# A Probabilistic Fixed Point Result Using Altering Distance Functions

Authors: Claudia Zaharia; Nataša Ćirović
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove a general fixed point theorem in Menger spaces for mappings satisfying a contractive condition of Ćirić type, formulated by means of altering distance functions. Thus, we extend some recent results of Choudhury and Das, Miheţ, and Babačev and also clarify some aspects regarding a theorem of Choudhury, Das, and Dutta.


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## Body

## 1. Introduction and Preliminaries

In [1], Menger introduced the concept of probabilistic metric space as a generalization of metric spaces, in which the distance between points is expressed by means of distribution functions. This idea has made probabilistic metric spaces suitable for modeling phenomena when the uncertainty regarding measurements is assumed as inherent to the measuring process, as, for instance, in the investigation of certain physical quantities and physiological thresholds [2]. Probabilistic metric space theory has become a very active field of research. In particular, fixed point theory in probabilistic structures has found relevant applications in studying the existence and uniqueness of solutions of random equations [3], as well as algorithm complexity analysis [4, 5], and convergence analysis for stochastic optimization algorithms [6].In the present paper, we establish a fixed point result for probabilistic contractions of Ćirić type, with the contractive condition stated by means of an altering distance function. Our theorem is obtained under very weak hypotheses, and thus it generalizes or improves several known results [7–10]. We also discuss the connections with a related theorem given by Choudhury et al. in [11], in order to explain the role of our assumptions.We begin by recalling some fundamental concepts of probabilistic metric space theory. For a comprehensive exposition on this topic we refer the reader to the monographs [2, 3].Definition 1.
A triangular norm (ort-norm) is a mapping T:[0,1]×[0,1]→[0,1] which is associative, commutative, and nondecreasing in each variable and satisfies T(x,1)=x for all x∈[0,1].Some basic examples areTM(x,y)=min(x,y), TP(x,y)=xy, and TL(x,y)=max(x+y-1,0) (the minimum, product, and Łukasiewicz t-norm, resp.). Another important class is that of t-norms of Hadžić type [12], that is, t-norms whose family of iterates {Tn(x)}n defined by T0(x)=1, Tn(x)=T(Tn-1(x),x)∀n≥1, is equicontinuous at x=1.We will denote byD+ the set of functions F:R→[0,1] which are nondecreasing and left continuous on R, such that F(0)=0 and limt→∞F(t)=1.Definition 2.
A Menger space is a triple(X,F,T) where X is a nonempty set, F is a mapping from X×X to D+, and T is a t-norm, such that the following conditions are satisfied:(i)
Fxy(t)=1 for all t>0 iff x=y.(ii)
Fxy(t)=Fyx(t), for all x,y∈X, t>0.(iii)
Fxy(t+s)≥T(Fxz(t),Fzy(s)), for all x,y,z∈X, t,s>0.
(Here and in the following,F(x,y) will be denoted by Fxy.)Let(X,F,T) be a Menger space such that supx∈(0,1)T(x,x)=1. The family {U(ε,λ)}ε>0,λ∈(0,1), where (1)Uε,λ=x,y∈X×X:Fxyε>1-λ,is a base for a Hausdorff uniformity on X, named strong uniformity. The corresponding, strong topology on X is introduced by the family of neighbourhoods of x∈XNx={Nx(ε,λ)}ε>0,λ∈(0,1), where Nx(ε,λ)={y∈X:Fxy(ε)>1-λ}, and this topology is metrizable [2].Definition 3.
Let(X,F,T) be a Menger space. A sequence (xn)n in X is said to be (i)
Cauchy if, for anyε>0, λ∈(0,1), there exists n0∈N such that Fxnxm(ε)>1-λ, for all m,n≥n0;(ii)
convergent tox∈X if, for any ε>0, λ∈(0,1), there exists n0∈N such that Fxnx(ε)>1-λ for all n≥n0.
X is said to be complete if every Cauchy sequence (xn)n⊂X is convergent in X.If thet-norm T is continuous, and the sequences (xn)n, (yn)n⊂X converge, respectively, to x and y∈X, then Fxnyn(t) converges to Fxy(t), for each continuity point t of Fxy [2].Definition 4.
Given a setA⊂X, the probabilistic diameter of A is the mapping DA:R→[0,1] defined by (2)DAt=sups<tinfx,y∈AFxys,t>00,t≤0.A is said to be probabilistically bounded if DA∈D+.The notion of contraction in a Menger space was introduced by Sehgal in [13].Definition 5 (see [13]).
Let(X,F,T) be a Menger space. A mapping f:X→X is said to be a probabilistic contraction (or Sehgal contraction) if there exists c∈(0,1) such that (3)Ffxfyt≥Fxytc,∀x,y∈X,t>0.Many significant contributions to the development of fixed point theory in probabilistic structures can be found in monograph [14]. It should be pointed out that the triangular norm by which the space is endowed plays a key role in the existence of fixed points of probabilistic contractions. It was shown by Radu [15] that the largest class of continuous t-norms T with the property that every Sehgal contraction on a complete Menger space (X,F,T) has a unique fixed point is that of t-norms of Hadžić type.The idea of using altering distance functions in order to obtain more general contractive conditions first appears in [16], in the setting of metric spaces. The corresponding concept of generalized probabilistic contraction was introduced by Choudhury and Das in [8] as follows.Definition 6 (see [8]).
A mappingφ:[0,∞)→[0,∞) is said to belong to the class Φ if it satisfies (i)
φ(t)=0 iff t=0;(ii)
φ is strictly increasing and limt→∞φ(t)=∞;(iii)
φ is continuous at t=0 and left continuous on (0,∞).The mappingsφ∈Φ will be called altering distance functions.Definition 7 (cf. [8]).
Let(X,F,T) be a Menger space. The mapping f:X→X is said to be a generalized probabilistic contraction of Choudhury-Das type if there exist φ∈Φ and c∈(0,1) such that (4)Ffxfyφt≥Fxyφtc,∀x,y∈X,t>0.It was proved in [8] that such contractions on a complete Menger space endowed with the strongest t-norm TM have a unique fixed point. The result was subsequently generalized by Miheţ [9] for the case of arbitrary continuous t-norms, under the supplementary assumption that the orbit of the mapping f at some x∈X is probabilistically bounded.Our aim is to prove a fixed point result for mappings satisfying the more general contractive condition(5)Ffxfyφt≥min⁡Fxyφtc,Fxfxφtc,Fyfyφtc,∀x,y∈X,t>0,for some φ∈Φ and c∈(0,1).


