# Soft Fixed Point Theorems for the Soft Comparable Contractions

Authors: Chi-Ming Chen; Zhi-Hao Xu; Erdal Karapinar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we introduce the notions of a softinf-comparable contraction and soft comparable Meir-Keeler contraction in a soft metric space. Furthermore, we prove two soft fixed point theorems which assure the existence of soft fixed points for these two types of comparable contractions. The obtained results not only generalize but also unify many recent fixed point results in the literature.


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## Body

## 1. Introduction and Preliminaries

It is the main feature of mathematical study to produce different methods and tools to perceive the behavior of systems that we have difficulty understanding with known methods. In particular, it may be necessary to deal with systems that contain uncertainties and to use inaccurate data in different situations. With this motivation, one of the mathematical tools used to deal with the necessities of systems established with uncertainty and to analyze the models created by the uncertainties and uncertainties already existing in the data is the Fuzzy Set Theory. Fuzzy sets were introduced by Zadeh [1] for dealing with the uncertainties on its own limits. Another mathematical tool to deal with the uncertainties is the soft set that was introduced by Molodtsov [2]. In this paper, we shall focus on the soft set theory. The topology based on the soft sets was defined by Cagman et al. [3]. They also considered the basic topological notions over soft sets. On the other hand, a soft real set and soft real number were proposed successfully by Das and Samanta [4]. Furthermore, the same authors in considered the notions of a soft metric and its topology, properly. After then, Abbas et al. [5] proved a fixed point theorem by introducing the notion of soft contraction mapping over the soft metric space. Application potential of the soft sets in various distinct research topics is very rich and wide, for example, the smoothness of functions, game theory, operation research, probability theory, and measurement theory. For more details on soft sets and application, we can refer to, e.g., [3, 4, 6–12].As usual,ℝ denotes real numbers and ℝ+≔0,∞. Furthermore, the letters ℤ,ℕ denote integers and natural numbers, respectively. The symbol Bℝ denotes the collection of all nonempty bounded subsets of ℝ.We shall denote an initial universeΩ. We set P as a set of parameters. As usual, 2Ω denotes the collection of all subsets of Ω. For a nonempty subset S of P, we consider a set-valued mapping T:S→2Ω for all τ∉A with Tτ=ϕ. We define a pair T,A on Ω as
(1)T,S=Tτ,τ:τ∈P.Here,T,S is called a soft set [2]. The symbol SΩ represents the collection of all soft sets on Ω.A soft setT,S on Ω is called null soft [11] (respectively, absolute soft set [11]) represented by, respectively, S~ifTτ=(respectively,Tτ=S) for allτ∈S. We presume that T1,S1 and T2,S2 are two soft sets on Ω. We define the intersection [11] of the mentioned two sets above as a soft set T3,S3, denoted by T1,S1∩~T2,S2=T3,S3, where S3=S1∩S2, and for each τ∈S3, T3τ=T1τ∩T2τ. As expected, we define the union of T1,S1 and T2,S2 [11] as a soft set T3,S3, denoted by T1,A∪~T2,B=T3,C where S3=S1∪S2 and for each τ∈S3,
(2)T3τ=T1τ,ifτ∈S1\S2,T2τ,ifτ∈S2\S1,T1τ∪T2τ,ifτ∈S1∩S2.We use the notationTc,P to indicate the complement [11] of soft set T,P on Ω where Tc:X→2Ω is a mapping given by Tcτ=Ω\Tτ for all τ∈P.A mappingT:P→Bℝ is called a soft real set [13]. The symbol ℝ+P is used to denote the set of all nonnegative soft real numbers. If T,P is a singleton soft set, then it is called a soft real number. Regarding the corresponding soft set, soft real numbers will be denoted as γ~, η~, ξ~, etc. In particular, 0¯ and 1~ are the soft real numbers where 0¯τ=0, 1~τ=1 for all τ∈P.For two soft real numbers, for allτ∈P, we have the following inequalities [13]:
(1)
γ~≤~η~ if γ~τ≤~η~τ(2)
γ~≥~η~ if γ~τ≥~η~τ(3)
γ~<~η~ if γ~τ<~η~τ(4)
γ~>~η~ if γ~τ>~η~τDefinition 1.
