An existence result for weakly homogeneous variational inequalities

TL;DR

This paper establishes an existence theorem for weakly homogeneous variational inequalities in finite-dimensional Hilbert spaces, using conditions that are easier to verify and extend previous results in the field.

Contribution

It provides a new, more accessible existence result for weakly homogeneous variational inequalities, expanding the theoretical framework and connecting with earlier work by Gowda and Sossa.

Findings

Conditions for existence are easier to verify.

The result extends previous theorems to a broader class of problems.

Comparison with coercivity and norm-coercivity results shows broader applicability.

Abstract

In this paper, what we concern about is the weakly homogeneous variational inequality over a finite dimensional real Hilbert space. We achieve an existence result {under} copositivity of leading term of the involved map, norm-coercivity of the natural map and several additional conditions. These conditions we used are easier to check and cross each other with those utilized in the main result established by Gowda and Sossa (Math Program 177:149-171, 2019). As a corollary, we obtain a result on the solvability of nonlinear equations with weakly homogeneous maps involved. Our result enriches the theory for weakly homogeneous variational inequalities and its subcategory problems in the sense that the main result established by Gowda and Sossa covers a majority of existence results on the subcategory problems of weakly homogeneous variational inequalities. Besides, we compare our {existence} result with the well-known coercivity result obtained for general variational inequalities and a norm-coercivity result obtained for general complementarity problems, respectively.