Nonlinear Analysis: Theory, Methods & Applications
Best proximity points: Convergence and existence theorems for -cyclic mappings
Section snippets
Introduction and preliminaries
The Banach Contraction Principle is a fundamental result in fixed point theory. Thus, several extensions of this result have appeared in the literature; see, e.g., [1] and the references cited therein. One of the most interesting extensions was given by Kirk et al. [2] as follows. Theorem 1 Let be nonempty closed subsets of a complete metric space ( ). Suppose that satisfies the following conditions: for , where ; for some and
Results for -cyclic -contractions
In this section, we give some basic definitions and concepts related to the main results of this paper.
A Banach space is said to be
- (a)
uniformly convex if there exists a strictly increasing function such that the following implication holds for all and :
- (b)
strictly convex if the following implication holds for all and :
Definition 1
Let be nonempty subsets of a metric space .
-cyclic -contraction mappings on reflexive Banach spaces
The following proposition is a general case of Proposition 1. The proof is as the proof of Theorem 2.2 of [10]. Proposition 2 Let be nonempty subsets of a metric space and let be a -cyclic -contraction mapping. For every the sequences and are bounded.
Proof Suppose . Since, by Corollary 1, the sequences and are both bounded or both unbounded. Suppose that the two sequences are unbounded. Fix and define
Acknowledgements
This work is dedicated to Professor Pasquale Vetro for his 65th birthday. The author is thankful to the anonymous referees for their valuable comments which have improved the presentation of the paper. This work has been supported by University of Palermo (Local University Project ex 60%).
References (10)
- et al.
Existence and convergence of best proximity points
J. Math. Anal. Appl.
(2006) - et al.
Best proximity points for cyclic Meir–Keeler contractions
Nonlinear Anal.
(2008) - et al.
The existence of best proximity points in metric spaces with the property UC
Nonlinear Anal.
(2009) - et al.
Convergence and existence results for best proximity points
Nonlinear Anal.
(2009) Contraction mappings and extensions
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