On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities

Lu-Chuan Ceng,Jen-Chih Yao,Adrian Petruşel

doi:10.3390/math7100925

Abstract

In a real Hilbert space, we denote CFPP and VIP as common fixed point problem of finitely many strict pseudocontractions and a variational inequality problem for Lipschitzian, pseudomonotone operator, respectively. This paper is devoted to explore how to find a common solution of the CFPP and VIP. To this end, we propose Mann viscosity algorithms with line-search process by virtue of subgradient extragradient techniques. The designed algorithms fully assimilate Mann approximation approach, viscosity iteration algorithm and inertial subgradient extragradient technique with line-search process. Under suitable assumptions, it is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the CFPP and VIP, which is the unique solution to a hierarchical variational inequality (HVI).

Highlights

Introduction and PreliminariesThroughout this article, we suppose that the real vector space H is a Hilbert one and the nonempty subset C of H is a convex and closed one
It is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the common fixed point problem (CFPP) and variational inequality problem (VIP), which is the unique solution to a hierarchical variational inequality (HVI)
We first provide an example of Lipschitzian, pseudomonotone self-mapping A satisfying the boundedness of A(C ) and strictly pseudocontractive self-mapping T1 with Ω = Fix( T1 ) ∩ VI(C, A) 6=

Summary

Introduction

Introduction and PreliminariesThroughout this article, we suppose that the real vector space H is a Hilbert one and the nonempty subset C of H is a convex and closed one. They proved that the sequences constructed by the suggested algorithms converge weakly to a point of Fix(S) ∩ VI(C, A). We always suppose that the following hypotheses hold: Tk is a ζ k -strictly pseudocontractive self-mapping on H for k = 1, ..., N s.t. ζ ∈ [0, 1) with ζ = max{ζ k : 1 ≤ k ≤ N }.

Results

Conclusion