On the solution stability of variational inequalities

Abstract

In the present paper, we will study the solution stability of parametric variational conditions

$${{0 \in f(\mu, x)+ N_{K(\lambda)}(x)},}$$

where M and Λ are topological spaces, ${f : M \times R^n \to R^n}$ is a function, ${K : \Lambda\to 2^{R^n}}$ is a multifunction and N _K(λ)(x) is the value at x of the normal cone operator associated with the set K(λ). By using the degree theory and the natural map we show that under certain conditions, the solution map of the problem is lower semicontinuous with respect to parameters (μ,λ). Our results are different versions of Robinson’s results [15] and proved directly without the homeomorphic result between the solution sets.