Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces

Lateef Olakunle Jolaoso,Yekini Shehu,Yeol Je Cho

doi:10.1186/s13660-021-02570-6

Abstract

In this paper, using a Bregman distance technique, we introduce a new single projection process for approximating a common element in the set of solutions of variational inequalities involving a pseudo-monotone operator and the set of common fixed points of a finite family of Bregman quasi-nonexpansive mappings in a real reflexive Banach space. The stepsize of our algorithm is determined by a self-adaptive method, and we prove a strong convergence result under certain mild conditions. We further give some applications of our result to a generalized Nash equilibrium problem and bandwidth allocation problems. We also provide some numerical experiments to illustrate the performance of our proposed algorithm using various convex functions and compare this algorithm with other algorithms in the literature.

Highlights

The theory of variational inequalities has been of great interest due to its wide applications in several branches of pure and applied sciences
We study the common solution of variational inequalities and fixed point problems in a real reflexive Banach space
We prove a strong convergence theorem for finding a common solution in the solution set of variational inequalities with pseudo-monotone and Lipschitz continuous operator, and the set of fixed points for a finite family of Bregman quasi-nonexpansive mappings in a reflexive Banach space

Summary

Introduction

The theory of variational inequalities has been of great interest due to its wide applications in several branches of pure and applied sciences. Thong and Vuong [63] introduced a modified Tseng extragradient method in which the operator is pseudo-monotone and there is no requirement for a prior estimate of the Lipschitz constant of the cost operator The stepsize of their algorithm is determined by a line search process, and they proved weak and strong convergence results for the variational inequalities in real Hilbert spaces. Cai et al [8] introduced a double projection algorithm for solving monotone variational inequalities in 2-uniformly convex real Banach spaces This method requires finding a prior estimate of the Lipschitz constant of the cost operator before its convergence is guaranteed.

Preliminaries

Applications

Conclusion