Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

Haiyan Gao,Lili Wang,Liangshi Zhao

doi:10.1186/s13660-020-02490-x

Abstract

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.

Highlights

Variational inequality is a powerful tool for studying problems arising in optimization, economics, differential equations, engineering and structural analysis, etc
In 2012, Yao–Postolache [17] introduced an iterative scheme for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings, and showed a few necessary and sufficient
The Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem are applied to prove the existence of a solution for the generalized nonlinear variational-like inequality

Summary

Introduction

Variational inequality is a powerful tool for studying problems arising in optimization, economics, differential equations, engineering and structural analysis, etc. In 1994, Yao [16] obtained the existence of solutions for generalized variational inequalities in Banach spaces. Liu–Ume–Kang [13] and Zeng [24] established some existence and uniqueness theorems of solutions for generalized nonlinear variational-like inequalities in reflexive Banach spaces by applying the minimax inequality due to Ding–Tan [9].

Results

Conclusion