910 119911 119908) = 119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (45)

then we see that 119891 and 120593 satisfy the conditions of Theorem 3with 120572 = 120572

1015840= 2119901 for 0 lt 119901 lt 1 and 120572 = 2

119901 for 119901 gt 2 So wefeel that (41) holds for all 119909 isin 119883

Theorem 5 Let 119883 and 119884 be a normed space and a completenormed space respectively Assume that 120601 119883 rarr R is amapping defined by

120601 (119909) = 119909119901 if 119909 = 0

0 if 119909 = 0(46)

for all 119909 isin 119883 and a fixed 119901 lt 0 Suppose that 119891 119883 rarr 119884 is amapping such that

1003817100381710038171003817119863119891 (119909 119910 119911 119908)

1003817100381710038171003817le 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (47)

for all 119909 119910 119911 119908 isin 119883 Then the mapping 119891 is a quadratic-additive mapping

Proof If we define a mapping 120593 1198834 rarr R by120593 (119909 119910 119911 119908) = 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (48)

then119891 and 120593 are fulfilled in the conditions ofTheorem 3 with120572 lt 2

119901lt 1 Based on the fact that 119873R(120593(119909 119910 0 0) 119905) is

continuous in 119909 119910 under the condition (ii) we arrive at thedesired conclusion

Discrete Dynamics in Nature and Society 7

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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

6 Discrete Dynamics in Nature and Society

for all 119909 isin 119883 It implies that 119889(119891 119869119891) le 12120572 by the definitionof 119889 Therefore according to Theorem 1 the sequence 119869119899119891converges to a unique fixed point 119865 119883 rarr 119884 of 119869 in the set119879 = 119892 isin 119878 | 119889(119891 119892) lt infin which is represented by

119865 (119909) = 1198731015840minus lim119899rarrinfin

(2119899minus1

(119891 (2minus119899119909) minus 119891 (minus2

minus119899119909))

+22119899minus1

(119891 (2minus119899119909) + 119891 (minus2

minus119899119909)))

(35)

for all 119909 isin 119883 Since

119889 (119891 119865) le1

1 minus 4120572119889 (119891 119869119891) le

1

2 (120572 minus 4) (36)

inequality (5) holdsNext we will show that 119865 is quadratic-additive mapping

As in the previous case we have inequality (23) for all119909 119910 119911 119908 isin 119883 and all 119899 isin N The first terms on the right-handside of inequality (23) tend to 1 as 119899 rarr infin by the definitionof 119865 Now consider that

1198731015840(119863119869119899119891 (119909 119910 119911 119908)

119905

2)

ge min 1198731015840 (22119899minus1119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908) 1199058)

1198731015840(22119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

1198731015840(2119899minus1

119863119891 (2minus119899119909 2minus119899119910 2minus119899119911 2minus119899119908)

119905

8)

1198731015840( minus 2119899minus1

119863119891 (minus2minus119899119909 minus2minus119899119910 minus2minus119899119911 minus2minus119899119908)

119905

8)

ge min 11987310158401015840 (120593 (119909 119910 119911 119908) 2minus2119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus2119899minus2120572119899119905)

11987310158401015840(120593 (119909 119910 119911 119908) 2

minus119899minus2120572119899119905)

11987310158401015840(120593 (minus119909 minus119910 minus119911 minus119908) 2

minus119899minus2120572119899119905)

(37)

which tends to 1 as 119899 rarr infin for all 119909 119910 119911 119908 isin 119883 Thereforeit follows from (23) that

1198731015840(119863119865 (119909 119910 119911 119908) 119905) = 1 (38)

for all 119909 119910 119911 119908 isin 119883 and 119905 gt 0 That is 119863119865(119909 119910 119911 119908) = 0 forall 119909 119910 119911 119908 isin 119883

In particular instead of the assumption ofTheorem 3 that(119883119873) is a fuzzy normed space it is enough to consider that119883is a linear spaceMoreover we can useTheorem3 to get a clas-sical result in the framework of normed spaces Let (119883 sdot )be a normed linear space Then we can define a fuzzy norm119873119883on119883 by following

119873119883 (119909 119905) =

0 119905 le 119909

1 119905 gt 119909 (39)

where 119909 isin 119883 and 119905 isin R and see [12]

Theorem 4 Let 119883 and 119884 be a normed space and a completenormed space respectively If a mapping 119891 119883 rarr 119884 satisfies

1003817100381710038171003817119863119891 (119909 119910 119911 119908)1003817100381710038171003817 le 119909

119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (40)for all 119909 119910 119911 119908 isin 119883 and a fixed 119901 isin (0 1) cup (2infin) then thereexists a unique quadratic-additive mapping 119865 119883 rarr 119884 suchthat

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le

119909119901

2 minus 2119901

if 0 lt 119901 lt 1

119909119901

2119901minus 4

if 119901 gt 2(41)

for all 119909 isin 119883

Proof Let119873119884be a fuzzy norm on 119884 Then we get

119873119884(119863119891 (119909 119910 119911 119908) 119905) =

0 if 119905 le 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

1 if 119905 gt 10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119863119891 (119909 119910 119911 119908)1003816100381610038161003816

1003816100381610038161003816

1003816100381610038161003816

(42)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R and

119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

= 0 if 119905 le 119909119901 + 1003817100381710038171003817119910

1003817100381710038171003817

119901+ 119911119901+ 119908

119901

1 if 119905 gt 119909119901 + 10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901

(43)

for all 119909 119910 119911 119908 isin 119883 and all 119905 isin R Therefore it follows from(40) that

