A Study on Mixed Variational-like Inequality Under Generalized Quasimonotonicity in Banach Spaces

Abstract

This paper studies nonlinear mixed variational-like inequalities under generalized quasimonotone mappings in Banach spaces. The main objective of this work is to relax the well-known KKM (Knaster–Kuratowski–Mazurkiewicz) condition which is extensively used in the literature to prove the existence of solutions for variational inequalities and equilibrium problems, and to establish the existence of solutions for the nonlinear mixed variational-like inequalities, which uncovers another approach of solving variational inequalities. Further, we propose an iterative algorithm to find approximate solutions to our problem and to study its convergence criteria. Finally, as an application, we find a gap function for nonlinear mixed variational-like inequalities, which uncovers another way of solving our variational-like inequalities using the methods of solutions and algorithms of the optimization problems.

Keywords

• variational-like inequalities
• auxiliary principle
• iterative scheme
• 0−diagonally concave
• gap function