On the Cauchy Problem for Differential Equations in a Banach Space over the Field of p-Adic Numbers. I

For the Cauchy problem for an operator differential equation of the form $y'(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the completion of an algebraic closure of the field of $p$-adic numbers, a criterion of correct solvability in the class of locally analytic vector-functions is established. It is shown how the Cauchy-Kovalevskaya theorem for $p$-adic partial differential equations may be obtained as a particular case from this criterion.