Evolution equations with nonlocal multivalued Cauchy problems

Abstract

We consider evolution equations in Banach spaces. Their linear parts generate a strongly continuous C0-semigroup of contractions. The nonlinear term is a Carathéodory function. When the semigroup is not compact the nonlinearity has an additional restriction, involving the Hausdorff measure of noncompactness. We provide solutions satisfying nonlocal, multivalued Cauchy conditions. Our approach involves a suitable degree argument. The duality mapping is used for guaranteeing the lack of fixed points of the associated homotopic fields along the boundary of their domain. We apply our results for the investigation of transport and diffusion equations for which we provide the existence of nonlocal solutions.