Long-Term Behavior of Solutions of the Difference Equation

We investigate the long-term behavior of solutions of the following difference equation: , , where the initial values , , and are real numbers. Numerous fascinating properties of the solutions of the equation are presented. 1. Introduction and Preliminaries Recently there has been great interest in studying nonlinear difference equations which do not stem from differential equations (see, e.g., [1–28] and the references therein). Standard properties which have been studied are boundedness [5, 9, 23–25], periodicity [2, 5, 9, 10, 27], asymptotic periodicity [3, 4, 8, 11–14, 16, 17, 19, 20, 23], and local and global stability [5, 9–11, 23–26], as well as existence of specific solutions such as monotone or nontrivial solutions [1, 6, 7, 13, 15, 18–22]. In this paper, we investigate the long-term behavior of solutions of the third-order difference equation where the initial values are real numbers. The difference equation (1.1) belongs to the class of equations of the form where , , and . The case , has been recently investigated in [8]. 2. The Equilibria and Periodic Solutions of (1.1) This section is devoted to the study of the equilibria and periodic solutions of (1.1). 2.1. Equilibria of (1.1) If is an equilibrium of (1.1), then it satisfies the equation Hence, (1.1) has exactly two equilibria, one positive and one negative, which we denote by and , respectively: (the golden number and its conjugate). 2.2. Periodic Solutions of (1.1) Here, we study the existence of periodic solutions of (1.1). For related results, see, for example, [2, 5, 9, 10, 27] and the references therein. The first two results are simple, but we will prove them for the completeness, the benefit of the reader, and since we use them in the sequel. Theorem 2.1. There are no eventually constant solutions of difference equation (1.1). Proof. If is an eventually constant solution of (1.1), then , for some , where is an equilibrium point. In this case, (1.1) gives , which implies Repeating this procedure, we obtain for . Hence, there are no eventually constant solutions. Theorem 2.2. Difference equation (1.1) has no nontrivial period two solutions nor eventually period two solutions. Proof. Assume that and , for every , and some , with . Then, we have From this and since , we obtain a contradiction, finishing the proof of the result. Theorem 2.3. There are no periodic or eventually periodic solutions of (1.1) with prime period three. Proof. If for some , we have If , , or , then from (2.6) we easily obtain contradictions in all these cases. Hence, we may assume that , , and . Equalities