Topological degree theories and nonlinear operator equations in Banach spaces Original Research Article

Let XX be a real Banach space and G1,G2G1,G2 two nonempty, open and bounded subsets of XX such that 0∈G20∈G2 and View the MathML sourceG2¯⊂G1. The problem View the MathML source(∗)Tx+Cx=0 is considered, where T:X⊃D(T)→XT:X⊃D(T)→X (or X∗X∗) is an accretive (or monotone) operator with 0∈D(T)0∈D(T) and T(0)=0T(0)=0, while C:X⊃D(C)→XC:X⊃D(C)→X (or X∗X∗) can be, e.g. one of the following types: (a) compact; (b) continuous and bounded with the resolvents of TT compact; (c) demicontinuous, bounded and of type (S+)(S+) with TT positively homogeneous of degree one; (d) quasi-bounded and satisfying a generalized (S+)(S+)-condition w.r.t. the operator TT, while TT is positively homogeneous of degree one. Solutions are sought for the problem (∗) lying in the set D(T+C)∩(G1∖G2)D(T+C)∩(G1∖G2). These solutions are nontrivial even when C(0)=0C(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,CT,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 Kartsatos, and other authors, involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator TT. An application in the field of partial differential equations is also included.