Solution sensitivity of a generalized variational inequality.
Let \(X\) denote a reflexive Banach space with the dual \(X^*\), \((Q,d)\) and \((M,d)\) be two metric spaces. Suppose that \(x_0\in X\), \(\xi\in Q\), and \(\mu\in M\) are some given points. Let \(F: X\times M\to 2^{X^*}\) and \(K:Q\to 2^X\) be two given set-valued mappings such that \(K(\cdot)\) has nonempty closed convex values. In this paper, the author considers the following parametric generalized variational inequality problem (PGVIP): find \(x=x(\mu,\xi)\) such that \[ 0\in F(x,\mu)+N_{K(\xi)}(x) \] where \((\mu,\xi)\in M\times Q\) is a pair of parameters. Under some suitable conditions, the author proves that PGVIP has a unique solution \(x=x(\mu,\xi)\). Moreover, if \(F\) is Lipschitz continuous and strongly monotone, then \(x=x(\mu,\xi)\) is a Hölder continuous function. Some results for the special cases of PGVIP and an application are also given in this paper.
scientific article; zbMATH DE number 1885694