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We consider a restriction of Vladimirov's fractional differentiation operator $D^\alpha$, $\alpha >0$, to complex-valued radial functions on a non-Archimedean field. In particular, it is found to possess such a right inverse $I^\alpha$ that the appropriate change of variables reduces equations with $D^\alpha$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. We study nonlinear equations of this kind, find conditions of their local and global solvability. Next, we begin an operator-theoretic investigation of the operator $I^\alpha$, and study a related analog of the Laplace transform.
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