Existence Results for Multivalued Compact Perturbations of ${m}$-Accretive Operators

TL;DR

This paper establishes existence results for solutions to perturbed m-accretive operator equations in Banach spaces, extending and improving previous results by various authors, including cases with multivalued and single-valued perturbations.

Contribution

It provides new existence theorems for multivalued and single-valued compact perturbations of m-accretive operators, generalizing prior work and including conditions for surjectivity and weak coercivity.

Findings

Existence of solutions under multivalued perturbations.

Surjectivity results for expansive and weakly coercive operators.

Generalizations of previous theorems by Kartsatos, Liu, and Morales.

Abstract

Let $X$ be a real Banach space with its dual $X^*$ and $G$ be a nonempty, bounded and open subset of $X$ with $0\in G$. Let $T: X\supset D(T)\to 2^{X}$ be an $m$-accretive operator with $0\in D(T)$ and $0\in T(0)$, and let $C$ be a compact operator from $X$ into $X$ with $D(T)\subset D(C)$. We prove that $f\in \overline{R(T)}+\overline{R(C)}$ if $C$ is multivalued and $f\in \overline{R(T+C)}$ if $C$ is single-valued, provided $Tx+Cx+\varepsilon x\not\ni f$ for all $x\in D(T)\cap \partial G$ and $\varepsilon >0.$ The surjectivity of $T+C$ is proved if $T$ is expansive and $T+C$ is weakly coercive. Analogous results are given if $T$ has compact resolvents and $C$ is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized.