On the perturbation theory of $m$-accretive operators in Banach spaces

Abstract:

Let $X$ be a real Banach space. Let $T:X\supset D(T)\to 2^{X}$ be $m$-accretive with $(T+I)^{-1}$ compact. Let $C:X\supset D(T)\to X$ be such that $C(I+\lambda T)^{-1}:X\to X$ is condensing for some $\lambda \in (0,1).$ Let $p\in X$ and assume that there exists a bounded open set $G\subset X$ and $z\in D(T)\cap G$ such that $C(D(T)\cap \overline {G})$ is bounded and \begin{equation*}\langle u+Cx-p,j\rangle \ge 0,\tag *{(*)}\end{equation*} for all $x\in D(T)\cap \partial G,~u\in Tx,~j\in J(x-z).$ Then $p\in (T+C)(D(T)\cap \overline {G}).$ A basic homotopy result of the degree theory for $I-A,$ with $A$ condensing and $D(A)$ possibly unbounded, is used to improve and/or extend recent results by Hirano and Kalinde.