Quasi-autonomous Evolution Inclusions

Abstract

This chapter deals with a kind of semilinear differential inclusions in general Banach spaces . Firstly, we study different types of generalized solutions including limit and weak solutions . Under appropriate assumptions, we show that the set of the limit solutions is a compact $R_\delta$ -set. When the right-hand side satisfies the one-sided Perron condition , a variant of the well-known lemma of Filippov-Pliś , as well as a relaxation theorem, are proved. Secondly, we study a kind of semilinear evolution inclusions . If the nonlinearity is one-sided Perron with sublinear growth, then we establish the relation between the solutions of the considered differential inclusion and the solutions of the relaxed one. A variant of the well known Filippov-Pliś lemma is also proved. Finally, we analyze the existence of pullback attractor for non-autonomous differential inclusions with infinite delays by using measures of noncompactness. As samples of applications, we apply the abstract results to control systems driven by semilinear partial differential equations and multivalued feedbacks.