On the Boundedness solutions of the difference equation $x_{n+1}=a x^\alpha_{n}+bx^\alpha_{n-1},0<\alpha \leq2$ and its application in medicine

TL;DR

This paper investigates the boundedness of solutions to a quadratic difference equation with parameters and explores its applications in medicine, contributing to the understanding of discrete dynamical systems in applied contexts.

Contribution

It provides a qualitative analysis of the difference equation's solutions and demonstrates its relevance to medical applications, expanding the understanding of rational dynamical systems.

Findings

Conditions for bounded solutions are identified.

The difference equation's behavior is characterized under various parameter settings.

Application examples in medicine illustrate practical relevance.

Abstract

Recently, mathematicians have been interested in studying the theory of discrete dynamical system, specifically difference equation, such that considerable works about discussing the behavior properties of its solutions (boundedness and unboundedness) are discussed and published in many areas of mathematics which involves several interesting results and applications in applied mathematics and physics ,One of the most important discrete dynamics which is become of interest for researchers in the field is the rational dynamical system .In this paper we may discuss qualitative behavior and properties of the difference equation $x_{n+1}=ax^2_{n}+bx^2_{n-1}$ with $a$ and $b$ are two parameters and we shall show its application to medicine.