(1)
The mappingϕ:ℝ+P→ℝ+P is called soft increasing, if(3)r~<~t~⇒ϕr~<~ϕt~.(2)
The mappingϕ:ℝ+P→ℝ+P is called soft continuous at a~∈~ℝ+P, if for every γ~>~0¯, there exists δ~>~0¯ such that 0¯<~x~−a~<~δ~ implies(4)ϕx~−ϕa~<~γ~.Moreover,ϕ:ℝ+P→ℝ+P is called soft continuous at every point a~ of ℝ+P, then we call ϕ as a continuous mapping.A soft setT,P on Ω is called a soft point [4, 14], denoted by xτ~, if there is a unique τ∈P such that Tτ=x for some τ∈P and Tω=ϕ for all ω∈P\τ.Definition 2 (see).
LetX~=T,P be an absolute soft set, and let SPX~ be the collection of all soft points of X~. A mapping d~:SPX~×SPX~→ℝ+P is called a soft metric on X~ if d~ satisfies the following conditions for all xτ1~,xτ2~,xτ3~∈X~:
M1d~xτ1~,xτ2~≥~0¯,
M2d~xτ1~,xτ2~=0¯ if and only if xτ1~=xτ2~,
M3d~xτ1~,xτ2~=d~xτ2~,xτ1~,
M4d~xτ1~,x~τ3≤~d~xτ1~,xτ2~+d~xτ2~,xτ3~.The tripleX~,d~,P is called a soft metric space, in short, s.m.s.For the sake of simplicity, we setM≔X~,d~,P.SupposeM is a s.m.s. and γ~ is a nonnegative soft real number. A soft open ball with the center xe~ and radius γ~ is defined by Bxe~,γ~=ye′~∈~X~:d~xe~,ye′~<~γ~. Analogously, a soft closed ball with center xe~ and radius γ~ is Bxe~,γ~=ye′~∈~X~:d~xe~,ye′~≤~γ~. We set that a soft setF,Pis soft open inX~with respect tod~ if and only if all soft points of F,E are interior points of F,P.In a soft metric spaceM, a sequence of soft points xλ,n~n is called convergent in M if there is a soft point yν~∈~X~ such that
(5)limn→∞d~xλ,n~,yν~=0¯.Furthermore, a sequencexλ,n~n is said to be a Cauchy in M if
(6)limi,j→∞d~xλ,i~,xλ,j~=0¯.Moreover, if each Cauchy sequence inX~ converges to some point of X~, then M is called complete soft metric space.LetN=Y~,σ~,P′ be another soft metric space. A soft mapping f,φ:M→N is soft continuous at a point xλ~∈~SPX~, if for each Bf,φx~λ,γ~ of N, there exists Bxλ~,δ~) such that
(7)fBxλ~,δ~⊂~Bf,φxλ~,γ~.In other words, for everyγ~>~0¯, there exists δ~>~0¯ such that d~xλ~,yμ~<~δ~ implies that σ~f,φxλ~,f,φxλ~<~γ~. Moreover, if f,φ is soft continuous for each point of SPX~, then it is called soft continuous mapping.


## 2. Soft Fixed Points for the Soft Inf-Comparable Contraction

In this section, we first introduce the notion of softinf-comparable mapping ψ:ℝ+P→ℝ+P.Definition 3 (see [15]).
LetP be a parameter set and ψ:ℝ+P→ℝ+P. We call ψ a soft inf-comparable mapping if it satisfies the following two axioms:
ψ1ψτ~<~τ~ for all τ~∈ℝ+P\0¯ and ψ0¯=0¯,
ψ2liminfτn~→τ~ψτn~<~τ~ for all τ~>~0¯.Lemma 4.
Letψ:ℝ+P→ℝ+P be a soft inf-comparable mapping. Then, limn→∞ψnτ~=0¯ for all τ~>0¯, where ψn denotes the n-th iteration of ψ.Proof.
Letτ~>~0¯ be fixed. If ψ0nτ~=0¯ for some n0∈ℕ, then we have
(8)ψn0+1τ~=ψψn0τ~=ψ0¯=0¯,which implies that
(9)ψn0+rτ~=0¯,forallr∈ℕ.
Thus, we conclude that(10)limn→∞ψnτ~=0¯.
Ifψnτ~>~0¯ for each n∈ℕ, then we take σn~=ψnτ~, and
(11)σn+1~=ψn+1τ~=ψψnτ~=ψσn~,for all n∈ℕ. By the condition (ψ1) of the soft inf-comparable mapping ψ, we have that for all n∈ℕ,
(12)σn+1~=ψσn~<~σn~.
