New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces

Abstract:

Let $X$ be a real Banach space and $G$ a bounded, open and convex subset of $X.$ The solvability of the fixed point problem $(*)~Tx+Cx \owns x$ in $D(T)\cap \overline {G}$ is considered, where $T:X\supset D(T)\to 2^{X}$ is a possibly discontinuous $m$-dissipative operator and $C: \overline {G}\to X$ is completely continuous. It is assumed that $X$ is uniformly convex, $D(T)\cap G \not = \emptyset$ and $(T+C)(D(T)\cap \partial G)\subset \overline {G}.$ A result of Browder, concerning single-valued operators $T$ that are either uniformly continuous or continuous with $X^{*}$ uniformly convex, is extended to the present case. Browder’s method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let $\Gamma = \{\beta :\mathcal {R}_{+}\to \mathcal {R}_{+}~;~\beta (r)\to 0\text { as }r\to \infty \}.$ The effect of a weak boundary condition of the type $\langle u+Cx,x\rangle \ge -\beta (\|x\|)\|x\|^{2}$ on the range of operators $T+C$ is studied for $m$-accretive and maximal monotone operators $T.$ Here, $\beta \in \Gamma ,~x\in D(T)$ with sufficiently large norm and $u\in Tx.$ Various new eigenvalue results are given involving the solvability of $Tx+ \lambda Cx\owns 0$ with respect to $(\lambda ,x)\in (0,\infty )\times D(T).$ Several results do not require the continuity of the operator $C.$ Four open problems are also given, the solution of which would improve upon certain results of the paper.