# Fixed Point Theorems for Contractive Selfmappings of a Bounded Metric Space

Authors: Youssef Touail; Driss El Moutawakil; Samia Bennani
Journal: Journal of Function Spaces
 (2019)

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## Abstract

The main purpose of this paper is to prove a new fixed theorem for selfmapping of a metric space(X,d). As applications, we get a new fixed point result for shrinking or contractive maps and a fixed point theorem for a new class of weakly contractive selfmappings of a bounded metric space (X,d), where the auxiliary function ϕ satisfies ϕ(0)=0 and inft>0ϕ(t)>0.


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## Body

## 1. Introduction

LetT:X→X be a mapping of a metric space (X,d). It is well known that T is called a shrinking or a contractive map if it satisfies the inequality d(Tx,Ty)<d(x,y) for each x,y∈X with x≠y. In [1], V. V. Nemytzk was the first mathematician who studied the problem of the existence of a fixed point of these mappings. Furthermore, it is mentioned in [2] that, to obtain a fixed point of such mappings, it is necessary either to add the assumption that there exists a point x∈X for which {Tnx} contains a convergent subsequence, or else to assume that the space is compact.In [3], the authors introduced the notion of weakly contractive mappings in Hilbert spaces and proved that any weakly contractive mapping defined on Hilbert spaces has a unique fixed point. Rhoades [4] proved that the corresponding result is also valid when Hilbert space is replaced by a complete metric space. Recall that a mapping T:X→X of a metric space (X,d) is said to be weakly contractive if d(Tx,Ty)≤d(x,y)-ϕ(d(x,y)), for all x,y∈X, where ϕ:[0,+∞[→[0,+∞[ is a continuous nondecreasing function such that ϕ(0)=0. Since then, weakly contractive mappings have been dealt with in a number of papers. Some of these works are noted in [1, 5–8].On the other hand, the authors [9] have introduced the concept of τ-distance functions in a general topological space (X,τ) and mentioned that metric spaces, symmetric spaces, probabilistic metric spaces, and topological vector spaces have all such functions. Moreover, they presented an application of this new concept to the fixed point theory by giving the following result (Corollary 4.1 [9]) which generalizes the well-known Banach’s fixed point theorem as followsTheorem 1 (Corollary 4.1 [9]).
Let(X,τ) be a Hausdorff topological space with a τ-distance p. Suppose that X is p-bounded and S-complete. Let T:X→X be a mapping satisfying: there exists k∈[0,1[ such that for all x,y∈X, we have p(Tx,Ty)≤kp(x,y).
ThenT has a unique fixed point.Recall that a sequence in X is p-Cauchy if it satisfies the usual metric condition with respect to p. The definition of a p-bounded S-complete space is presented in Definition 3.1 [9] as follows.Definition 2 (Definition 3.1 [9]).
Let(X,τ) be a topological space with a τ-distance p. (1)
X is S-complete if for every p-Cauchy sequence(xn), there exists x in X with limp(x,xn)=0.(2)
X is p-Cauchy complete if for every p-Cauchy sequence(xn), there exists x in X with limxn=x with respect to τ.(3)
X is said to be p-bounded ifsup{p(x,y)/x,y∈X}<∞.For more information, we refer the reader to [9]. Our purpose in this paper is to present a new fixed point result for shrinking maps and a fixed point theorem for a new class of weakly contractive selfmappings of a bounded metric space (X,d) by using Theorem 1, where the auxiliary function ϕ satisfies ϕ(0)=0 and inft>0⁡ϕ(t)>0.


