Solvability of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators

Abstract

Let $X$ be a real reflexive Banach space and $X^{\ast }$ be its dual space. Let $G_1$ and $G_2$ be open subsets of $X$ such that ${\bar{G}}_2\subset G_1$, $0\in G_2$, and $G_1$ is bounded. Let $L:X\supset D(L)\to X^{\ast }$ be a densely defined linear maximal monotone operator, $A:X\supset D(A)\to 2^{X^{\ast }}$ be a maximal monotone and positively homogeneous operator of degree $\gamma >0$, $C:X\supset D(C)\to X^{\ast }$ be a bounded demicontinuous operator of type $(S_{+})$ w.r.t. $D(L)$, and $T:{\bar{G}}_1\to 2^{X^{\ast }}$ be a compact and upper-semicontinuous operator whose values are closed and convex sets in $X^{\ast }$. We first take $L=0$ and establish the existence of nonzero solutions of $Ax+Cx+Tx\ni 0$ in the set $G_1\setminus G_2.$ Secondly, we assume that $A$ is bounded and establish the existence of nonzero solutions of $Lx+Ax+Cx\ni 0$ in $G_1\setminus G_2.$ We remove the restrictions $\gamma \in (0,1]$ for $Ax+Cx+Tx\ni 0$ and $\gamma =1$ for $Lx+Ax+Cx\ni 0$ from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions.