Differential equations in Ward's calculus

TL;DR

This paper explores solving differential equations within Ward's calculus framework using fixed point methods, connecting formal power series, classical calculus, and combinatorics.

Contribution

It introduces methods for solving differential equations in Ward's calculus using fixed point theorems and links to classical calculus via Sheffer's expansion.

Findings

Solutions expressed as fixed points of contractive maps

Connection established between Ward's calculus and classical differential calculus

Examples illustrating combinatorial applications

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 $\K[[x]]$. The solutions of that equations are giving in terms of fixed points for certain contractive maps in our metric framework. Our main tools are Banach's Fixed Point Theorem, Fundamental Calculus Theorem 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.