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

Abstract

A more comprehensive and unified theory is developed for the solvability of the inclusions S ⊂ R ( A + B ) ¯ S \subset \overline {R(A + B)} , int S ⊂ R ( A + B ) S \subset R(A + B) , where A : X ⊃ D ( A ) → 2 Y A:X \supset D(A) \to {2^Y} , B : X ⊃ D ( B ) → Y B:X \supset D(B) \to Y and S ⊂ X S \subset X . Here, X X is a real Banach space and Y = X Y = X or Y = X ∗ Y = {X^*} . Mainly, A A is either maximal monotone or maccretive, and B B is either pseudo-monotone or compact. Cases are also considered where A A has compact resolvents and B B is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators A A and B 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.