Let be the infinitesimal generator of a compact semigroup of linear operators on a Banach space . The authors establish the existence of mild solutions to the nonlocal Cauchy problem
under some conditions on and , where and are given functions. They assume a Lipschitz condition on with respect to , but they do not require any compactness assumption on , opposed to S. Aizicovici and M. McKibben [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649–668 (2000; Zbl 0954.34055)] and L. Byszewski and H. Akca [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65–72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for , but do not require any Lipschitz condition on .