Solution Forms and Stability Behavior for Selected Nonlinear Systems of Difference Equations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-339
Published: Oct 15, 2026
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E. M. Elsayed, R. Alsaedi, M. K. Alheis, Solution Forms and Stability Behavior for Selected Nonlinear Systems of Difference Equations, Int. J. Anal. Appl., 24 (2026), 339.

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E. M. Elsayed, R. Alsaedi, M. K. Alheis

Abstract

In this paper, we investigate the qualitative behavior and derivation of explicit closed-form solutions for the fourth-order nonlinear rational difference equation \(z_{m+1}=\frac{\alpha z_{m-2}z_{m-3}}{\beta z_{m}+\gamma z_{m-3}},\qquad m=0,1,2,\ldots ,\) where the parameters \(\alpha ,\beta ,\gamma\) and the initial conditions \(z_{-3},z_{-2},z_{-1},z_{0}\) are non-zero real numbers. Our analysis begins with the determination of the equilibrium point and the establishment of sufficient conditions for local asymptotic stability using linearization methods. Furthermore, we explore the global attractivity of the equilibrium point using monotonicity and comparison techniques. The boundedness of positive solutions is analyzed, providing criteria under which the sequence remains bounded. A core contribution of this work is the derivation of exact closed-form representations for the solutions in four special cases involving different sign combinations of the parameters. The results demonstrate that the structure of this higher-order equation leads to rich dynamical outcomes including stability, boundedness, and explicit solution behavior.

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