On the Cauchy Problem for Differential Equations in a Locally Convex Space over the Field of Complex $$p$$-Adic Numbers

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 sequentially complete locally convex Hausdorff space over the field of complex $$p$$ -adic numbers, the criterion of well-posedness 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. Finally, the Cauchy problem of the form $$y^{(m)}(z)=Ay(z)$$ is also studied.