Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

Lu-Chuan Ceng,Tu-Yan Zhao,Dan-Qiong Wang,Yun-Ling Cui,Long He,Hui-Ying Hu

doi:10.1186/s13660-021-02683-y

Abstract

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

Highlights

Throughout this paper, suppose that C is a nonempty closed convex subset of a real Hilbert space (H, · ) with the inner product ·, ·
(i) The problem of finding a solution of general system of variational inequalities (GSVI) (1.2) with the common fixed point problem (CFPP) constraint of a countable family of -uniformly Lipschitzian pseudocontractions and an asymptotically nonexpansive mapping in [3] is extended to develop our problem of finding a solution of the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP
(ii) The problem of finding a solution of the equilibrium problem with the VIP and CFPP constraints in [16] is extended to develop our problem of finding a solution of the MBEP with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP

Summary

Introduction

Throughout this paper, suppose that C is a nonempty closed convex subset of a real Hilbert space (H, · ) with the inner product ·, ·. In 2009, by using the viscosity approximation method, Chang et al [16] introduced an iterative algorithm for finding an element in the common solution set of the common fixed point problem (CFPP) of a countable family of nonexpansive self-mappings {Tk}∞ k=1 on C, the VIP for an α-inverse-strongly monotone mapping A, and the EP(C, ) for bifunction on C, that is, for any initial x1 ∈ H, the sequence {xk} is generated by. Let denote the common solution set of the fixed point problem (FPP) of asymptotically nonexpansive mapping T : C → C with {θk} and GSVI (1.2) for two inverse-strongly monotone mappings B1, B2. Some strong convergence results for the proposed algorithms are established under the suitable assumptions, and applied for finding a common solution of the GSVI, VIP, and FPP, where VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively. Our results improve and extend some corresponding results in the earlier and very recent literature; see, e.g., [3, 16, 22, 24]

Preliminaries

Main results

Conclusions