Topological Methods in Nonlinear Analysis

A set-valued approach to hemivariational inequalities

Alexandru Kristály and Csaba Varga

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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:

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.

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Topol. Methods Nonlinear Anal., Volume 24, Number 2 (2004), 297-307.

First available in Project Euclid: 23 June 2016

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Kristály, Alexandru; Varga, Csaba. A set-valued approach to hemivariational inequalities. Topol. Methods Nonlinear Anal. 24 (2004), no. 2, 297--307.

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