Differential equations in Ward's calculus

Abstract

In this paper, we solve some differential equations in the $D_h$ derivative in Ward's sense. We use a special metric in the formal power series ring $K\left[\right[x\left]\right]$. The solutions of those equations are given in terms of fixed points for certain contractive maps in our metric framework. Our main tools are Banach's fixed point theorem, the fundamental theorem of calculus, and Barrow's rule for Ward's calculus. Later, we return to the usual differential calculus via Sheffer's expansion of some kind of operators. Finally, we give some examples related, in some sense, to combinatorics.