Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces

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Abstract

In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of fixed-point theorems of Meir–Keeler and Kirk. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120]. We also give a simple proof of this characterization.

Introduction

Throughout this paper we denote by N the set of all positive integers.

In 1969, Meir and Keeler [7] proved the following very interesting fixed-point theorem, which is a generalization of the Banach contraction principle [2]. See also [8], [9].

Theorem 1 Meir and Keeler [7]

Let (X,d) be a complete metric space and let T be a mapping on X. Assume that for every ε>0, there exists δ>0 such that εd(x,y)<ε+δimpliesd(Tx,Ty)<εfor x,yX. Then T has a unique fixed point.

On the other hand, in 2003, Kirk [5] introduced the notion of asymptotic contraction on a metric space, and proved a fixed-point theorem for such contractions. Asymptotic contraction is an asymptotic version of Boyd–Wong contraction [3]. See also [1].

Definition 1 Kirk [5]

Let (X,d) be a metric space and let T be a mapping on X. Then T is called an asymptotic contraction on X if there exists a continuous function ϕ from [0,) into itself and a sequence {ϕn} of functions from [0,) into itself such that

  • (i)

    ϕ(0)=0,

  • (ii)

    ϕ(r)<r for r(0,),

  • (iii)

    {ϕn} converges to ϕ uniformly on the range of d, and

  • (iv)

    for x,yX and nN, d(Tnx,Tny)ϕn(d(x,y)).

Theorem 2 Kirk [5]

Let (X,d) be a complete metric space and let T be a continuous, asymptotic contraction on X with {ϕn} and ϕ in Definition 1. Assume that there exists xX such that the orbit {Tnx:nN} of x is bounded, and that ϕn is continuous for nN. Then there exists a unique fixed point zX. Moreover, limnTnx=z for all xX.

Jachymski and Jóźwik showed that the continuity of T is needed in Theorem 2; see Example 1 in [4]. They also proved a result similar to Theorem 2. The assumption in [4] is that ϕ is upper semicontinuous with limt(t-ϕ(t))= and T is uniformly continuous.

In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of both Theorems 1 and 2. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [6]. We also give a simple proof of this characterization.

Section snippets

Meir–Keeler contraction

In this section, we discuss Meir–Keeler contraction.

Definition 2

Let (X,d) be a metric space. Then a mapping T on X is said to be a Meir–Keeler contraction (MKC, for short) if for any ε>0, there exists δ>0 such that εd(x,y)<ε+δimpliesd(Tx,Ty)<εfor all x,yX.

In [6], Lim introduced the notion of an L-function and characterized MKC.

Definition 3

Lim [6]

A function ϕ from [0,) into itself is called an L-function if ϕ(0)=0, ϕ(s)>0 for s(0,), and for every s(0,) there exists δ>0 such that ϕ(t)s for all t[s,s+δ].

We give a

ACMK

In this section, we discuss the following notion, which is a generalization of both asymptotic contraction and MKC.

Definition 4

Let (X,d) be a metric space. Then a mapping T on X is said to be an asymptotic contraction of Meir–Keeler type (ACMK, for short) if there exists a sequence {ϕn} of functions from [0,) into itself satisfying the following:

  • (A1)

    limsupnϕn(ε)ε for all ε0.

  • (A2)

    For each ε>0, there exist δ>0 and νN such that ϕν(t)ε for all t[ε,ε+δ].

  • (A3)

    d(Tnx,Tny)<ϕn(d(x,y)) for all nN and x,yX with xy.

We

Fxed-point theorems

In this section, we prove a fixed-point theorem which is a generalization of both Theorems 1 and 2.

Theorem 3

Let (X,d) be a complete metric space. Let T be an ACMK on X. Assume that T is continuous for some N. Then there exists a unique fixed point zX. Moreover, limnTnx=z for all xX.

Proof

Let {ϕn} be as in Definition 4. We note d(Tnx,Tny)ϕn(d(x,y))for all x,yX and nN. We first showlimnd(Tnx,Tny)=0for allx,yX.Fix x,yX. Let T0 be the identity mapping on X. In the case of Tx=Ty for some N{0}, (2)

Acknowledgements

The author wishes to express his gratitude to Professor W. A. Kirk for giving the historical comment.

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The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

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