Abstract

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

Highlights

  • 1 Introduction Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points

  • Important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets Ai ⊂ X of a metric space (X, d), or a Banach space (X, )

  • It has to be noticed that every nonexpansive mapping [, ] is a -strict pseudocontraction and that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [ ]

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Summary

Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. If H is a real Hilbert space with an inner product ·, · and a norm · and A is a nonempty closed convex subset of H, T : A → A is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some β ∈ [ , ) if lim sup sup Tnx – Tny – αn x – y – β I – Tn x – I – Tn y ≤. T : A → A is asymptotically β-strictly pseudocontractive in the intermediate sense if lim sup – βn( + μn) d Tnx, Tny – (αn + βn)d (x, y) ≤ ; ∀x, y ∈ A for βn = β ∈ [ , ); ∀n ∈ N and some real sequences {αn}, {μn} being, in general, dependent on the initial points, i.e., αn = αn(x, y), μn = μn(x, y) and. →A is an asymptotically strict contraction as expected since ξn → as n → ∞; ∀x, y ∈ A from ( . )

Note also that if μ
If hold with
Taking x
If the conditions of Property are modified as αn
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