Mapping theorems involving compact perturbations and compact resolvents of nonlinear operators in Banach spaces

Abstract

where it is assumed that the “resolvent” L, = (T + (l/n)l)-i, II = 1, 2,... exists and is defined at least on the range of the operator f C. Such a class of operators T includes all m-accretive operators (cf. Kato ] 5 I). The equations (G,) can be solved by a degree theory argument if we assume, among other things, that L, is compact and C is continuous and bounded, or that L, is continuous and C compact. In most of our results we do not explicitly assume the strong continuity of the operator T. Due to thi-s fact, our theorems complement and extend various results of Browder [ 2 ], Petryshyn (6-81, Petryshyn and Tucker [9] as well as Kartsatos [4] and Ward [9]. 0 ur results are particularly related to Theorem 2 of Petryshyn [7], where operators T are studied with D(T) =X and range a Banach space Y. However, these spaces are assumed to possess