119873119884(119863119891 (119909 119910 119911 119908) 119905)

ge 119873R (119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 119905)

(44)

If we define a mapping 120593 1198834 rarr R by

120593 (119909 119910 119911 119908) = 119909119901+10038171003817100381710038171199101003817100381710038171003817

119901+ 119911119901+ 119908

119901 (45)

then we see that 119891 and 120593 satisfy the conditions of Theorem 3with 120572 = 120572

1015840= 2119901 for 0 lt 119901 lt 1 and 120572 = 2

119901 for 119901 gt 2 So wefeel that (41) holds for all 119909 isin 119883

Theorem 5 Let 119883 and 119884 be a normed space and a completenormed space respectively Assume that 120601 119883 rarr R is amapping defined by

120601 (119909) = 119909119901 if 119909 = 0

0 if 119909 = 0(46)

for all 119909 isin 119883 and a fixed 119901 lt 0 Suppose that 119891 119883 rarr 119884 is amapping such that

1003817100381710038171003817119863119891 (119909 119910 119911 119908)

1003817100381710038171003817le 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (47)

for all 119909 119910 119911 119908 isin 119883 Then the mapping 119891 is a quadratic-additive mapping

Proof If we define a mapping 120593 1198834 rarr R by120593 (119909 119910 119911 119908) = 120601 (119909) + 120601 (119910) + 120601 (119911) + 120601 (119908) (48)

then119891 and 120593 are fulfilled in the conditions ofTheorem 3 with120572 lt 2

119901lt 1 Based on the fact that 119873R(120593(119909 119910 0 0) 119905) is

continuous in 119909 119910 under the condition (ii) we arrive at thedesired conclusion

Discrete Dynamics in Nature and Society 7

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for giving usefulsuggestions and for the improvement of this paper Thefirst author was supported by the Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (no2013R1A1A2A10004419)

References

[1] S M Ulam A Collection of Mathematical Problems Inter-science New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T Aoki ldquoOn the stability of the linear transformation in Banachspacesrdquo Journal of the Mathematical Society of Japan vol 2 pp64ndash66 1950

[4] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[5] L Cadariu and V Radu ldquoFixed points and the stability ofJensenrsquos functional equationrdquo Journal of Inequalities in Pure andApplied Mathematics vol 4 no 1 article 4 2003

[6] L Cadariu and V Radu ldquoFixed points and the stability ofquadratic functional equationsrdquoAnalele Universitatii de Vest dinTimisoara Seria Matematica-Informatica vol 41 no 1 pp 25ndash48 2003

[7] L Cadariu and V Radu ldquoOn the stability of the Cauchy func-tional equation a fixed point approachrdquo Grazer MathematischeBerichte vol 346 pp 43ndash52 2004

[8] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[9] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[10] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[11] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[12] A K Mirmostafaee and M S Moslehian ldquoFuzzy almostquadratic functionsrdquo Results in Mathematics vol 52 no 1-2 pp161ndash177 2008

[13] A K Mirmostafaee and M S Moslehian ldquoFuzzy versions ofHyers-Ulam-Rassias theoremrdquo Fuzzy Sets and Systems vol 159no 6 pp 720ndash729 2008

[14] L Cadariu and V Radu ldquoFixed points and generalized stabilityfor functional equations in abstract spacesrdquo Journal of Mathe-matical Inequalities vol 3 no 3 pp 463ndash473 2009

[15] L Cadariu and V Radu ldquoFixed points and stability for func-tional equations in probabilistic metric and random normedspacesrdquo Fixed Point Theory and Applications vol 2009 ArticleID 589143 2009

[16] D Mihet ldquoThe probabilistic stability for a functional equationin a single variablerdquo Acta Mathematica Hungarica vol 123 no3 pp 249ndash256 2009

[17] D Mihet and V Radu ldquoOn the stability of the additive Cauchyfunctional equation in random normed spacesrdquo Journal ofMathematical Analysis and Applications vol 343 no 1 pp 567ndash572 2008

[18] D Mihet ldquoThe fixed point method for fuzzy stability of theJensen functional equationrdquo Fuzzy Sets and Systems vol 160no 11 pp 1663ndash1667 2009

[19] D Mihet ldquoThe stability of the additive Cauchy functionalequation in non-Archimedean fuzzy normed spacesrdquo Fuzzy Setsand Systems vol 161 no 16 pp 2206ndash2212 2010

[20] A K Mirmostafaee M Mirzavaziri and M S MoslehianldquoFuzzy stability of the Jensen functional equationrdquo Fuzzy Setsand Systems vol 159 no 6 pp 730ndash738 2008

[21] A K Mirmostafaee and M S Moslehian ldquoStability of additivemappings in non-Archimedean fuzzy normed spacesrdquo FuzzySets and Systems vol 160 no 11 pp 1643ndash1652 2009

[22] Y-H Lee ldquoOn the quadratic additive type functional equa-tionsrdquo International Journal of Mathematical Analysis vol 7 no37mdash40 pp 1935ndash1948 2013

[23] J B Diaz and B Margolis ldquoA fixed point theorem of thealternative for contractions on a generalized complete metricspacerdquo Bulletin of the American Mathematical Society vol 74pp 305ndash309 1968

[24] I A Rus Principles and Applications of Fixed PointTheory 1979C-N Dacia Eds 1979 (Romanian)

[25] D Mihet and C Zaharia ldquoProbabilistic (quasi)metric versionsfor a stability result of BakerrdquoAbstract and Applied Analysis vol2012 Article ID 269701 10 pages 2012

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