Some properties of “well–posed” variational. inequalities governed by linear operators

Abstract

In a previous work [6], we gave the following definition of “well-posed” (in the sense of Tikhonov) variational inequalities (V.I.). Given a mono tone operator A:X→X′ on a Banach space X, a closed convex subset K of X and geX′, the V.I.: find XeK s.tA for every yeK , is “well-posed” if and diam(Te)→0, where In the present work we study the properties of “well-posed” V .I. in the special case when A is linear bounded operator defined on a Hilbert space X. In particular, we find that the “well-posedness” on every closed convex set is equivalent to the coercivity of A, which is in turn equivalent to the existence and uniqueness of the solution on every closed convex set of X. Moreover, we show that when K=X our definition is equivalent to the well-posedness in the classical) sense of Hadamard (for analogous results see [6],[7]). Eventualy we show that well-posedness guarantees existence and uniqueness in a more general case than shown in [6], and we study the relationship between the “well-posedness” of a V....