Existence of monotone solutions for semilinear differential inclusions (Q1288185)

This paper deals with the problem \[ \begin{gathered} \frac{du}{dt} (t) \in Au(t)+F(u(t)),\quad t\geq 0,\tag{1}\\ u(0)=\xi,\tag{2} \end{gathered} \] where \(A\: D(A)\subset X\to X\). The main result of the paper is as follows: Let \(X\) be a reflexive and separable Banach space, \(D\) a nonempty, locally weakly closed set in \(X\) and ``\(\preceq\)'' a preorder on \(D\) characterized by the set-valued mapping \(P\: D\to 2^D\), \(P(\xi)=\{\eta\in D\); \(\xi\preceq\eta\}\) whose graph is weakly-weakly sequentially closed in \(D\times D\). Let \(A: D(A)\subset X\to X\) be the infinitesimal generator of a \(C_0\)-semigroup \(S(t): X\to X\), \(t\geq 0\) and \(F\: D\to 2^X\) a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper-semicontinuous. Then a necessary and sufficient condition in order that \(P\) be admissible with respect to (1)--(2) is the bounded \(w\)-monotonicity condition.