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Research ArticleSome Theorems about (ð, ð, ð, ð)-Contraction inFuzzy Metric Spaces
Parvin Azhdari
Department of Statistics, Islamic Azad University, Tehran North Branch, Tehran, Iran
Correspondence should be addressed to Parvin Azhdari; par [email protected]
Received 4 July 2015; Revised 17 September 2015; Accepted 21 September 2015
Academic Editor: Gunther JaÌger
Copyright © 2015 Parvin Azhdari. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We previously proved a number of fixed point theorems for some kinds of contractions like ðð-contraction and ð â ð» contraction
in fuzzy metric spaces. In this paper, we discuss the problem of existence of fixed point for (ð, ð, ð, ð)-contraction in fuzzy metricspaces in sense of George and Veeramani.
1. Introduction
Probabilistic metric spaces are generalizations of metricspaces which have been introduced by Menger [1]. Georgeand Veeramani [2] modified the concept of the fuzzy metricspaces, whichwere introduced byKramosil andMichalek [3].Fixed point theory for contraction type mappings in fuzzymetric spaces is closely related to the fixed point theory forthe same type of mappings in probabilistic metric spaces.Hicks introduced ð¶-contraction [4]. Radu in [5] extendedð¶-contraction to the generalized ð¶-contraction. Mihet in[6] presented the notion of a ð-contraction of (ð, ð)-type.He also introduced the class of (ð, ð, ð, ð)-contraction infuzzy metric spaces [7] which is a generalization of the(ð, ð)-contraction [8]. Ciric also proved some new resultsfor Banach contractions and Edelstin contractive mappingson fuzzy metric spaces [9]. We obtained the fixed point for(ð, ð, ð, ð)-contraction in probabilistic metric spaces and weintroduced the generalized (ð, ð, ð, ð)-contraction too [10].The outline of the paper is as follows. In Section 2, we brieflyrecall some basic concepts. In Section 3, two fixed pointtheorems about (ð, ð, ð, ð)-contractions are proved.
2. Preliminary Notes
Since the definitions of the notion of fuzzy metric space areclosely related to the definition of generalizedMenger spaces,we review a number of definitions from probabilistic metric
space theory used in this paper as an example and Schweizerand Sklarâs definition can be considered. For more details, werefer the reader to [11, 12].
Amappingð¹ : (ââ,â) â [0, 1] is called a distance dis-tribution function if it is nondecreasing and left continuouswith ð¹(0) = 0. The class of all distance distribution functionsis denoted by Î
+. A probabilistic metric space is an ordered
pair (ð, ð¹) where ð is a nonempty set and ð¹ is a mappingfromðÃð to Î
+. The value of ð¹ at (ð, ð) â ðÃð is denoted
by ð¹ðð(â ) satisfying the following conditions:
(1) âð¥ ⥠0, ð¹ðð(ð¥) = 1 iff ð = ð.
(2) âð, ð â ð, âð¥ ⥠0, ð¹ðð(ð¥) = ð¹
ðð(ð¥).
(3) If ð¹ðð(ð¥) = 1, ð¹
ðð(ðŠ) = 1 then ð¹
ðð(ð¥ + ðŠ) =
1 for all ð, ð, ð â ð and for all ð¥, ðŠ ⥠0.
A binary operation ð : [0, 1] Ã [0, 1] â [0, 1] iscalled a triangular norm (abbreviated ð¡-norm) if the followingconditions are satisfied:
(1) ð(ð, 1) = ð for all ð â [0, 1].
(2) ð(ð, ð) = ð(ð, ð) for every ð, ð â [0, 1].
(3) ð ⥠ð, ð ⥠ð â ð(ð, ð) ⥠ð(ð, ð) for all ð, ð, ð â[0, 1].
(4) ð(ð(ð, ð), ð) = ð(ð, ð(ð, ð)) for every ð, ð, ð â [0, 1].
Hindawi Publishing CorporationJournal of MathematicsVolume 2015, Article ID 873807, 5 pageshttp://dx.doi.org/10.1155/2015/873807
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2 Journal of Mathematics
Definition 1. AMenger space is a triple (ð, ð¹, ð)where (ð, ð¹)is a probabilisticmetric space;ð is a ð¡-norm and the followinginequality holds:
ð¹ðð
(ð¥ + ðŠ) ⥠ð (ð¹ðð
(ð¥) , ð¹ðð(ðŠ))
âð, ð, ð â ð, âð¥, ðŠ ⥠0.
