Simultaneous extragradient iterative method to a split equality variational inequality problem and a multiple-sets split equality fixed point problem for multi-valued demicontractive mappings

K R Kazmi ,Mohd Furkan ,R Ali

doi:10.28919/afpt/4028

Abstract

This paper deals with a strong convergence theorem for a simultaneous extragradient iterative method to approximate a common solution to a split equality variational inequality problem and a multiple-sets split equality fixed point problem for two countable families of multi-valued demicontractive mappings in real Hilbert spaces. Further, we give a numerical example to justify the main result. The method and results presented in this paper extend and unify some recent known results in the literature.

Highlights

Let H1, H2 and H3 be real Hilbert spaces, let C ⊆ H1 and Q ⊆ H2 be nonempty, closed and convex sets
The SpFP(1) in finite dimensional Hilbert spaces was introduced by Censor and Elfving [5] for modeling inverse problem which arises from retrievals and in medical image reconstruction [4]
Censor et al [7] proposed the following multiple-sets split feasibility problem, which arises in applications such as intensity modulated radiation therapy [20]: N

Summary

INTRODUCTION

Let H1, H2 and H3 be real Hilbert spaces, let C ⊆ H1 and Q ⊆ H2 be nonempty, closed and convex sets. In 2014, Wu et al [23] introduced and studied the following multiple-sets split equality problem for finite families of multi-valued quasi-nonexpansive mappings:. Find x∗ ∈ Fix(Ti) and y∗ ∈ Fix(Si) such that Ax∗ = By∗, i=1 i=1 where N is a positive integer, and {Ti}Ni=1 : H1 ⇒ CB(H1), {Si}Ni=1 : H2 ⇒ CB(H2) are families of multi-valued quasi-nonexpansive mappings. Chidume [11] introduced and studied the following multiple-sets split equality fixed point problem (in short, MSSpEFPP) for countable families of multi-valued demicontractive mappings:. Motivated by the ongoing research work in this direction, we propose and analyze a simultaneous extragradient iterative method to approximate a common solution to SpEVIP(9)-(10) and MSSpEFPP(8) for countable families of multi-valued demicontractive mappings in real Hilbert spaces. The method and results presented in this paper extend and unify some recent known results in the literature; see for instance [6, 9, 11, 17, 24]

PRELIMINARIES

SIMULTANEOUS EXTRAGRADIENT ITERATIVE ALGORITHMS

MAIN RESULTS

NUMERICAL EXAMPLE