Nonlinear Split Ordered Variational Inequality Problems

TL;DR

This paper extends the theory of variational inequality problems to nonlinear split ordered contexts on partially ordered vector spaces, providing solvability results without requiring continuity of the involved mappings.

Contribution

It introduces a new framework for nonlinear split ordered variational inequalities and proves their solvability using fixed point theorems under order-monotonic conditions.

Findings

Proves solvability of nonlinear split ordered variational inequalities.

Establishes fixed point theorems applicable to these problems.

Demonstrates applications to various split variational inequality problems.

Abstract

The concept of nonlinear split ordered variational inequality problems on partially ordered vector spaces is a natural extension of linear split vector variational inequality problems on Banach spaces. The results about nonlinear split ordered variational inequality problems are immediately applied to solving nonlinear split vector optimization problems. In this paper, we prove the solvability of some nonlinear split ordered variational inequality problems by using some fixed point theorems on partially ordered spaces, in which the considered mappings may not be required to have any type of continuity and they just satisfy some order-monotonic conditions. As applications of the results about nonlinear split ordered variational inequality problems, we study solvability of some nonlinear split variational inequality problems and linear split variational inequality problems on partially ordered vector spaces and partially ordered Banach spaces.