Multivalued evolution equations with nonlocal initial conditions in Banach spaces

Abstract

We prove the existence of integral solutions to the nonlocal Cauchy problem \(u{\prime}(t)\epsilon-Au(t)+F(t,u(t)), 0{\leq}\,t\, {\leq}\,T; u(0)=g(u)\) in a Banach space X, where \(A : D\,(A)\, {\subset}\, X \rightarrow X\) is m-accretive and such that –A generates a compact semigroup, \(F : [0,T] \times X \rightarrow 2^{X}\) has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and \(g :C \left([0,T]; \overline{D\,(A)}\right) \rightarrow \overline{D\,(A)}\). The case when A depends on time is also considered.