On solvability of the variational inequality with \(+\)-coercive multivalued mapping (Q2735858)

Let \(X\) be a reflexive Banach space, \(X^*\) be its topological dual space, \(A:X\to 2^{X^*}\) be a multivalued mapping and \(K\) be a closed convex and bounded set in \(X\). One considers the variational inequality NEWLINE\[NEWLINE[A(y), \xi-y]_+\geq \langle f,\xi-y \rangle, \quad\forall \xi\in K,\tag{1}NEWLINE\]NEWLINE where NEWLINE\[NEWLINE[A(y),\xi]_+= \sup_{d\in A(y)} \langle d,y\rangle, \quad y,\xi\in XNEWLINE\]NEWLINE is the upper support function associated with \(A\). The main results of this paper concern the existence of the solution of (1) under some relaxed restrictions on properties of the operator \(A\). The obtained results are illustrated by the enclosed example for quasilinear elliptic operator with a Signorini-like boundary condition which contains some multivalued operator. The advances relative to precedings investigations are given.