Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem

Abstract

The aim of this paper is to give three theorems about the existence and uniqueness of mild, strong, and classical solutions of a nonlocal Cauchy problem for a semilinear evolution equation. The method of semigroups and the Banach theorem about the fixed point are used to prove the existence and uniqueness of solutions of the problem considered. The results obtained are a continuation of those reported previously by this author and can be applied in physics with better effects than the classical Cauchy problem for the semilinear equation. The theorems proved in this paper generalize some results obtained by A. Pazy (“Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, Berlin/New York, 1983).