Solvability of Variational Inequalities on Hilbert Lattices

This paper provides a systematic solvability analysis for (generalized) variational inequalities on separable Hilbert lattices. By contrast to a large part of the existing literature, our approach is lattice-theoretic, and is not based on topological …xed point theory. This allows us to establish the solvability of certain types of (generalized) variational inequalities without requiring the involved (set-valued) maps be hemicontinuous or monotonic. Some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work, and appear to have scope for applications. This is illustrated by means of several applications to …xed point theory, optimization and game theory. Mathematics Subject Classi…cation (1991): 90C33, 54H25. Keywords: Variational inequalities, Hilbert lattices, …xed point theorems.