## 2. Main Results

Theorem 3.
LetT:X→X be a mapping of a bounded complete metric space (X,d) such that infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}>0. Then T has a unique fixed point.Proof.
Letα=infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}. It is clear that, for all x≠y∈X, one has d(Tx,Ty)≤d(x,y)-α, and, therefore ed(Tx,Ty)≤ked(x,y), where k=e-α.
Let us consider the functionp:X×X→[0,+∞[ defined by (1)px,y=edx,yifx≠y0ifx=yAs mentioned in [9] (Example 2.4.), the function p is a τd-distance on X where τd is the usual metric topology.
On the other hand, the mappingT satisfies on (X,τd) the following contraction:(2)pTx,Ty≤kpx,y,wherek=e-α∈0,1. According to Corollary 4.1. in [9], we deduce that T has a unique fixed point in X.Corollary 4.
LetT:X→X be a shrinking mapping of a bounded complete metric space (X,d) such that infx≠y∈X{d(x,y)-d(Tx,Ty)}≠0. Then T has a unique fixed point.Example 5.
LetX={1;2} with the usual metric d(x,y)=x-y. Define T by T1=1 and T2=1. We have(3)d1,2-dT1,T2=1Then T satisfies all assumptions of Corollary 4 and T has the unique fixed point which is equal to 1.Example 6.
LetX=B¯(0,1), the unit closed ball of a real Banach space, with the metric d(x,y)=1ifx≠y0ifx=y.
Let us consider the mappingT defined by Tx=0, for all x∈X.
ThenT:X→X is a shrinking mapping of the complete bounded metric space (X,d) and(4)dx,y-dTx,Ty=dx,y=1,forallx≠y∈X.Therefore T satisfies all assumptions of Corollary 4 and T has the unique fixed point which is equal to 0.Remarks 7.
Obviously, for a shrinking mappingT:X→X of a metric space (X,d), one can ask does there exist a relationship between compactness and the condition infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}≠0. The answer is negative. Indeed, in the first example, the space X is compact and infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}≠0. However, for X=[0,1] and Tx=1/2x, the mapping T is shrinking and infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}=0. Furthermore, in the second example, the space (X,d) is not compact and infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}≠0.Definition 8.
LetT:X→X be a mapping of a metric space (X,d). T will be said a E-weakly contractive maps if d(Tx,Ty)≤d(x,y)-ϕ(d(x,y)), for all x,y∈X, where ϕ:[0,+∞[→[0,+∞[ is a function satisfying ϕ(0)=0 and inft>0⁡ϕ(t)>0.As a second application of Theorem3., we get the following new fixed point for E- weakly contractive selfmappings of a bounded metric space (X,d).Theorem 9.
LetT:X→X be a E-weakly contractive mapping of a bounded complete metric space (X,d). Then T has a unique fixed point.Proof.
From the Definition8., it is clear that, for all x≠y∈X, we have(5)0<inft>0ϕt≤ϕdx,y≤dx,y-dTx,Ty,which implies that infx≠y∈X⁡{d(x,y)-d(Tx,Ty)}>0. According to Theorem 3. the mapping T has a unique fixed point in X.Example 10.
LetX={1;2} with the usual metric d(x,y)=x-y. Define T and ϕ by T1=1, T2=1 and ϕ(t)=1ift≠00ift=0.
ThenT satisfies all assumptions of Theorem 9 and T has the unique fixed point 1. Note that ϕ is not continuous at 0.Example 11.
LetX={1;2} with the usual metric d(x,y)=x-y. Define T and ϕ by T1=2, T2=1, and ϕ(t)=0 for all t∈[0,+∞[.
Then(X,d) is a bounded complete metric space and d(Tx,Ty)≤d(x,y)-ϕ(d(x,y)) for all x,y∈X. However, T is not a E-weakly contractive since inft>0ϕ(t)=0 and T has no fixed point. Therefore the condition that inft>0⁡ϕ(t)≠0 is essential.Example 12.
LetX=[0,+∞[ with the usual metric d(x,y)=x-y. Define T and ϕ by Tx=ln(1+ex) and ϕ(t)=t(1-sup{T′(x)∣x∈[0,+∞[}) for all x,t∈[0,+∞[, and w here T′ is the derived function of T.
ThenT is a E-weakly contractive map with no fixed point on X since (X,d) is a complete unbounded metric space and inft>0ϕ(t)=0.


## 3. Application

In this section, we investigate the existence and uniqueness of a solution for the nonlinear integral equation:(6)xt=ft+∫0tKs,xsds,where x∈C[0,τ] and the space of all continuous functions from [0,τ] into R, with τ>0.K : [ 0 , τ ] × R → R is a continuous mapping and f:[0,τ]→R is a given function.LettingX=C[0,τ] endowed by the metric d:X×X→R+ defined by (7)dx,y=supt∈0,Txt-yt,obviously (X,d) is a complete metric space.Consider the mappingT:X→X defined as follows:(8)Txt=ft+∫0tKs,xsdsfor any x∈X.Note that (6) has a solution if and only if T has a fixed point.Under the above assumptions we have the following theorem.Theorem 13.
If there existsM>0 such that (9)Ks,xs-Ks,ys≤1τxs-ys-M,for all s∈[0,τ] and x,y∈X such that x≠y. Then the nonlinear integral equation (6) has a unique solution.Proof.
Assuming thatx,y∈X and t∈[0,τ], then we have (10)Txt-Tyt=∫0tKs,xsds-∫0tKs,ysds=∫0tKs,xs-Ks,ysds≤∫0tKs,xs-Ks,ysds≤dx,y-Mhence(11)dTx,Ty≤dx,y-Mfor all x,y∈X. Then infx≠y{d(x,y)-d(Tx,Ty)}≥M>0, which implies by Theorem 3 that there exists a unique solution of the nonlinear equation (6).



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