On the qualitative and quantitative analysis for two fourth-order difference equations
F. Hilal Gumus, Raafat Abo-Zeid
article (Uluslararası, Hakemli) | Yazar(lar): F. Hilal Gumus, Raafat Abo-Zeid | Yıl: 2024 | Dergi/Kurum: JOURNAL OF APPLIED MATHEMATICS AND COMPUTING | İndeks: WOS.SCI | DOI: 10.1007/s12190-024-02010-w | Anahtar Kelimeler: Qualitative analysis, Periodicity, Stability, Difference equations, Boundedness character — Özet:
Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations x(n+1)=x(n-2)x(n-3 )/ax(n)+bx(n-3),n >= 0, and x(n+1)=x(n-2)x(n-3)/-ax(n)+bx(n-3), n >= 0, with positive arbitrary real parametersa,band arbitrary real initial conditions, aswell as study the qualitative behaviors for each. For the first equation, we show thatevery admissible solution converges to a period-3 solution whena+b=1. For thesecond equation, we show that every admissible solution converges to zero ifb>2whenb(2) >= 4a. Whenb2<4a, we show the existence of periodic solutions undercertain conditions. We introduce the forbidden sets as well as provide some illustrativeexamples for the above-mentioned equations.