## 2. Main Results

In order to prove our results, we will need the following lemma from [10].Lemma 8 (see [10]).
Let(X,F,T) be a Menger space; φ∈Φ and c∈(0,1). If x,y∈X are such that (6)Fxyφt≤Fxyφtc,∀t>0,then x=y.For eachx∈X, we will denote by O(f,x) the orbit of the mapping f at x; that is, O(f,x)={fn(x):n∈N}.Theorem 9.
Let(X,F,T) be a complete Menger space with T a continuous t-norm. Suppose f:X→X is a mapping satisfying the contractive condition (5), for some φ∈Φ and c∈(0,1). If there exists x∈X such that O(f,x) is probabilistically bounded, then f has a unique fixed point in X.Proof.
Letx be as in the statement of the theorem. We will show that the sequence (xn)n, xn=fn(x), is Cauchy.
From (5) it follows that(7)Fxnxn+1φt≥min⁡Fxn-1xnφtc,Fxnxn+1φtc,∀t>0.By replacing t with t/c above we get(8)Fxnxn+1φtc≥min⁡Fxn-1xnφtc2,Fxnxn+1φtc2,∀t>0.Therefore(9)Fxnxn+1φt≥min⁡Fxn-1xnφtc,Fxn-1xnφtc2,Fxnxn+1φtc2,and, inductively, (10)Fxnxn+1φt≥min⁡Fxn-1xnφtc,Fxn-1xnφtc2,…,Fxn-1xnφtcp,Fxnxn+1φtcp,for all t>0 and for any positive integer p. Since φ is strictly increasing, we obtain (11)Fxnxn+1φt≥min⁡Fxn-1xnφtc,Fxnxn+1φtcp,∀t>0,for all p. By letting p→∞ it follows that (12)Fxnxn+1φt≥Fxn-1xnφtc,∀t>0.Consequently, for all n∈N and t>0, (13)Fxnxn+1φt≥Fx0x1φtcn→n→∞1.
Next, letm be a positive integer. We prove by induction on n that(14)Fxnxn+mφcnt≥min⁡Fx0x1φt,Fx0xmφt,∀t>0,for all n∈N. The case n=0 is immediate. Suppose now that inequality (14) holds for some n∈N. Then (15)Fxn+1xn+m+1φcn+1t=Ffxnfxn+mφcn+1t≥min⁡Fxnxn+mφcnt,Fxnxn+1φcnt,Fxn+mxn+m+1φcnt≥min⁡Fx0x1φt,Fx0xmφt,Fx0x1φt,Fx0x1φtcm,for all t>0. By the monotonicity of φ, it follows that Fx0x1φt/cm≥Fx0x1(φ(t)), for all t>0, whence (16)Fxn+1xn+m+1φcn+1t≥min⁡Fx0x1φt,Fx0xmφt,∀t>0.
As such, we conclude that inequality (14) holds for all m,n∈N, or equivalently (17)Fxnxn+mφt≥min⁡Fx0x1φtcn,Fx0xmφtcn,∀m,n∈N,t>0.
Now, letε>0 and λ∈(0,1). Given that φ is continuous at 0, there exists t>0 with φ(t)<ε. Also, since DO(f,x)∈D+, there exists n0∈N such that DO(f,x)φt/cn>1-λ, for all n≥n0.
Thus,(18)Fxnxn+mε≥Fxnxn+mφt≥min⁡Fx0x1φtcn,Fx0xmφtcn≥DOf,xφtcn>1-λ,for all n≥n0,  m∈N, and therefore (xn)n is a Cauchy sequence. Accordingly, there exists x∗∈X, x∗=limn→∞xn.
Next, we will prove thatx∗ is a fixed point of f. Specifically, we will show that (19)Fx∗fx∗φt≥Fx∗fx∗φtc,for all t>0.
By the contractive condition (5), (20)Fxn+1fx∗φt≥min⁡Fxnxn+1φtc,Fx∗fx∗φtc,Fxnx∗φtc,for all n∈N and t>0. If t is such that Fx∗f(x∗) is continuous at φ(t), then (19) follows by letting n→∞ in the above inequality and taking into account relation (13). If Fx∗f(x∗) is not continuous at φ(t), let (tm)m be a strictly increasing sequence converging to t such that Fx∗f(x∗) is continuous at φ(tm), for all m∈N. As above, we infer that Fx∗f(x∗)(φ(tm))≥Fx∗f(x∗)φtm/c, ∀m∈N, whence, for m→∞, we obtain (19). By Lemma 8 we conclude that x∗=f(x∗).
