Existence results for mixed hemivariational-like inequalities involving set-valued maps

Nicusor Costea, Ariana Pitea

Let $X$ and $Y$ be Banach spaces such that there exists a linear continuous operator $\gamma:X\rightarrow Y$ and let $K$ be a nonempty, closed and convex subset of $X$. We study inequality problems of the type

Find $u\in K$ such that:

\[ \exists u^\ast\in A(u):\ \langle u^\ast,\eta(v,u)\rangle+J^0(\gamma(u);\gamma(\eta(v,u)))\geq \psi(u,v), \forall v\in K, \]

where $A:X\rightarrow 2^{X^\ast}$ is a set-valued map, $J:Y\rightarrow \mathbb{R}$ is a locally Lipschitz functional, $\eta:X\times X\rightarrow X$ and $\psi:X\times X\rightarrow \mathbb{R}$ is a prescribed bifunction.

The presence of the set-valued map ensures that not one, but types of solution can be defined, while the presence of the bifunction does not allow the inequality to be equivalently written as an inclusion. We prove that the inequality possesses at least one solution (strong solution) provided the set-valued map is upper semicontinuous (lower semicontinuous, respectively).