Unique solvability of weakly homogeneous generalized variational inequalities

TL;DR

This paper investigates conditions under which generalized variational inequalities with weakly homogeneous mappings have unique solutions, introducing weaker assumptions than strong monotonicity and deriving new theoretical results.

Contribution

It provides novel sufficient conditions for unique solvability of generalized variational inequalities involving weakly homogeneous mappings, expanding existing theoretical frameworks.

Findings

Established new conditions for unique solvability.

Derived results using exceptional family of elements.

Extended theories to important subclasses.

Abstract

An interesting observation is that most pairs of weakly homogeneous mappings have no strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper focuses on studying the unique solvability of the generalized variational inequality with a pair of weakly homogeneous mappings. By using a weaker condition than the strong monotonicity and some additional conditions, we achieve several results on the unique solvability of the underlying problem. These results are exported by making use of the exceptional family of elements or derived from new obtained Karamardian-type theorems or established under the exceptional regularity condition. They are new even when the problem comes down to its important subclasses studied in recent years.