Leray-Schauder type continuation theorems for some nonlinear equations in Banach spaces

Abstract

LET X BE a real reflexive Banach space and X* denote its dual Banach space. In this paper we are interested in obtaining Leray-Schauder and Borsuk-Ulam type Continuation theorems for a family of equations of the form w E Au + Tu + C,u where A : D(A) C X-+ 2x’ is a maximal monotone mapping with weakly-closed-graph in X x X*, T: D(T) C C-, 2x’ a maximal monotone mapping satisfying condition (S+) and {C,}, t E [0, 11, a continuous family of compact mappings from X into X*. It should be noted that if A : D(A) C X+ 2x’ is a linear maximal monotone mapping then its graph G(A) is a weakly-closed subset of X x X”. Our results of this paper generalize an earlier result of Gupta [ll] where a Leray-Schauder type continuation theorem is obtained for equations of the form w E Au + Tu + C,u where A:D(A)CX-+2X* is linear maximal monotone, T:X+ X* is a single-valued, everywhere defined bounded monotone mapping satisfying condition (S+) and {C,}, t E [0, 11, a continuous family of compact mappings from X into X’. So our theorems of this paper extend earlier results of [ll] in several ways e.g. we allow A to be nonlinear, effective domain D(T) of T may not be all of X, T can be multivalued and unbounded. We note that equations of the form w E Au + Tu arise in the study of general elliptic boundary value problems defined on a bounded domain $2 contained in an Euclidean space W”. For such elliptic problems the mapping A is defined by the top-order terms of the differential expression and the boundary conditions and the mapping T is defined by lower order terms. Accordingly our results are of the nature of stability results for compact perturbations of elliptic boundary value problems. Also A turns out to be a maximal monotone mapping when the top order terms of the differential equation are linear and satisfy Leray-Lions conditions and the boundary conditions are defined by some maximal monotone graph in R*. Such an A is nonlinear but usually has a weakly-closed graph. We should like to point out that continuation theorems of this type have been studied earlier by Browder [4] Gupta [12] and Brezis & Browder [2] for equations of Hammerstein type viz equations w = (I + KN + C,)(u) where K: X* + X is a continuous monotone linear mapping, N: X-, X’ is a bounded monotone mapping satisfying condition (S+) and {C,}, t E [0, 11, a continuous family of compact mappings from X into X. Now, equations of the form w E (I+KN+ CJ( u are prototypes of what are called integral equations of Hammerstein type, ) while equations of the form w E Au + Tu + C,u model boundary value problems given by partial differential operators.