Abstract
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.
