On a Rational $(P+1)$th Order Difference Equation with Quadratic Term

Abstract

In this paper, we derive the forbidden set and determine the solutions of the difference equation that contains a quadratic term \begin{equation*} x_{n+1}=\frac{x_{n}x_{n-p}}{ax_{n-(p-1)}+bx_{n-p}},\quad n\in\mathbb{N}_0, \end{equation*} where the parameters $a$ and $b$ are real numbers, $p$ is a positive integer and the initial conditions $x_{-p}$, $x_{-p+1}$, $\cdots$, $x_{-1}$, $x_{0}$ are real numbers.

Analytical Study of Nonlinear Systems of Higher-Order Difference Equations: Solutions, Stability, and Numerical Simulations

“…Furthermore, we can obtain numbers that satisfy some conditions of interest, for example, Richard and Raoul numbers. Moreover, our paper builds upon the contributions of Ghezal and Zemmouri [12] and Abo-Zeid et al [6,7] by synthesizing and advancing existing knowledge in the field of nonlinear difference equations and systems. By incorporating recent advancements in theoretical analysis and numerical validation techniques, our study pushes the boundaries of understanding in this domain and opens up new avenues for exploration.…”

“…However, while several methods exist for solving linear difference equations, the landscape of nonlinear systems remains largely uncharted. Despite recent attempts to simplify complex nonlinear systems into linear forms, there persists a significant gap in analytical approaches for addressing systems of difference equations, posing a challenge for researchers striving to deepen their understanding of their behavior and properties (see, [6,7]).…”