(1)
If (ð, ð¹, ð) is a Menger space with ð satisfyingsup0â€ð 0, ð â (0, 1)},where ð
ð,ð= {(ð¥, ðŠ) â ð à ð, ð¹
ð¥,ðŠ(ð) > 1 â ð}, is a base
for a uniformity on ð and is called the ð¹-uniformity onð. A topology on ð is determined by this ð¹-uniformity,(ð â ð)-topology.
Definition 2. A sequence (ð¥ð)ðâN is called an ð¹-convergent
sequence to ð¥ â ð if, for all ð > 0 and ð â (0, 1), there existsð = ð(ð, ð) â N such that ð¹
ð¥ðð¥(ð) > 1 â ð, for all ð ⥠ð.
Definition 3. A sequence (ð¥ð)ðâN is called an ð¹-Cauchy
sequence if, for all ð > 0 and ð â (0, 1), there exists ð =ð(ð, ð) â N such that ð¹
ð¥ðð¥ð+ð
(ð) > 1 â ð, for all ð ⥠ð, for allð â N.
A probabilistic metric space (ð, ð¹, ð) is called ð¹-sequentially complete if every ð¹-Cauchy sequence is ð¹-convergent.
It is helpful to mention that George and Veeramani[2] have extended fuzzy metric spaces (GV-fuzzy metricsspaces).
Definition 4. The3-tuple (ð,ð, ð) is said to be a fuzzymetricspace if ð is a continuous ð¡-norm and ð is a fuzzy set onð2
à (0,â) satisfying the following conditions:
(1) âð¡ > 0, âð, ð â ð, ð(ð, ð, ð¡) > 0.(2) âð¡ > 0, âð, ð â ð, ð(ð, ð, ð¡) = 1 iff ð = ð.(3) âð¡ > 0, âð, ð â ð, ð(ð, ð, ð¡) = ð(ð, ð, ð¡).(4) âð¡, ð > 0, âð, ð, ð â ð, ð(ð(ð, ð, ð¡),ð(ð, ð, ð )) â€
ð(ð, ð, ð¡ + ð ).(5) âð, ð â ð, ð(ð, ð, â ) : [0,â] â [0, 1] is
continuous.
Gregori and Romaguera introduced the next definition[13].
Definition 5. Let (ð,ð, ð) be a fuzzy metric space. Asequence {ð¥
ð} in ð is said to be point convergent to ð¥ â ð
(shown as ð¥ð
ð
â ð¥) if there exists ð¡ > 0 such that
limðââ
ð(ð¥ð, ð¥, ð¡) = 1. (2)
Now, we recall the definition of the generalized ð¶-contraction [5]; let M be the family of all the mappings ð :ð â ð such that the following conditions are satisfied:
(1) âð¡, ð ⥠0 : ð(ð¡ + ð ) ⥠ð(ð¡) + ð(ð ).(2) ð(ð¡) = 0 â ð¡ = 0.(3) ð is continuous.
Definition 6. Let (ð, ð¹) be a probabilisticmetric space andð :ð â ð. A mapping ð is a generalized ð¶-contraction if thereexist a continuous, decreasing function â : [0, 1] â [0,â]such that â(1) = 0, ð
1, ð2â M, and ð â (0, 1) such that the
following implication holds for every ð, ð â ð and for everyð¡ > 0:
â â ð¹ð,ð
(ð2(ð¡)) < ð
1(ð¡)
â â â ð¹ð(ð),ð(ð)
(ð2(ðð¡)) < ð
1(ðð¡) .
(3)
If ð1(ð ) = ð
2(ð ) = ð and â(ð ) = 1 â ð for every ð â [0, 1], we
obtain Hicksâs definition.Mihet in [7] showed that if ð : ð â ð is a (ð, ð, ð, ð)-
contraction and (ð,ð, ð) is a complete fuzzy metric space,then ð has a unique fixed point, and CÌiricÌ et al. presentedthe theorem of fixed and periodic points for nonexpansivemappings in fuzzy metric spaces [14].
The comparison functions from the class ð of all map-pings ð : (0, 1) â (0, 1) have the following properties:
(1) ð is an increasing bijection.(2) âð â (0, 1), ð(ð) < ð.
Since every comparison mapping of this type is continuous,if ð â ð, then, for every ð â (0, 1), lim
ðââðð
(ð) = 0.