Finally, we prove thatx∗ is the only fixed point of f in X. To that end, let y∈X be such that f(y)=y. Then, using (5), we get(21)Fx∗yφt≥min⁡Fx∗yφtc,Fx∗x∗φtc,Fyyφtc=Fx∗yφtc,∀t>0.Once again, by Lemma 8, it follows that y=x∗.Corollary 10.
If(X,F,T) is a complete Menger space with T a continuous t-norm of Hadžić type, and f:X→X satisfies condition (5) for some c∈(0,1) and some φ∈Φ such that limt→∞(φ(t)-φ(ct))=∞, then f has a unique fixed point in X.Proof.
We will show that, for everyx∈X, O(f,x) is probabilistically bounded. To do so, let x∈X be arbitrary and define (xn)n by xn=fn(x) for all n≥0. We will prove by induction on n that (22)Fx0xnφt≥TnFx0x1φt-φct,∀t>0,for all n≥1. The case n=1 is trivial. Suppose now that the relation holds for some n≥1. Then (23)Fx0xn+1φt≥TFx0x1φt-φct,Fx1xn+1φct≥TFx0x1φt-φct,min⁡Fx0xnφt,Fx0x1φt,Fxnxn+1φt,∀t>0.Given that (24)Fxnxn+1φt≥Fx0x1φtcn≥Fx0x1φt,∀t>0,Fx0x1φt≥Fx0x1φt-φct≥TnFx0x1φt-φct,∀t>0,from the induction hypothesis we obtain(25)Fx0xn+1φt≥TFx0x1φt-φct,TnFx0x1φt-φct=Tn+1Fx0x1φt-φct,∀t>0,which proves our claim.
Now, sincelimt→∞Fx0x1(φ(t)-φ(ct))=1 and the family {Tn}n is equicontinuous at 1, it follows that DO(f,x)∈D+.By settingφ(t)=t in the above corollary we get the following.Corollary 11.
Let(X,F,T) be a complete Menger space with T being a continuous t-norm of Hadžić type and let f:X→X be a mapping such that(26)Ffxfyt≥min⁡Fxytc,Fxfxtc,Fyfytc,for all x,y∈X, t>0. Then f has a unique fixed point in X.Remark 12.
In paper [10], Babačev proved a fixed point result for mappings satisfying the contractive condition (27)Ffxfyφt≥min⁡Fxyφtc,Fxfxφtc,Fyfyφtc,Fxfy2φtc,Fyfx2φtc,∀t>0,for some altering distance function φ and some c∈(0,1), in Menger spaces with the t-norm TM. We note that, by applying the triangle inequality, (28)Fxfy2φtc≥min⁡Fxyφtc,Fyfyφtc,Fyfx2φtc≥min⁡Fxyφtc,Fyfyφtc,so this condition essentially reduces to (5). Therefore Theorem 9 improves the result in [10], as well as Ćirić’s result in [7] (which can be obtained from that of Babačev for φ(t)=t).Also, in [11], Choudhury et al. gave the following related theorem.Theorem 13 (see [11]).
