Existence of monotone solutions for semilinear differential inclusions

Abstract

Let X be a reflexive and separable Banach space, \(A:D(A)\subset X\to X\) the generator of a C 0-semigroup \(S(t):X\to X, t\ge 0, D\) a locally weakly closed set in \(X,F:D\to 2^X\) a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper semicontinuous. Let \(\preceq\) be a preorder on D, characterized by the set-valued map \(P:D\to 2^D\), defined by \(P(\xi )=\lbrace \eta\in D;\ \ \xi\preceq\eta \rbrace \) whose graph is weakly\(\times\)weakly sequentially closed. The main result of the paper is:¶¶Theorem. Under the general assumptions above a necessary and a sufficient condition in order that for each \( \xi \in D \) there exists at least one mild solution u of¶¶\( {du\over dt}(t) \in Au(t) + F(u(t))\ \ t\ge 0\)¶¶satisfying \(u(0)=\xi\) and \(u(s)\preceq u(t)\) for each s≤ t is the so called “bounded w-monotonicity condition” below.¶\((Bw\cal MC)\) There exists a locally bounded function \({\cal M} : D \to R_+^* \) such that for each \( \xi \in D \) there exists \(y \in F(\xi)\) such that for each \( \delta > 0\) and each weak neighborhood V of 0, there exist \(t\in (\,0,\delta \,]\) and \(p \in V\) with \(\Vert p\Vert\le {\cal M}(\xi )\) and satisfying \(S(t)\xi +t(y+p)\in P(\xi )\).