Keeping (ψ2) in mind and considering that the soft sequenceσn∈ℕis bounded from below and also that the soft sequence is strictly decreasing, one can find an ν~≥~0¯ such that
(13)limn→∞σn~=ν~.
We assert thatν~=0¯. If not, suppose that ν~>~0¯, then we find
(14)ν~=limn→∞σn+1~=limn→∞infψσn~=limσn~→ν~infψσn~<~ν~,
a contradiction. So we obtain thatlimn→∞ψnτ~=0¯.We introduce the notion of softinf-comparable contraction, as follows:Definition 5.
LetM be a soft metric space and let ψ:ℝ+P→ℝ+P be a soft inf-comparable mapping. A mapping f,φ:M→M is called a soft inf-comparable contraction if for each soft points xp~,yτ~∈SPX~,
(15)d~f,φxp~,f,φyτ~≤~ψmaxd~xp~,yτ~,d~xp~,f,φxp~,d~yτ~,f,φyτ~.Example 6.
SetR=ℝ~,d~,P where the soft metric is expressed as
(16)dφp,τ=maxp,τ,dx,y=x−y,d~xp~,yτ~=35dφp,τ+dx,y,with P=0,∞, φt=2/3t for t∈0,∞.
Letψ:ℝ+P→ℝ+P be denoted by
(17)ψω~=56ω~.and let fx=2/5x. Consequently, we find
(18)d~f,φxp~,f,φyτ~=d~25x2/3p~,25y2/3τ~=25maxp,τ+25x−y,d~xp~,yτ~=35maxp,τ+x−y,d~xp~,12x1/3p~=35maxp,13p+x−12x=35p+12x,d~yτ~,12y1/3τ~=35maxτ,13τ+y−12y=35τ+12y.
As a result,f,φ forms a soft inf-comparable contraction on R.We say that a soft pointxτ~∈SPX~ is a soft fixed point of a self-soft-mapping f,φ if f,φxτ~=xτ~.Theorem 7.
Letψ:ℝ+P→ℝ+P be a soft inf-comparable mapping. Let f,φ:M→M be a soft inf-comparable contraction on a complete soft metric space M. Then, a soft mapping f,φ possesses a soft fixed point.Proof.
Letxτ00~∈SPX~ be given. For each n∈ℕ∪0, we put
(19)xτn+1n+1~=f,φxτnn~=fn+1xτ00~φn+1τ0.
Then, we have for eachn∈ℕ∪0(20)d~xτnn~,xτn+1n+1~=d~f,φxτn−1n−1~,f,φxτnn~≤~ψmaxd~xτn−1n−1~,xτnn~,d~xτn−1n−1~,f,φxτn−1n−1~,d~xτnn~,f,φxτnn~=ψmaxd~xτn−1n−1~,xτnn~,d~xτn−1n−1~,xτnn~,d~xτnn~,xτn+1n+1~.
Sinceψ:ℝ+P→ℝ+P is a soft inf-comparable mapping, we can conclude that for each n∈ℕ∪0,
(21)d~xτnn~,xτn+1n+1~≤~ψd~xτn−1n−1~,xτnn~.
By induction, we obtain that(22)d~xτnn~,xτn+1n+1~≤~ψd~xτn−1n−1~,xτnn~≤~ψ2d~xτn−2n−2~,xτn−1n−1~≤~⋯≤~ψnd~xτ00~,xτ11~.
By Lemma4, we obtained that
(23)limn→∞d~xτnn~,xτn+1n+1~=0¯.
In what follows, we check whether the sequencexτnn~ is Cauchy: for each ε~, there is n0∈ℕ such that if n,k≥n0, then
(24)d~xτkrkr~,xτnrnr~<~ε~.
Suppose, on the contrary, that the statement∗ is false. Then, there exists ε~>~0¯ such that, for any r∈ℕ, there are nr,kr∈ℕ with nr>kr≥r satisfying that
(1)
nr is even and kr is odd(2)
d~xτkrkr~,xτnrnr~≥~ε~(3)
nr is the smallest even number such that condition 2 holds
By1 and 2, we conclude that
(25)ε~≤~d~xτkrkr~,xτnrnr~≤~d~xτkrkr~,xτnr−2nr−2~+d~xτnr−2nr−2~,xτnr−1nr−1~+d~xτnr−1nr−1~,xτnrnr~≤~ε~+d~xτnr−2nr−2~,xτnr−1nr−1~+d~xτnr−1nr−1~,xτnrnr~.