Let(X,F,T) be a complete Menger space with continuous t-norm T, and let a,b,c be positive numbers with a+b+c<1, and φ∈Φ. Suppose that f:X→X satisfies the inequality(29)Ffxfyφt≥min⁡Fxyφt1a,Fxfxφt2b,Fyfyφt3c,for all x,y∈X, t>0, and t1,t2,t3>0 with t1+t2+t3=t. Then f has a unique fixed point.As indicated in [11], a mapping satisfying the contractive condition (29) must also verify our condition (5). Namely, suppose that (29) holds. Let ε=1-(a+b+c)/3∈(0,1) and let t1=(a+ε)t, t2=(b+ε)t, t3=(c+ε)t. It follows that (30)Ffxfyφt≥min⁡Fxyφta+εa,Fxfxφtb+εb,Fyfyφtc+εc,∀t>0.Due to the monotonicity of φ, the above relation implies that (31)Ffxfyφt≥min⁡Fxyφtk,Fxfxφtk,Fyfyφtk,for all t>0, where k=maxa/a+ε,b/b+ε,c/c+ε∈(0,1).Note that Theorem13 only requires that the t-norm by which the space is endowed is continuous. Unfortunately, we can show that this assumption alone is not sufficient to guarantee the existence of fixed points for contractions of this type.Specifically, let(X,F,TL) be a complete Menger space and let f be a Sehgal contraction on X with contraction constant k<1/3. Then(32)Ffxfyt≥Fxytk≥min⁡Fxytk,Fxfxtk,Fyfytk≥min⁡Fxyt1k,Fxfxt2k,Fyfyt3k,for all x,y∈X, t>0, and t1,t2,t3>0 with t1+t2+t3=t. Thus, f satisfies the conditions of Theorem 13 with a=b=c=k and φ(t)=t. However, a well-known counterexample of Sherwood ([17], Corollary 1 of Theorem 3.5) shows that there exist Sehgal contractions on complete Menger spaces endowed with the t-norm TL having no fixed point.It should be mentioned that a similar observation regarding continuity can be made with respect to Theorem3.1 in [18], where the class of contractions considered also includes Sehgal contractions.Finally, we illustrate the applicability of Theorem9 with the following example.Example 14.
LetX=[0,1] and T(a,b)=ab. Define Fxy(t)=t/t+1x-y for all x,y∈X and t>0. (X,F,T) is a complete Menger space. We will only show that the triangle inequality is verified.Assume thatt>s>0 and x,y,z∈X. Since the function t/t+1 is increasing, it holds that (33)TFxyt,Fyzs=tt+1x-yss+1y-z≤tt+1x-y+y-z≤tt+1x-z≤t+st+s+1x-z=Fxzt+s.Letφ(t)=t2/2t+1 for all t>0, c=1/2, and (34)fx=x2,x∈0,1,0,x=1.One can easily check that φ is an altering distance function and that O(f,x) is probabilistically bounded for every x∈X.We will prove that condition (5) of Theorem 9 is satisfied. The following three cases are possible:(1)
Ifx,y∈[0,1), then for all t>0 we have (35)Ffxfyφt=tt+12fx-fy=tt+1x-y≥2t2t+12x-y=Fxyφ2t.(2)
Ifx=y=1, then (36)Ffxfyφt=1=Fxyφ2tfor allt>0.(3)
Ifx∈[0,1) and y=1, then, for all t>0, (37)Ffxfyφt=tt+1x≥tt+1≥2t2t+12=2t2t+12y-fy=Fyfyφ2t.Thus the condition (5) is satisfied in this case as well.However, note that by settingx=2/3 and y=1 we obtain (38)Ffxfyφt=tt+12/3<2t2t+12/3=Fxyφ2tfor all t>0; therefore, f does not satisfy the stronger condition (4).By applying Theorem9 we conclude that the function f has a unique fixed point. It is easy to see that this point is x=0.



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*Source: 919202-2015-08-10.xml*