Best proximity points: Convergence and existence theorems for p-cyclic mappings

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Abstract

We introduce a new class of mappings, called p-cyclic φ-contractions, which contains thep-cyclic contraction mappings as a subclass. Then, convergence and existence results of best proximity points for p-cyclic φ-contraction mappings are obtained. Moreover, we prove results of the existence of best proximity points in a reflexive Banach space. These results are generalizations of the results of Al-Thagafi and Shahzad (2009) [8].

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 A1,,Ap be nonempty closed subsets of a complete metric space (X,d ). Suppose that T:i=1pAii=1pAi satisfies the following conditions:

  • (i)

    T(Ai)Ai+1 for i=1,,p, where Ap+1=A1 ;

  • (ii)

    d(Tx,Ty)kd(x,y) for some k]0,1[ and

Results for p-cyclic φ-contractions

In this section, we give some basic definitions and concepts related to the main results of this paper.

A Banach space X is said to be

  • (a)

    uniformly convex if there exists a strictly increasing function δ:[0,2][0,1] such that the following implication holds for all x,y,pX,R>0 and r[0,2R]: xpR,ypR,xyrx+y2p(1δ(rR))R;

  • (b)

    strictly convex if the following implication holds for all x,y,pX and R>0: xpR,ypR,xyx+y2p<R.

Definition 1

Let A1,,Ap be nonempty subsets of a metric space (X,d).

p-cyclic φ-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 A1,,Ap be nonempty subsets of a metric space (X,d) and let T:i=1pAii=1pAi be a p-cyclic φ-contraction mapping. For every x0Aithe sequences {Tpnx0} and {Tpn+1x0} are bounded.

Proof

Suppose x0Ai. Since, by Corollary 1d(Tpnx0,Tpn+1x0)d(A1,A2), the sequences {Tpnx0} and {Tpn+1x0} are both bounded or both unbounded. Suppose that the two sequences are unbounded. Fix n1N and define en,k=

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%).

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