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

Authors:	Athanassios G Kartsatos

Institution:	Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Abstract:	Let be a real Banach space and a bounded, open and convex subset of 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 -dissipative operator and $C: \overline{G}\to X$ is completely continuous. It is assumed that 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 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 is studied for -accretive and maximal monotone operators 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 Four open problems are also given, the solution of which would improve upon certain results of the paper.

Keywords:	$m$-accretive operator maximal monotone operator compact perturbation compact resolvent eigenvalues for nonlinear operators fixed point theory degree theory

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