### A set-valued approach to hemivariational inequalities

DOI: http://dx.doi.org/10.12775/TMNA.2004.029

#### Abstract

Let $X$ be a Banach space, $X^*$ its dual

and let $T\colon X\to L^p(\Omega ,\mathbb {R}^k)$ be a linear, continuous

operator, where

$p, k\ge 1$, $\Omega $ being a bounded open set in

$\mathbb {R}^N$. Let $K$ be a subset of $X$, ${\mathcal

A}\colon K\rightsquigarrow X^*$, $G\colon K\times X\rightsquigarrow

\mathbb {R}$ and $F\colon \Omega \times \mathbb {R}^k\times

\mathbb {R}^k\rightsquigarrow \mathbb {R}$ set-valued maps with

nonempty values. Using mainly set-valued analysis, under suitable

conditions on the involved maps,

we shall guarantee solutions

to the following inclusion problem:

{\it Find $u\in K$ such that, for every } $v\in K$ $$\sigma

({\mathcal A}(u),v-u)+G(u,v-u)+ \int_\Omega

F(x,T{u}(x),T{v}(x)-T{u}(x))dx \subseteq \mathbb {R}_+.$$ In

particular, well-known variational and hemivariational

inequalities can be derived.

and let $T\colon X\to L^p(\Omega ,\mathbb {R}^k)$ be a linear, continuous

operator, where

$p, k\ge 1$, $\Omega $ being a bounded open set in

$\mathbb {R}^N$. Let $K$ be a subset of $X$, ${\mathcal

A}\colon K\rightsquigarrow X^*$, $G\colon K\times X\rightsquigarrow

\mathbb {R}$ and $F\colon \Omega \times \mathbb {R}^k\times

\mathbb {R}^k\rightsquigarrow \mathbb {R}$ set-valued maps with

nonempty values. Using mainly set-valued analysis, under suitable

conditions on the involved maps,

we shall guarantee solutions

to the following inclusion problem:

{\it Find $u\in K$ such that, for every } $v\in K$ $$\sigma

({\mathcal A}(u),v-u)+G(u,v-u)+ \int_\Omega

F(x,T{u}(x),T{v}(x)-T{u}(x))dx \subseteq \mathbb {R}_+.$$ In

particular, well-known variational and hemivariational

inequalities can be derived.

#### Keywords

Measurable set-valued maps; variational-hemivariational inequalities

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