Lettingr→∞, we obtain that
(26)limr→∞d~xτkrkr~,xτnrnr~=ε~.
On the other hand,(27)ε~≤~d~xτkr−1kr−1~,xτnr−1nr−1~≤~d~xτkr−1kr−1~,xτnr−3nr−3~+d~xτnr−3nr−3~,xτnr−2nr−2~+d~xτnr−2nr−2~,xτnr−1nr−1~≤~ε~+d~xτnr−3nr−3~,xτnr−2nr−2~+d~xτnr−2nr−2~,xτnr−1nr−1~.
Lettingr→∞, we obtain that
(28)limr→∞d~xτkr−1kr−1~,xτnr−1nr−1~=ε~.
By the above arguments, we obtain that(29)d~xτkrkr~,xτnrnr~=d~f,φf,φ,f,φxτnr−1nr−1~≤~ψmaxd~xτkr−1kr−1~,xτnr−1nr−1~,d~xτkr−1kr−1~,f,φxτkr−1kr−1~,d~xτnr−1nr−1~,f,φxτnr−1nr−1~≤~ψmaxd~xτkr−1kr−1~,xτnr−1nr−1~,d~xτkr−1kr−1~,xτkrkr~,d~xτnr−1nr−1~,xτnrnr~.
Takinglimr→∞inf, we get ε~<~ε~. This implies a contradiction. So the sequence xτnn~ is Cauchy.
SinceM is complete, there exists xτ∗~∈X~ such that
(30)xτnn~→xτ∗~asn→∞,that is,
(31)d~xτnn~,xτ∗~→0¯asn→∞.
Notice also that(32)d~f,φxτ∗~,xτ∗~≤~d~f,φxτnn~,f,φxτ∗~+d~f,φxτnn~,xτ∗~<~ψmaxd~xτnn~,xτ∗~,d~xτnn~,f,φxτnn~,d~xτ∗~,f,φxτ∗~+d~xτn+1n+1~,xτ∗~<~ψmaxd~xτnn~,xτ∗~,d~xτnn~,xτn+1n+1~,d~xτ∗~,f,φxτ∗~+d~xτn+1n+1~,xτ∗~.
Takingn→∞, we get that
(33)d~f,φxτ∗~,xτ∗~≤~ψmax0¯,0¯,,d~xτ∗~,f,φxτ∗~+0¯<~d~f,φxτ∗~,xτ∗~,and this is a contradiction unless d~f,φxτ∗~,xτ∗~=0¯. Thus, f,φx~τ∗=x~λ∗ completes the proof.Example 8.
Consider Example6. All hypotheses of Theorem 7 are fulfilled. Thus, we can conclude that 00¯ is a fixed soft point of the soft inf-comparable Meir-Keeler contraction f,φ.


## 3. Observation on the Soft Comparable Meir-Keeler Contractions

We start this section by recalling the Meir-Keeler contraction in the standard setting.Definition 9.
(see). A self-mappingg on a metric space X,d is called a Meir-Keeler contraction if the following is fulfilled: for any η>0, there is γ>0 such that
(34)η≤dx,y<η+γ⇒dgx,gy<ηforallx,y∈X.The mappingϕ:ℝ+P×ℝ+P×ℝ+P→ℝ+P is said to be soft comparable, if the following two axioms are fulfilled:(ϕ1) ϕ is a soft increasing and soft continuous function in each coordinate(ϕ2) for ω~∈ℝ+P\0¯, ϕω~,ω~,ω~<~ω~, and ϕω1~,ω2~,ω3~=0¯ if and only if ω1~=ω2~=ω3~=0¯Now, we introduce the notion of soft comparable Meir-Keeler contraction.Definition 10.
Letϕ:ℝ+P×ℝ+P×ℝ+P→ℝ+P be soft comparable. A self-soft-mapping f,φ on a soft metric space M is called a soft comparable Meir-Keeler contraction if for each soft real number η~>~0¯, there is γ~>~0¯ such that
(35)η~≤~ϕd~xp~,yτ~,d~xp~,f,φxp~,d~yτ~,f,φyτ~<~η~+γ~⇒d~f,φxp~,f,φyτ~<~η~,for each soft points xλ~,yμ~∈SPX~.Example 11.
SetR=ℝ~,d~,P where the soft metric is expressed as
(36)dφp,τ=maxp,τ,dx,y=x−y,d~xp~,yτ~=dφp,τ+dx,y,with P=0,∞, φt=1/3t for t∈0,∞.
