On an Operator Equation Involving Mappings of Monotone Type

Let X be a real reflexive Banach space and A: X — 2A a maximal monotone mapping such that the graph G(A) of A is weaklyclosed in X X X and 0 £/4(0). Also, let T: X — 2A be a quasi-bounded coercive mapping of type (M) such that the effective domain D(T) of T contains a dense linear subspace X of X. Then it is shown that for each * f I co e X there exists a u e X such that co e Au + Tu and the subset \u 6 X| co e Au + Tu\ is a weakly-compact subset of A". An application to an elliptic nonlinear boundary value problem of Neumann type is given. The aim of this paper is to prove a theorem on the existence of solutions for an equation of type Au + Tu = f, where A and T are nonlinear mappings with their domains in a real reflexive Banach space X and range in the dual Banach space X . We shall also give an application of this result to the problem of existence of solutions of a nonlinear elliptic boundary value problem of Neumann type in L (Q), where fi is a bounded domain in an Euclidean space R" with smooth boundary. We may mention that our results also apply to boundary value problems of variational type for quasilinear elliptic systems with strongly nonlinear lower order terms. These applications will appear in a subsequent paper elsewhere. We employ the following definitions: If X is a real reflexive Banach space and X is its dual Banach space, we denote by (co, x) the duality pairing between an element gj in X and x in X. For a multivalued mapping T: X —• 2 (the set of subsets of X ), we denote by D(T), the effective domain of T, the subset of X defined by D(T) = {x e X| V * Tx 4 0\. A mapping T: X — 2 is said to be monotone if its graph G(T) = \[x, o)]| x £ D(T), co £ Tx\ is a monotone subset of X x X in the sense that (cjj co2, *j xj > 0 for all [xj, coj £ G(T), [x 2, co J £ G(T). A monotone mapping T: X —> 2 is said to be maximal monotone if its graph G(T) is maximal among all the monotone subsets of X x X in the sense of inclusion. Definition 1. Let X be a reflexive Banach space and XQ a dense linear x * subspace of X. Let T: X —> 2 be a given mapping with effective domain D(T) such that X. C D(T). Then T is said to be of type (M) with respect to X if the following hold: Received by the editors July 24, 1974. AMS (MOS) subject classifications (1970). Primary 47H15, 47H05; Secondary 47F05, 35J40.