Operator approach to the Cauchy-Kovalevskaya theorem

Abstract

For operator differential equations in a Banach space, we present the conditions for initial data which are necessary and sufficient for the Cauchy problem to have a solution in the class of analytic, entire, or exponential-type entire vector functions. In the case where an operator differential equation is a system of partial differential equations, the sufficient condition obtained coincides with the well-known Cauchy-Kovalevskaya theorem on the solvability of the Cauchy problem in the class of analytic functions.