Definition 7. Let (ð, ð¹) be a probabilistic space, ð â ð, andlet ð be a map from (0,â) to (0,â). A mapping ð : ð âð is called a (ð, ð, ð, ð)-contraction on ð if it satisfies thefollowing condition:
ð¹ð¥,ðŠ
(ð) > 1 â ð
â ð¹ð(ð¥),ð(ðŠ)
(ð (ð)) > 1 â ð (ð) ,
ð¥, ðŠ â ð, ð > 0, ð â (0, 1) .
(4)
In the rest of the paper, we suppose thatð is an increasingbijection.
Example 8. Letð = {0, 1, 2, . . .} and (for ð¥ Ìž= ðŠ)
ð(ð¥, ðŠ, ð¡) =
{{{{
{{{{
{
0, if ð¡ †2âmin(ð¥,ðŠ),
1 â 2âmin(ð¥,ðŠ)
, if 2âmin(ð¥,ðŠ) < ð¡ †1,
1, if ð¡ > 1.
(5)
Suppose that ðŽ : ð â ð, ðŽ(ð) = ð + 1, and theÅukasiewicz ð¡-norm defined by ð
ð¿(ð, ð) = max{ð + ð â 1, 0}.
Then (ð,ð, ðð¿) is a fuzzy metric space [15].
Assume ð¥, ðŠ, ð, ð are such thatð(ð¥, ðŠ, ð) > 1 â ð:
(i) If 2âmin(ð¥,ðŠ) < ð †1, then 1 â 2âmin(ð¥,ðŠ) > 1 â ð.This indicates that 1â 2âmin(ð¥+1,ðŠ+1) > 1â (1/2)ð; thatis,
ð(ðŽð¥,ðŽðŠ, ð) > 1 â1
2ð. (6)
(ii) If ð > 1, then ð(ðŽð¥,ðŽðŠ, ð) = 1; hence againð(ðŽð¥,ðŽðŠ, ð) > 1 â (1/2)ð. Thus, the mapping ðŽis a (ð, ð, ð, ð)-contraction on ð with ð(ð) = ð andð(ð) = (1/2)ð.
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Journal of Mathematics 3
3. Main Results
In this section,we recall some contraction through the severaldefinitions and then we prove the existence of fixed pointfor these contractions. It would be interesting to comparedifferent types of contractionmapping in fuzzymetric spaces.It is useful to note that the concept of ð
ð-contraction has been
introduced by Mihet [6].As we mentioned, existence of convergent sequence is
sometimes a difficult condition. Gregori and Romaguerapresented another type of convergence that is called ð-convergence [13]. A GV-fuzzy metric space (ð,ð, ð) withthe point convergence is a space with convergence in senseof FreÌchet too.
In [16] we showed the existence of fixed point on ðµ-contraction and ð¶-contraction in the case of ð-convergencesubsequence. Furthermore, we have proved a theorem for ð
ð-
contraction [17].First, we introduce ðâð» contraction; then we review the
fixed point theorem by ð-convergent subsequence.Let (ð,ð, ð) be a fuzzy metric space and ð â ð. A
mapping ð : ð â ð is called a ð â ð» contraction on ðif the following condition holds:
ð(ð¥, ðŠ, ð) > 1 â ð
â ð(ð (ð¥) , ð (ðŠ) , ð (ð)) > 1 â ð (ð) ,
ð¥, ðŠ â ð, ð â (0, 1) .
(7)
Consider the mapping ð : (0, 1) â (0, 1); we say that theð¡-norm ð is ð-convergent if, âð¿ â (0, 1), âð â (0, 1), âð =ð (ð¿, ð) â ð;
â€ð
ð=1(1 â ð
ð +ð
(ð¿)) > 1 â ð, âð ⥠1. (8)
A theorem for ð â ð» contraction on a GV-fuzzy metricspace is as follows [18].
Theorem 9. Let (ð,ð, ð) be a GV-fuzzy metric space andsup0â€ð0, ð(ð¥, ðŽ(ð¥), ð¿) > 1 â ð¿ and the sequence ðŽð(ð¥) has aconvergent subsequence. If ð is ð-convergent and ââ
ð=1ðð
(ð¿)
is convergent, then ðŽ will have a fixed point.
In this paper, due to the next theorem, the existence offixed point for (ð, ð, ð, ð)-contraction is proved under thenew conditions.
