On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general Banach spaces

Abstract

Sufficient conditions on the existence of mild solutions for the following semilinear nonlocal evolution inclusion with upper semicontinuous nonlinearity: u ′ ( t ) ∈ A ( t ) u ( t ) + F ( t , u ( t ) ) , 0 < t ⩽ d , u ( 0 ) = g ( u ) , are given when g is completely continuous and Lipschitz continuous in general Banach spaces, respectively. An example concerning the partial differential equation is also presented.