Topological degree theories and nonlinear operator equations in Banach spaces

Let X be a real Banach space and G(1), G(2) two nonempty, open and bounded subsets of X such that 0 is an element of G(2) and (G) over bar (2)) subset of G(1). The problem (*) Tx + Cx = 0 is considered, where T : X superset of D(T ) -> X (or X*) is an accretive (or monotone) operator with 0 is an element of D(T) and T(0) 0, while C : X superset of D(C) -> X (or X*) can be, e.g. one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type (S+) with T positively homogeneous of degree one; (d) quasi-bounded and satisfying a generalized (S+)-condition w.r.t. the operator T, while T is positively homogeneous of degree one. Solutions are sought for the problem (*) lying in the set D(T + C) boolean AND (G(1) \ G(2)). These solutions are nontrivial even when C(0) = 0. The degree theories of Leray and Schauder, Browder, and Skrypnik are used, as well as the degree theory by Kartsatos and Skrypnik for densely defined operators T, C. The last three degree theories do not assume any compactness conditions. The excision and additivity properties of these degree theories are employed, and the main results are significant extensions or generalizations of previous results by Krasnoselskii, Guo, Ding and Kansatos, and other authors, involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator T. An application in the field of partial differential equations is also included. (c) 2007 Elsevier Ltd. All rights reserved.