Solvability of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators

TL;DR

This paper investigates the solvability of inclusion problems involving perturbations of positively homogeneous maximal monotone operators in Banach spaces, extending existing results to broader conditions and applying findings to PDEs.

Contribution

It generalizes solvability conditions for inclusions with positively homogeneous maximal monotone operators, removing previous restrictions on the degree of homogeneity, and applies results to PDEs.

Findings

Established existence of solutions for specific inclusion problems in Banach spaces.

Extended solvability results to cases with broader homogeneity degrees.

Applied theoretical results to elliptic and parabolic PDEs with Dirichlet boundary conditions.

Abstract

Let $X$ be a real reflexive Banach space and $X^*$ 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 $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, $A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone and positively homogeneous operator of degree $\gamma>0$, $C:X\supset D(C)\to X^*$ be a bounded demicontinuous operator of type $(S_+)$ w.r.t. $D(L)$, and $T:\bar G_1\to 2^{X^*}$ be a compact and upper-semicontinuous operator whose values are closed and convex sets in $X^*$. 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.