Behavior of solutions of a second order rational difference equation

• Main Author: Abo-Zeid Raafat
• Format: Article
• Language: English
• Published: University of Kragujevac, Faculty of Technical Sciences Čačak, Serbia 2019-01-01
• Series: Mathematica Moravica
• Subjects: 2010 Mathematics Subject Classification Primary: 39A20; Secondary: 39A21 Key words and phrases Difference equation; forbidden set; periodic solution; unbounded solution Full paper
• Online Access: https://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2019/1450-59321901011A.pdf

In this paper, we solve the difference equation xn+1 = α xnxn-1 - 1 , n = 0, 1, . . . , where α > 0 and the initial values x-1, x0 are real numbers. We find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when α > 2 3 √ 3 , under certain conditions there exist solutions, that are either periodic or converging to periodic solutions. We show also the existence of dense solutions in the real line. Finally, we show that when α < 2 3 √ 3 , one of the negative equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero.