Homotopy invariance of parameter-dependent domains and perturbation theory for maximal monotone and $m$-accretive operators in Banach spaces

Abstract

Let $X, Y$ be real Banach spaces and $B_r(0)$ the open ball in $Y$ with center at zero and radius $r> 0.$ Let $G$ be an open subset of $X.$ One of the problems that constantly arise in the study of perturbations of maximal monotone and $m$-accretive operators is the verification of the invariance of a degree function on parameter-dependent domains. For example, these domains can be of the type $A_tG, $ $t\in [0,1],$ where $A_t:X\to 2^Y$ has a continuous inverse $A_t^{-1}:Y\to X$ and $D(A_t)\cap G \neq \emptyset.$ Browder has shown that such degrees are invariant in various situations where the operator $A_t$ is a local homeomorphism, but no such result is known (or can be easily derived from known results) when $A_t$ is a discontinuous (possibly densely defined and/or multi-valued) mapping. The main purpose of this paper is to show that, under certain natural conditions, the Leray-Schauder degree $d(I-H(t),A_tG\cap B_m(0),0)$ is well defined and constant for all $t\in [0,1]$ and all sufficiently large $m,$ where $H(t):Y\to Y$ is an appropriate compact homotopy. This fundamental result can be used to improve various results in the perturbation theory of $m$-accretive and maximal monotone operators. In particular, results by Hirano and Kartsatos, involving saddle point conditions in real Hilbert spaces, are improved. Ideas from the main theory are also used to create new degree theories for perturbed $m$-accretive operators as well as demicontinuous accretive operators defined on closures of bounded open subsets of $X$ when $X^*$ is uniformly convex.