# Monotone-iterative technique for an initial value problem for difference equations with non--instantaneous impulses

## TL;DR

This paper introduces a monotone iterative method for solving initial value problems involving nonlinear difference equations with non-instantaneous impulses, providing a convergent approximation scheme with explicit formulas.

## Contribution

It develops a novel monotone iterative algorithm for nonlinear difference equations with non-instantaneous impulses, including explicit solution formulas and convergence proofs.

## Key findings

- The iterative sequences converge to minimal and maximal solutions.
- Explicit formulas for successive approximations are derived.
- The method effectively handles impulses over finite intervals.

## Abstract

In this paper a special type of difference equations is investigated. The impulses start abruptly at some points and their action continue on given finite intervals. This type of equations is used to model a real process. An algorithm, namely, the monotone iterative technique is suggested to solve the initial value problem for nonlinear difference equations with non-instantaneous impulses approximately. An important feature of our algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with with non-instantaneous impulses, and a formula for its explicit form is given. It is proved both sequences are convergent and their limits are minimal and maximal solutions of the considered problem.