Variational Inequality in Hilbert Spaces and their Applications

Abstract

The study of variational inequalities frequently deals with a mapping F from a vector 0 space X or a convex subset of X into its dual X . Let H be a real Hilbert space and a(u, v) be a real bilinear form on H. Assume that the linear and continuous 0 mapping A : H −→ H determines a bilinear form via the pairing a(u, v) = hAu, vi.0 Given K ⊂ H and f ∈ H . Then, Variational inequality(VI) is the problem of finding u ∈ K such that a(u, v − u) ≥ hf, v − ui, for all v ∈ K. In this work, we outline some results in theory of variational inequalities. Their relationships with other problems of Nonlinear Analysis and some applications are also discussed.