Qualitative behavior of a higher-order nonautonomous rational difference equation
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Qualitative behavior of a higher-order nonautonomous rational difference equation Mohamed Amine Kerker1
· Elbahi Hadidi1 · Abdelouahab Salmi1
Received: 26 March 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This paper is concerned with the following nonautonomous rational difference equation yn+1 =
αn + yn , n = 0, 1, . . . , αn + yn−k
where {αn }n≥0 is a convergent sequence of positive numbers, k is a positive integer and the initial values y−k , . . . , y0 are nonnegative real numbers. We give sufficient conditions under which the unique equilibrium y¯ = 1 is globally asymptotically stable. Furthermore, the condition under which every positive solution is oscillatory about the equilibrium point is given. Our results are illustrated through some examples. Keywords Nonautonomous difference equation · Oscillation · Boundedness · Global asymptotic stability Mathematics Subject Classification 39A21 · 39A30
1 Introduction Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delayed differential equations having applications in economy, ecology, physics, engineering, etc [3,9]. The study of properties of nonlinear differ-
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Mohamed Amine Kerker [email protected] Elbahi Hadidi [email protected] Abdelouahab Salmi [email protected]
1
Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria
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M. A. Kerker et al.
ence equations has been an area of intense interest in the last few decades. There has been a lot of work concerning the global behavior of solutions of rational difference equations (for example, see [1,4–8,10,12–19]). In [10], Kocic and Ladas studied the (k + 1)th order difference equation yn+1 =
a + byn , n ∈ N, A + yn−k
(1.1)
a, b, A are nonnegative real numbers and k is a positive integer. They showed that the positive equilibrium point of the Eq. (1.1) is globally asymptotically stable. They showed also that every positive solution of Eq. (1.1) is oscillatory about the positive equilibrium point. Recently, Dekkar et al. [2] considered a nonautonomous analogues of Eq. (1.1). Precisely, they studied the following difference equation yn+1 =
αn + yn , n ∈ N, αn + yn−k
(1.2)
where {αn }n≥0 is a periodic sequence of positive numbers with period T , where T , k ≥ 1 are nonnegative integers and the initial values y−k , y−k+1 , . . . , y0 are nonnegative real numbers. They showed that the unique positive equilibrium point y¯ = 1 of Eq. (1.2) is globally asymptotically stable, and all positive solutions are oscillatory about the equilibrium point. They also proposed the following three open problems: – Open problem 1 Investigate the global character of solutions of Eq. (1.2) where {αn }n≥0 is a convergent sequence. – Open problem 2 Investigate the global character of solutions of Eq. (1.2) where {αn }n≥0 is a bounded sequence. – Open problem 3 Investigate the global character of the equation yn+1 =
αn + yn , n ∈ N, βn + yn−k
where {
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