Theorem 10. Let (ð,ð, ð) be a GV-fuzzy metric space andsup0â€ð 0 and ð¿ â(0, 1), ð(ð¥, ðŽ(ð¥), ð) > 1 â ð¿ and the sequence ðŽð(ð¥) has aconvergent subsequence. If ð is ð-convergent andââ
ð=1ðð
(ð) isconvergent, then ðŽ will have a fixed point.
Proof. Let, for every ð â N, ð¥ð= ðŽð
(ð¥) = ðŽ(ð¥ðâ1
) and ð¥ =ðŽ0
(ð¥).
By the assumptionð(ð¥,ðŽ(ð¥), ð) > 1 â ð¿, so
ð(ðŽ (ð¥) , ðŽ2
(ð¥) , ð (ð)) > 1 â ð (ð¿) (9)
and by induction for every ð â N
ð(ðŽð
(ð¥) , ðŽð+1
(ð¥) , ðð
(ð)) > 1 â ðð
(ð¿) . (10)
We show that the sequence ðŽð(ð¥) is a Cauchy sequence; thatis, for every ð > 0 and ð â (0, 1) there exists an integerð0= ð0(ð, ð) â N such that, for all ð¥, ðŠ â ð, ð ⥠ð
0, ð â
N, ð(ðŽð(ð¥), ðŽð+ð(ð¥), ð) > 1 â ð.Let ð > 0 and ð â (0, 1) be given; since the
series ââð=1
ðð
(ð) converges, there exists ð0(ð) such that
ââ
ð=ð0
ðð
(ð) < ð.Then, for every ð ⥠ð0,
ð(ðŽð
(ð¥) , ðŽð+ð
(ð¥) , ð) ⥠ð(ðŽð
(ð¥) , ðŽð+ð
(ð¥) ,
â
â
ð=ð
ðð
(ð)) ⥠ð(ðŽð
(ð¥) , ðŽð+ð
(ð¥) ,
ð+ðâ1
â
ð=ð
ðð
(ð)) ⥠ð
â â â ð (ððŽð
(ð¥) , ðŽð+1
(ð¥) , ðð
(ð) ,
ð (ðŽð+1
(ð¥) , ðŽð+2
(ð¥) , ðð+1
(ð)) , . . . ,
ð (ðŽð+ðâ1
(ð¥) , ðŽð+ð
(ð¥) , ðð+ðâ1
(ð))) .
(11)
Let ð1= ð1(ð) be such that â€â
ð=ð1
(1 â ðð
(ð¿)) > 1 â ð. Sinceð is ð-convergent, such a number ð
1exists. By using (10), we
obtain for every ð ⥠max(ð0, ð1) andð â N
ð(ðŽð
(ð¥) , ðŽð+ð
(ð¥) , ð) ⥠â€ð+ðâ1
ð=ð(1 â ð
ð
(ð¿))
⥠â€â
ð=ð(1 â ð
ð
(ð¿)) ⥠1 â ð.
(12)
Suppose {ð¥ðð
} is a convergent subsequence of {ð¥ð}which con-
verges to ðŠ0. Then, for every ð¡ > 0, lim
ðââð(ð¥ðð
, ðŠ0, ð¡) = 1.
Let ð > 0 be given. Since sup0â€ð 0 such that ð((1 â ð1), (1 â ð
1)) > 1 â ð. Since
{ð¥ð} and then {ð¥
ðð
} are Cauchy sequence, we can take ð2large
enough such thatð(ð¥ðð
, ð¥ðð+1, ð¡) > 1âð
1andð(ð¥
ðð
, ðŠ0, ð¡) >
1 â ð1, for every ð ⥠ð
2; then ð(ð¥
ðð+1, ðŠ0, 2ð¡) ⥠ð((1 â
ð1), (1 â ð
1)) > 1 â ð which implies that ð¥
ðð+1
â ðŠ0. By
(ð, ð, ð, ð)-contraction condition fromð(ð¥ðð
, ðŠ0, ð) > 1 â ð,
we have
ð(ðŽ(ð¥ðð
) , ðŽ (ðŠ0) , ð (ð)) > 1 â ð (ð) > 1 â ð
for every ð > 0.(13)
It means ðŽ(ð¥ðð
) = ð¥ðð+1
â ðŽ(ðŠ0). Since the convergence is
ð¹-convergence in aGV-fuzzymetric space, we getðŽ(ðŠ0) = ðŠ0
which means ðŠ0is a fixed point.
We mention an example for Theorem 13.
