Global behavior of a rational second order difference equation
Mehmet Gumus, Raafat Abo-Zeid
article (Uluslararası, Hakemli) | Yazar(lar): Mehmet Gumus, Raafat Abo-Zeid | Yıl: 2020 | Dergi/Kurum: JOURNAL OF APPLIED MATHEMATICS AND COMPUTING | İndeks: WOS.SCI | DOI: 10.1007/s12190-019-01276-9 | Anahtar Kelimeler: Difference equation, Forbidden set, Convergence, Unbounded solution — Özet: In this paper, we solve
In this paper, we solve the difference equation
x(n+1) = alpha/1 - x(n)x(n-1), n = 0, 1, ...,
where alpha > 0 and the initial values x(-1), x(0) are real numbers. We find invariant sets and discuss the global behavior of the solutions of that equation. We show that when alpha < 2/3 root 3, one of the positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Also, when alpha = 2/3 root 3, the unique positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Finally, we show that when alpha > 2/3 root 3, under certain conditions there exist solutions that are either periodic or converging to periodic solutions and give some examples. We show also the existence of dense solutions in the real line.