Ranges of perturbed maximal monotone and 𝑚-accretive operators in Banach spaces

Guan, Zhengyuan; Kartsatos, Athanassios G.

doi:10.1090/S0002-9947-1995-1297527-2

Ranges of perturbed maximal monotone and $m$-accretive operators in Banach spaces
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Trans. Amer. Math. Soc. 347 (1995), 2403-2435 Request permission

Abstract:

A more comprehensive and unified theory is developed for the solvability of the inclusions $S \subset \overline {R(A + B)}$, int $S \subset R(A + B)$, where $A:X \supset D(A) \to {2^Y}$, $B:X \supset D(B) \to Y$ and $S \subset X$. Here, $X$ is a real Banach space and $Y = X$ or $Y = {X^*}$. Mainly, $A$ is either maximal monotone or maccretive, and $B$ is either pseudo-monotone or compact. Cases are also considered where $A$ has compact resolvents and $B$ is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators $A$ and $B$. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbation of maximal monotone operators, and the Leray-Schauder degree theory.

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