On an operator equation involving mappings of monotone type

Abstract:

Let $X$ be a real reflexive Banach space and $A:X \to {2^{{X^{\ast }}}}$ a maximal monotone mapping such that the graph $G(A)$ of $A$ is weakly-closed in $X \times {X^{\ast }}$ and $0\epsilon A(0)$. Also, let $T:X \to {2^{{X^{\ast }}}}$ be a quasi-bounded coercive mapping of type $({\text {M}})$ such that the effective domain $D(T)$ of $T$ contains a dense linear subspace ${X_0}$ of $X$. Then it is shown that for each $\omega \epsilon {X^{\ast }}$ there exists a $u\epsilon X$ such that $\omega \epsilon Au + Tu$ and the subset $\{ u\epsilon X|\omega \epsilon Au + Tu\}$ is a weakly-compact subset of $X$. An application to an elliptic nonlinear boundary value problem of Neumann type is given.