Nonlocal cauchy problems for semilinear evolution equations involving almost sectorial operators

Of concern is the following Cauchy problem for a semilinear evolution equation with a nonlocal initial condition (Equation Presented) where A is an almost sectorial operator (not necessarily densely defined) and F, H are given functions. The existence and uniqueness of mild and classical solutions for the Cauchy problem, under various hypotheses, are proved. Then, as an immediate application of this result, we investigate the existence and uniqueness of classical solutions for the following nonlinear nonlocal Cauchy problem (Equation Presented) where G is a given functional. This paper is a continuation of the investigation of the earlier articles [22, 23] concerning the theory of almost sectorial operators and its applications.