On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

Lu-Chuan Ceng,Yekini Shehu,Jen-Chih Yao

doi:10.3390/math9243322

Abstract

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Highlights

Let h·, ·i be the inner product and k · k induced norm of a real Hilbert space H
Inspired by the above research works, we propose two Mann-type subgradientlike extragradient algorithms with linear-search process for solving a hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of family nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces
T : C → C is an asymptotically nonexpansive mapping and Ti : C → C is a nonexpansive mapping for i = 1, . . . , N such that the sequence { Tn }∞

Summary

Introduction

Let h·, ·i be the inner product and k · k induced norm of a real Hilbert space H. Given a convex closed set C ⊂ H with C 6= ∅. Let PC be the nearest point projection from H onto. C. Given T : C → H, we denote the set Fix( T ) = { x ∈ C : x = Tx } by Fix( T ) the fixed points set of T. We say that S : C → C is asymptotically nonexpansive if there exists a sequence {θn } ⊂ [0, +∞) with limn→∞ θn = 0 such that the following is the case. S is called nonexpansive if θn = 0 ∀n ≥ 1.

Methods

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Conclusion