Fixed point solutions for variational inequalities in image restoration over q-uniformly smooth Banach spaces

Pongsakorn Sunthrayuth,Poom Kumam

doi:10.1186/1029-242x-2014-473

Abstract

In this paper, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudo-contractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space which admits a weakly sequentially continuous generalized duality mapping. Then we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudo-contractions which is a solution of some variational inequalities. Furthermore, we apply our results to study some strong convergence theorems in and spaces with . Our results mainly improve and extend the results announced by Ceng et al. (Comput. Math. Appl. 61:2447-2455, 2011) and many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples for support our main theorem in the end of the paper. MSC:47H09, 47H10, 47H17, 47J25, 49J40.

Highlights

Let C, C, . . . , Cn be nonempty, closed, and convex subsets of a real Hilbert space H such that n i= Ci = ∅.The problem of image recovery in a Hilbert space setting by using convex of metric projections PCi, may be stated as follows: the original unknown image z is known a priori to belong to the intersection of {Ci}ni= ; given only the metric projectionsPCi of H onto Ci for i =, . . . , n recover z by an iterative scheme
The problems of image recovery have been studied in a Banach space setting by Kitahara and Takahashi [ ]
In this paper, motivated by the above facts, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudocontractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping

Summary

Introduction

Let C , C , . . . , Cn be nonempty, closed, and convex subsets of a real Hilbert space H such that n i=. In this paper, motivated by the above facts, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudocontractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping. ([ ]) Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. ([ ]) Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth and strictly convex Banach space X. ([ ]) Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping jq from X into X∗.

Implicit iteration scheme

Explicit iteration scheme

Some applications

Implicit iteration schemes

Methods