On a class of variational-hemivariational inequalities involving set valued mappings

Abstract

Using the KKM technique, we establish some existence results for variational-hemivariational inequalities involving monotone set valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive several sufficient conditions for the existence of solutions in the case of unbounded subsets.

Rothe method for parabolic variational–hemivariational inequalities

“…We pass to the limit in (8). Since ιu k τε is bounded in U , then by H(J)(ii) we have that ξ k τε is bounded in U * .…”

“…We prove the convergence of the semidiscrete scheme and at the same time the existence of solutions for a new class of problems. Indeed, although there is a rather considerable literature on stationary variational-hemivariational inequalities (see for example [3,8,13,16,24]), there are only a few papers on evolution ones (see [4][5][6]) and in these papers a method of sub-and supersolutions is used, i.e., there is a need to assume the existence of sub-and supersolutions to obtain the existence of solution. Here we provide an alternative proof of existence that does not need this hypothesis.…”

On a Class of Variational-Hemivariational Inequalities Involving Upper Semicontinuous Set-Valued Mappings

“…Goeleven et al [26] and Liu [27] proved the existence of solutions using the method of the first eigenfunction. For more related works regarding the existence of solutions for hemivariational inequalities, we refer to [1,3,6,[14][15][16][28][29][30] and the references therein.…”

“…which is studied by some researchers (see, for example, [1,2]). Problem (P) includes some models as special cases.…”

Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces

“…In 2010, applying KKM theorem, Costea and Lupu [13] extended the result above from the case of single-valued to that of set-valued. If F : K → 2 X * is a set-valued mapping, then problem (1) reduces to the following variational-hemivariational inequality problem:…”