On the existence of mild solutions of semilinear evolution differential inclusions

Abstract

In this paper we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion:and with initial data [0,d] is a family of linear operators in the Banach space E generating an evolution operator and F is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii,

Structure of the solution set to impulsive functional differential inclusions on the half-line

“…The moments of the impulse effects for the impulsive problems can be chosen in various ways: randomly, fixed beforehand, determined by the state of a system. For some recent works on impulsive differential problems, concerning the aspects we deal with, we refer to [2][3][4][5][6].…”

An existence theorem for a non-autonomous second order nonlocal multivalued problem

“…First we present the following result, which is analogous to the one proved in [10] or in [4], respectively for the Cauchy operator and for the generalized Cauchy operator.…”

On the evolution differential inclusions under a noncompact evolution system

“…Fan et al [11] studied the nonlocal impulsive differential equations when A(t) is governed by a compact semigroup, which is extended to impulsive differential inclusions scenario by Ji and Li [15] under weaker conditions. In [2,10,12], the measure of noncompactness is used to discuss some classes of differential and integral equations. Among the previous results, the multifunction F is usually supposed to be Lipschitz continuous or compact.…”

“…The C 0 −semigroup T (t) is not compact on X. Compared with the results in [10,12], we need not define a twocomponent measure of noncompactness and the Banach space X is not separable. This is due to the careful analysis to the combination of the properties of semicompact sets and Hausdorff measure of noncompactness.…”