Abstract
We consider the higher-order nonlinear difference equation with the parameters, and the initial conditions are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).
Highlights
Introduction and PreliminariesOur aim in this paper is to investigate the global behavior of solutions of the following nonlinear difference equation: xn 1 p xn qxn−k r xn−k, n1.1 where the parameters p, q, r and the initial conditions x−k, . . . , x0 are nonnegative real numbers, k ∈ {1, 2, . . .}.In 2002, Kulenovicand Ladas 1 proposed the following open problem.Open Problem 1
The positive equilibrium of 1.1 is globally asymptotically stable
Summary
Our aim in this paper is to investigate the global behavior of solutions of the following nonlinear difference equation: xn 1 p xn qxn−k r xn−k. For the global behavior of solutions of some related equations, see 3–9. I The equilibrium y is called locally stable or stable if for every ε > 0, there exists δ > 0 such that for all y−k, . Let p ≥ 2 be a positive integer and assume that every positive solution of equation xn 1 α A βxn Bxn γ xn−k Cnn−k. 1.17 is a continuous function satisfying the following properties: a f u, v is nonincreasing in each of its arguments, b if m, M ∈ a, b × a, b is a solution of the system m f M, M , M f m, m , 1.18 m M. 1.16 has a unique equilibrium y ∈ a, b , and every solution of 1.16 converges to y
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