Letϕ:ℝ+P×ℝ+P×ℝ+P→ℝ+P be denoted by
(37)ϕω1~,ω2~,ω3~=34·maxω1~,ω2~,ω3~,where
(38)ω1~=d~xp~,yτ~,ω2~=d~xp~,f,φxp~,ω3~=d~yτ~,f,φyτ~.
Letfx=1/2x. Then,
(39)d~f,φxp~,f,φyτ~=d~12x1/3p~,12y1/3τ~=13maxp,τ+12x−y,ω~1=d~xp~,yτ~=maxp,τ+x−y,ω2~=d~xp~,12x1/3p~=maxp,13p+x−12x=p+12x,ω3~=d~yτ~,12y1/3τ~=maxτ,13τ+y−12y=τ+12y.
So we can conclude that(40)ϕω1~,ω2~,ω3~,t4~=34·maxω1~,ω2~,ω3~≥~34ω1~=34maxp,τ+∣x−y∣.Consequently, a soft mappingf,φ forms a soft comparable Meir-Keeler contraction on R.We establish the following fixed point results for the soft comparable Meir-Keeler contraction.Theorem 12.
LetM be a complete soft metric space, and let ϕ:ℝ+P×ℝ+P×ℝ+P→ℝ+P be a soft comparable. Let f,φ:M→M be a soft comparable Meir-Keeler contraction on M. Then, f,φ possesses a soft fixed point.Proof.
Letxτ00~∈SPX~ be given. For each n∈ℕ∪0, we put
(41)xτn+1n+1~=f,φxτnn~=fn+1xτ00~φn+1τ0.
So, for eachn∈ℕ∪0 we have
(42)d~xτnn~,xτn+1n+1~=d~f,φxτn−1n−1~,f,φxτnn~≤~ϕd~xτn−1n−1~,xτnn~,d~xτn−1n−1~,f,φxτn−1n−1~,d~xτnn~,f,φxτnn~=ϕd~xτn−1n−1~,xτnn~,d~xτn−1n−1~,xτnn~,d~xτnn~,xτn+1n+1~.
Ifd~xτk−1k−1~,xτkk~<~d~xτkk~,xτk+1k+1~ for some k∈ℕ, then by the above inequality and the conditions of the function ϕ, we have
(43)d~xτnk~,xτk+1k+1~≤~ϕd~xτk−1k−1~,xτkk~,d~xτk−1k−1~,xτkk~,d~k~τkn,xτk+1k+1~<~d~xτkk~,xτk+1k+1~,
which implies a contradiction. Hence, for eachn∈ℕ, we find
(44)d~xτnn~,xτn+1n+1~<~d~xτk−1k−1~,xτkk~.
Thus, the sequenced~xτnn~,xτn+1n+1~ is decreasing and converges to a soft real number, say γ~≥~0¯. In other words, d~xτnn~,xτn+1n+1~→γ~, as n→∞.
Notice thatγ~=infd~xτnn~,xτn+1n+1~:n∈ℕ∪0. We claim that γ~=0¯. Suppose, on the contrary, that γ~>~0¯. Since f,φ is a soft comparable Meir-Keeler contraction, corresponding to γ~, there exists η~>~0¯ and k∈ℕ such that
(45)γ~≤~ϕd~xτkk~,xτk+1k+1~,d~xτkk~,xτk+1k+1~,d~xτk+1k+1~,xτk+2k+2~<~γ~+η~⇒d~xτk+1k+1~,xτk+2k+2~=d~f,φxτkk~,f,φxτk+1k+1~<~γ~.
This is a contradiction sinceγ~=infd~xτnn~,xτn+1n+1~:n∈ℕ∪0. Thus, we obtain that d~xτnn~,xτn+1n+1~→0¯, as n→∞.
As a next step, we check whether the sequencexτnn~ is Cauchy in M. Suppose, on the contrary, it is not. Thus, there exists a soft real number ε~>~0¯ such that for any k∈ℕ, there are mk,nk∈ℕ with nk>mk≥k satisfying
(46)d~xτmkmk~,xτnknk~≥~ε~.
Further, corresponding tomk≥k, we can choose nk in such a way that it is the smallest integer with nk>mk≥k and d~xτmkmk~,xτnknk~≥~ε~. Therefore,
(47)d~xτmkmk~,xτnk−2nk−2~<~ε~.