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4 Journal of Mathematics
Example 11. Let (ð,ð, ð) be a complete fuzzy metric spacewhere ð = {ð¥
1, ð¥2, ð¥3, ð¥4}, ð(ð, ð) = min{ð, ð}, and
ð(ð¥, ðŠ, ð¡) is defined as
ð(ð¥1, ð¥2, ð¡) = ð (ð¥
2, ð¥1, ð¡) =
{{{{
{{{{
{
0, if ð¡ †0,
0.9, if 0 < ð¡ †3,
1, if ð¡ > 3.
ð (ð¥1, ð¥3, ð¡) = ð (ð¥
3, ð¥1, ð¡) = ð (ð¥
1, ð¥4, ð¡)
= ð (ð¥4, ð¥1, ð¡) = ð (ð¥
2, ð¥3, ð¡)
= ð (ð¥3, ð¥2, ð¡) = ð (ð¥
2, ð¥4, ð¡)
= ð (ð¥4, ð¥2, ð¡) = ð (ð¥
3, ð¥4, ð¡)
= ð (ð¥4, ð¥3, ð¡) =
{{{{
{{{{
{
0, if ð¡ †0,
0.7, if 0 < ð¡ †6,
1, if 6 < ð¡
(14)
ðŽ : ð â ð is given by ðŽ(ð¥1) = ðŽ(ð¥
2) = ð¥
2and ðŽ(ð¥
3) =
ðŽ(ð¥4) = ð¥
1. If we take ð(ð) = ð/2, ð(ð) = ð/2, then ðŽ is a
(ð, ð, ð, ð)-contraction if we take ð = 0.1, ð¿ = 0.2, and ð¥ = ð¥1
as ðŽ(ð¥) = ðŽ(ð¥1) = ð¥2. Therefore,
ð(ð¥,ðŽ (ð¥) , 0.1) = ð(ð¥1, ð¥2, 0.1) = 0.9 > 1 â 0.2
= 0.8.
(15)
On the other hand, ðŽ(ð¥1) = ð¥
2, ðŽ(ð¥
2) = ð¥
2, ðŽ2
(ð¥3) =
ðŽ(ð¥1) = ð¥
2, and ðŽ2(ð¥
4) = ðŽ(ð¥
1) = ð¥
2. So ðŽð(ð¥) is a con-
vergent sequence and hence has a convergent subsequenceto ð¥2. It is clear that ð is ð-convergent and ââ
ð=1ðð
(ð) =
ââ
ð=1(ð/2)ð
= ð/(2 â ð). It means ð¥2is a unique fixed point
for ðŽ.
For more information, reader can refer to [10].
Definition 12. Let (ð,ð, ð) be a GV-fuzzy metric space andðŽ : ð â ð. The mapping ðŽ is a generalized (ð, ð, ð, ð)-contraction if there exist a continuous, decreasing functionâ : [0, 1] â [0,â] such that â(1) = 0, ð
1, ð2â M, and
ð â (0, 1) such that the following implication holds for everyð¢, V â ð and for every ð > 0:
â â ð (ð¢, V, ð2(ð)) < ð
1(ð)
â â â ð(ðŽ (ð¢) , ðŽ (V) , ð2(ð (ð))) < ð
1(ð (ð)) .
(16)
If ð1(ð) = ð
2(ð) = ð, and â(ð) = 1 â ð for every ð â [0, 1],
we obtain the Mihet definition.
In the next theorem, we will prove a theorem for existingfixed point in the generalized (ð, ð, ð, ð)-contraction.
Now, we prove the new theorem for this kind of contrac-tion.
Theorem 13. Let (ð,ð, ð) be a GV-fuzzy metric space with t-norm ð such that sup
0â€ð 0, and â,ð1, ð2are given as
in Definition 6. Let ð¥ð= ðŽð
(ð¥) = ðŽ(ð¥ðâ1
) for every ð â Nand (ðŽ0(ð¥) = ð¥). First, we show that the sequence ðŽð(ð¥) isa Cauchy sequence. We prove that for every ðŒ > 0 and ðœ â(0, 1) there exists an integer ð = ð(ð, ð) â N such that, forevery ð¥, ðŠ â ð and ð ⥠ð,ð(ðŽð(ð¥), ðŽð(ðŠ), ðŒ) > 1 â ðœ.
By the assumption, there is a ð â (0, 1) such that â(0) <ð1(ð).
