Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces
Introduction
Throughout this paper we denote by 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 be a complete metric space and let T be a mapping on X. Assume that for every , there exists such that for . 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 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 from into itself and a sequence of functions from into itself such that , for , converges to uniformly on the range of d, and for and ,
Theorem 2 Kirk [5]
Let be a complete metric space and let T be a continuous, asymptotic contraction on X with and in Definition 1. Assume that there exists such that the orbit of x is bounded, and that is continuous for . Then there exists a unique fixed point . Moreover, for all .
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 is upper semicontinuous with 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 be a metric space. Then a mapping T on X is said to be a Meir–Keeler contraction (MKC, for short) if for any , there exists such that for all .
In [6], Lim introduced the notion of an L-function and characterized MKC. Definition 3 A function from into itself is called an L-function if , for , and for every there exists such that for all .Lim [6]
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 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 of functions from into itself satisfying the following: for all . For each , there exist and such that for all . for all and with .
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 be a complete metric space. Let T be an ACMK on X. Assume that is continuous for some . Then there exists a unique fixed point . Moreover, for all . Proof Let be as in Definition 4. We note for all and . We first showFix . Let be the identity mapping on X. In the case of for some , (2)
Acknowledgements
The author wishes to express his gratitude to Professor W. A. Kirk for giving the historical comment.
References (9)
On a fixed point theorem of Kirk
J. Math. Anal. Appl.
(2005)- et al.
On Kirk's asymptotic contractions
J. Math. Anal. Appl.
(2004) Fixed points of asymptotic contractions
J. Math. Anal. Appl.
(2003)On characterizations of Meir–Keeler contractive maps
Nonlinear Anal.
(2001)
Cited by (0)
- 1
The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.