Global Attractivity and Periodic Character of Difference Equation of Order Four

We investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence , where the parameters , and are positive real numbers and the initial conditions , and are positive real numbers. 1. Introduction Our goal in this paper is to investigate the global stability character, boundedness, and the periodicity of solutions of the recursive sequence where the parameters and are positive real numbers and the initial conditions , and are positive real numbers. Recently there has been a lot of interest in studying the global attractivity, the boundedness character and the periodicity nature of nonlinear difference equations, see for example [1–15]. The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order, recently many researchers have investigated the behavior of the solution of difference equations—for example, in [3] Elabbasy et al. investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence: In [5] Elabbasy and Elsayed investigated the global stability character and boundedness of solutions of the recursive sequence Elsayed [11] investigated the global character of solutions of the nonlinear, fourth-order, rational difference equation Saleh and Aloqeili [16] investigated the difference equation Yang et al. [17] investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequence For some related work see [16–26]. Here, we recall some basic definitions and some theorems that we need in the sequel. Let be some interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation has a unique solution . Definition 1.1 ？？(Equilibrium Point). A point is called an equilibrium point of (1.8) if That is, for is a solution of (1.8), or equivalently, is a fixed point of . Definition 1.2 ？？(Periodicity). A sequence is said to be periodic with period if for all . Definition 1.3 ？？(Stability). (i) The equilibrium point of (1.8) is locally stable if for every？？ ,？？there exists ？？such that for all ,？？ ,？？ , with we have (ii) The equilibrium point of (1.8) is locally asymptotically stable if is locally stable solution of (1.8) and there exists , such that for all , with we have (iii)