Fromð(ð¥, ðŠ,ð2(ð)) ⥠0 it follows that
â â ð (ð¥, ðŠ,ð2(ð)) †â (0) < ð
1(ð ) , (18)
which implies that â â ð(ðŽ(ð¥), ðŽ(ðŠ), ð2(ð(ð))) < ð
1(ð(ð)),
and by continuing in this way we obtain that for every ð â N
â â ð (ðŽð
(ð¥) , ðŽð
(ðŠ) ,ð2(ðð
(ð))) < ð1(ð (ð)) . (19)
Suppose ð0(ðŒ, ðœ) is a natural number such that ð
2(ðð
(ð)) <
ðŒ aðð ð1(ðð
(ð)) < â(1 â ðœ) for every ð ⥠ð0(ðŒ, ðœ). Then
ð > ð0(ðŒ, ðœ) implies that
ð(ðŽð
(ð¥) , ðŽð
(ðŠ) , ðŒ)
⥠ð(ðŽð
(ð¥) , ðŽð
(ðŠ) ,ð2(ðð
(ð))) > 1 â ðœ.
(20)
Now let ðŠ = ðŽð(ð¥); then we obtain
ð(ðŽð
(ð¥) , ðŽð+ð
(ð¥) , ðŒ) > 1 â ðœ,
âð ⥠ð0, âð â N,
(21)
which means that ðŽð(ð¥) is a Cauchy sequence.Suppose {ð¥
ð} has a convergent subsequence {ð¥
ðð
} whichis convergent to ðŠ
0. Then, for every ð¡
0> 0,
limðââ
ð(ð¥ðð
, ðŠ0, ð¡0) = 1. (22)
Let ð > 0 be given since sup0â€ð 0such that ð((1 â ð¿), (1 â ð¿)) > 1 â ð. Since {ð¥
ð} and then {ð¥
ðð
}
are Cauchy sequences, we can take ð0large enough such that
ð(ð¥ðð
, ð¥ðð+1, ð¡0) > 1 â ð¿, andð(ð¥
ðð
, ðŠ0, ð¡0) > 1 â ð¿, âð ⥠ð
0;
then ð(ð¥ðð+1, ðŠ0, 2ð¡0) ⥠ð((1 â ð¿), (1 â ð¿)) > 1 â ð which
implies that
ð¥ðð+1
â ðŠ0. (23)
Let ð > 0 be such that ð2(ð(ð)) < ð and ð â (0, 1) such that
ð1(ð(ð)) < â(1 â ð). Sinceð
1andð
2are continuous at zero
andð1(0) = ð
2(0) = 0, such numbers, ð and ð, exist.
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Journal of Mathematics 5
If we take ð¡0
= ð2(ð) in relation (22), then we have
ð(ð¥ðð
, ðŠ0, ð2(ð)) ⥠0 so
â â ð(ð¥ðð
, ðŠ0, ð2(ð)) †â (0) < ð
1(ð) , (24)
and this implies that
â â ð(ðŽ(ð¥ðð
) , ðŽ (ðŠ0) , ð2(ð (ð))) < ð
1(ð (ð))
< â (1 â ð) .
(25)
Thus
ð(ðŽ(ð¥ðð
) , ðŽ (ðŠ0) , ð)
> ð(ðŽ(ð¥ðð
) , ðŽ (ðŠ0) , ð2(ð (ð))) > 1 â ð;
(26)
then ðŽ(ð¥ðð
) = ð¥ðð+1
â ðŽ(ðŠ0), and since convergence is ð¹-
convergence in a GV-fuzzy metric space, then ðŽ(ðŠ0) = ðŠ
0
which means ðŠ0is a fixed point.
By an example, we describe Theorem 13 more.
Example 14. Let (ð,ð, ð) and the mappings ðŽ,ð, and ðbe the same as in Example 11. Set â(ð) = ðâð â ðâ1for every ð â [0, 1] and ð
1(ð) = ð
2(ð) = ð. It is
obvious that ð is continuous on (0,â). The mapping ðŽis generalized (ð, ð, ð, ð)-contraction and lim
ðââðð
(ð¿) =
limðââ
(ð¿/2ð
) = 0 for every ð¿ â (0,â). On the other hand,there exists ð â (0, 1) such that â(0) = 1 â 1/ð < ð andð(0) = ð(0) = 0. By the previous example, ðŽð(ð¥) has aconvergent subsequence. So ð¥
2is the unique fixed point for
ðŽ.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The author thanks the reviewers for their useful comments.
References
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