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

Abstract

Let $X$ be a real Banach space with its dual $X^{\ast }$ 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 $Tx+Cx+\epsilon x\not\ni f$ for all $x\in D(T)\cap \partial G$ and $\epsilon >0.$ 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.

arXiv commentThe paper needs a significant adjustment because of similar existing results