Let X be a reflexive and separable Banach space, \documentclass[12pt]{minimal}
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$A:D(A)\subset X\to X$\end{document} the generator of a C0-semigroup \documentclass[12pt]{minimal}
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$S(t):X\to X, t\ge 0, D$\end{document} a locally weakly closed set in \documentclass[12pt]{minimal}
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$X,F:D\to 2^X$\end{document} a nonempty, closed, convex and bounded valued mapping which is weakly-weakly upper semicontinuous. Let \documentclass[12pt]{minimal}
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``$\preceq$''\end{document} be a preorder on D, characterized by the set-valued map \documentclass[12pt]{minimal}
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$P:D\to 2^D$\end{document}, defined by \documentclass[12pt]{minimal}
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$P(\xi )=\lbrace \eta\in D;\ \ \xi\preceq\eta \rbrace $\end{document} whose graph is weakly\documentclass[12pt]{minimal}
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$\times$\end{document}weakly sequentially closed. The main result of the paper is:¶¶Theorem.Under the general assumptions above a necessary and a sufficient condition in order that for each\documentclass[12pt]{minimal}
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$ \xi \in D $\end{document}there exists at least one mild solution u of¶¶\documentclass[12pt]{minimal}
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$$ {du\over dt}(t) \in Au(t) + F(u(t))\ \ t\ge 0$$\end{document}¶¶satisfying\documentclass[12pt]{minimal}
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$u(0)=\xi$\end{document}and\documentclass[12pt]{minimal}
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$u(s)\preceq u(t)$ \end{document}for each s≤ t is the so called “bounded w-monotonicity condition” below.¶\documentclass[12pt]{minimal}
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$(Bw\cal MC)$\end{document}There exists a locally bounded function\documentclass[12pt]{minimal}
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${\cal M} : D \to R_+^* $\end{document}such that for each\documentclass[12pt]{minimal}
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$ \xi \in D $\end{document}there exists\documentclass[12pt]{minimal}
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$y \in F(\xi)$\end{document}such that for each\documentclass[12pt]{minimal}
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$ \delta > 0$\end{document}and each weak neighborhood V of 0, there exist\documentclass[12pt]{minimal}
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$t\in (\,0,\delta \,]$\end{document}and\documentclass[12pt]{minimal}
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$p \in V$\end{document}with\documentclass[12pt]{minimal}
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$\Vert p\Vert\le {\cal M}(\xi )$\end{document}and satisfying\documentclass[12pt]{minimal}
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$S(t)\xi +t(y+p)\in P(\xi )$\end{document}.
