## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Andrei Khrennikov and Ekaterina Yurova Axelsson

### Generalization of Hensel's lemma: finding the roots of $p$-adic Lipschitz functions.

Abstract

We consider the problem of finding the roots of $p$-adic functions. In the case, where the function is defined by a polynomial with integer $p$-adic coefficients, using Hensel's lifting lemma helps us find the roots of the $p$-adic function. We generalize Hensel's lifting lemma for a wider class of $p$-adic functions, namely, the functions which satisfy the Lipschitz condition with constant $p^{k}$, $k>=0$, in particular, the functions of this class may be non-differentiable. The paper also presents an iterative procedure for finding approximate (in $p$-adic metric) values of the root of $p^{k}$-Lipschitz functions, thus generalizing the $p$-adic analogue of Newton's method for such a class of functions.