Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term

Abstract

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi-hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder's fixed point theorem and a version of the KKM principle due to Ky Fan (Math Ann 266: [519][520][521][522][523][524][525][526][527][528][529][530][531][532][533][534][535][536][537] 1984).Keywords Quasi-hemivariational inequality · Set-valued operator · Lower semicontinuous set-valued operator · Clarke's generalized gradient · Generalized monotonicity · KKM mapping Mathematics Subject Classification (2000) 47J20 · 47H04 · 49J53 · 54C60 · 47H05 Introduction and preliminariesThe study of inequality problems captured special attention in the last decades, one of the most recent and general type of inequalities being the hemivariational inequalities. The notion of hemivariational inequality was introduced by P.D. Panagiotopoulos at the beginning of the 1980s (see e.g. [27,28]) as a variational formulation for several classes of mechanical problems with nonsmooth and nonconvex energy super-potentials. In the case of convex super-potentials, hemivariational inequalities reduce to variational inequalities which were studied earlier by many authors (see e.g. Fichera [13] or Hartman and Stampacchia [18]).

A system of evolutionary problems driven by a system of hemivariational inequalities

“…Theorem 3.1 extends the results in Costea and Radulescu [5], Liu and Zeng [18], and Liu et al [19]. Many important situations are incorporated in Theorem 3.1.…”

The Ekeland variational principle for equilibrium problems revisited and applications

“…In this direction, we consider quasi-hemivariational inequalities introduced in [13]. Quasi-hemivariational inequalities are generalization of multivalued variational inequalities and several connections with equilibrium problems are obtained, see also [14][15][16]10] for recent and old investigations on the subject. Our approach is based on a result of [15] stating that a point is a solution of a quasi-hemivariational inequality if and only if it is a fixed point of a suitable multivalued mapping.…”

Existence Results and Optimal Control for a Class of Quasi Mixed Equilibrium Problems Involving the (f, g, h)-Quasimonotonicity

“…Proof The idea of the proof comes from Costea and Radulescu [14]. Arguing by contradiction, let us assume that problem (3.1) has no solution.…”

“…[14] studied the following quasi-hemivariational inequality problem: find x ∈ K and x * ∈ F(x) such that…”