So, we derive that(48)ε~≤~d~xτmkmk~,xτnknk~≤~d~xτmkmk~,xτnk−2nk−2~+d~xτnk−2nk−2~,xτnk−1nk−1~+d~xτnk−1nk−1~,xτnknk~<~ε~+d~xτnk−2nk−2~,xτnk−1nk−1~+d~xτnk−1nk−1~,xτnknk~,
for allk∈ℕ. As k→∞, the inequality above yields that
(49)limk→∞d~xτmkmk~,xτnknk~=ε~.
On the other hand, we have(50)ε~≤~d~xτmkmk~,xτnknk~≤~d~xτmkmk~,xτmk+1mk+1~+d~xτmk+1mk+1~,xτnk+1nk+1~+d~xτnk+1nk+1~,xτnknk~≤~d~xτmkmk~,xτmk+1mk+1~+d~xτmk+1mk+1~,xτmkmk~+d~xτmkmk~,xτnknk~+d~xτnknk~,xτnk+1nk+1~+d~xτnk+1nk+1~,xτnknk~.
Lettingk→∞ in the above inequality, we get
(51)limk→∞d~xλmk+1mk+1~,xλnk+1nk+1~=ε~.
Sincef,φ is a soft comparable Meir-Keeler contraction, we have
(52)d~xτmk+1mk+1~,xτnk+1nk+1~=d~f,φxτmkmk~,f,φxτnknk~<~ϕd~xτmkmk~,xτnknk~,d~xτmkmk~,xτmk+1mk+1~,d~xτnknk~,xτnk+1nk+1~.
Moreover, since(53)d~xτmkmk~,xτnk+1nk+1~≤~d~x~τmkmk,xτmk+1mk+1~+d~xτmk+1mk+1~,xτnk+1nk+1~,d~xτnknk~,xτmk+1mk+1~≤~d~xτnknk~,xτnk+1nk+1~+d~xτnk+1nk+1~,xτmk+1mk+1~.
Takingk→∞ in the above inequalities, we get that
(54)ε~≤~ϕε~,ε~,ε~<~ε~,and this is a contradiction. Thus, the sequence xλnn~ is Cauchy.
Keeping the completeness ofM in mind, one can find xτ∗~∈X~ such that
(55)xτnn~→xτ∗~asn→∞,that is,
(56)d~xτnn~,xτ∗~→0¯asn→∞.
And, we also have(57)d~f,φxτ∗~,xτ∗~≤~d~f,φxτnn~,f,φxτ∗~+d~f,φxτnn~,xτ∗~<~ϕd~xτnn~,xτ∗~,d~xτnn~,f,φxτnn~,d~xτ∗~,f,φxτ∗~+d~xτn+1n+1~,xτ∗~<~ϕd~xτnn~,xτ∗~,d~xτnn~,xτn+1n+1~,d~xτ∗~,f,φxτ∗~+d~xτn+1n+1~,xτ∗~.
Takingk→∞ in the inequality above,
(58)d~f,φxτ∗~,xτ∗~≤~ϕ0¯,0¯,d~f,φxτ∗~+0¯≤~ϕd~f,φxτ∗~,xτ∗~,d~f,φxτ∗~,xτ∗~,d~f,φxτ∗~,xτ∗~<~d~f,φxτ∗~,xτ∗~,
a contradiction unlessd~f,φxτ∗~,xτ∗~=0¯. Thus, f,φx~τ∗=x~τ∗ which completes the proof.Example 13.
Consider Example11. One can easily check all hypotheses of Theorem 12. Consequently, we conclude that 00¯ is a fixed soft point of the soft comparable Meir-Keeler contraction f,φ.We next introduce the notion of soft generalized Meir-Keeler contraction, as follows:Definition 14.
LetX~,d~,P be a soft metric space. A mapping f,φ:X~,d~,P→X~,d~,P is called a soft generalized Meir-Keeler contraction if for any soft real number η~>~0¯, there exists γ~>~0¯ such that for each soft point xp~,yτ~∈SPX~,
(59)η~≤~maxd~xp~,f,φxp~,d~yτ~,f,φyτ~,d~xp~,f,φyτ~<~η~+γ~⇒d~f,φxp~,f,φyτ~<~η~.It is clear that the soft generalized Meir-Keeler contraction is a comparable soft Meir-Keeler contraction; we can easily conclude the following corollary.Corollary 15.
A soft generalized Meir-Keeler contraction onf,φwhich is a complete soft metric spaceMpossesses a fixed soft point.



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