Variational inequalities with the duality operator in Banach spaces

Abstract

We study variational inequality by way of metric projection in Banach spaces. The main method is to use a topological degree theory for the class of operators of monotone type in Banach spaces. More precisely, some variational inequality associated with the duality operator is considered. As applications, the problem is discussed in the Lebesgue spaces $L^p$ and the Sobolev spaces $W^{1,2}$.