Journal of Difference Equations

Volume 2015, Article ID 486985, 12 pages

http://dx.doi.org/10.1155/2015/486985

## Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

^{1}School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia^{2}Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, Riyadh 11426, Saudi Arabia

Received 4 January 2015; Accepted 17 February 2015

Academic Editor: Mustafa R. S. Kulenović

Copyright © 2015 S. Atawna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This is a continuation part of our investigation in which the second order nonlinear rational difference equation , , where the parameters and , , , , are positive real numbers and the initial conditions , are nonnegative real numbers such that , is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

*Dedicated to Gerry Ladas on the occasion of his retirement*

#### 1. Introduction

In part 1 of this investigation [1], we have established the global stability of the hyperbolic equilibrium solution of the second order rational difference equation:where the parameters and , , , , are positive real numbers and the initial conditions , are nonnegative real numbers such that . Our aim in this part is on the global attractivity of the nonhyperbolic equilibrium solution of (1).

The periodic character of positive solutions of (1) has been investigated by the authors in [2]. They showed that the period-two solution is locally asymptotically stable if it exists.

Many rational difference equations were studied extensively in [3]. A systematic study of the second order rational difference equation of form (1), where the parameters , , , , , and the initial conditions , are nonnegative real numbers, was considered in the monograph of Kulenovic and Ladas [4]. They presented the known results up to . Next, Kulenovic and Ladas [4] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of form (1). Furthermore, they posed several open problems and conjectures related to this equation and its functional generalization.

Even after a sustained effort by many researchers such as [5–9], there were some difference equations of form (1) that have not been investigated till .

Amleh et al. in [10, 11] give an up-to-date account on recent developments related to (1) up to . Furthermore, they reposed several open problems and conjectures related to this equation.

Camouzis and Ladas in [12] summarize the progress up to . Recently, the work done by many researchers such as [13–21] have solved many open problems and conjectures proposed in [4, 10–12] related to (1) and have led to the development of some general theory about difference equation. However, as confirmed by Professor Kulenovic (personal communication, August, 24, 2014), the case remains open.

Our approach handles the aforementioned case as well as other cases. Furthermore, the results in this paper, together with the established results in [1, 2, 4], give a complete picture of the nature of solutions of the second order rational difference equation of form (1).

It is worth mentioning that there are very few results in the literature regarding the stability of nonhyperbolic equilibrium solution of a general difference equation of the formWe believe that our result is an important stepping stone in understanding the behavior of solutions of rational difference equation of form (1) which provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of form (2).

The transformation reduces (1) to the following equation: where are positive real numbers and the initial conditions , are nonnegative real numbers.

That being said, the remainder of this paper is organized as follows. In the next section, a brief description of some definitions and results from the literature that are needed to prove the main results in this paper is given. Section 3 gives necessary and sufficient conditions for (4) to have nonhyperbolic solution. Next, Section 4 examines the existence of intervals which attract all solutions of (4) and shows that the nonhyperbolic equilibrium solution of (4) is globally asymptotically stable. In Section 5 we consider several numerical examples generated by MATLAB to illustrate the results of the previous sections and to support our theoretical discussion. Finally, we conclude in Section 6 with suggestions for future research.

#### 2. Preliminaries

For the sake of self-containment and convenience, we recall the following definitions and results from [4].

Let be a nondegenerate interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation of form (2) has a unique solution .

A constant sequence, for all where , is called an equilibrium solution of (2) if

*Definition 1. *Let be an equilibrium solution of (2).(i) is called locally stable if for every , there exists such that, for all , with , we have (ii) is called locally asymptotically stable if it is locally stable, and if there exists , such that, for all , with , we have (iii) is called a global attractor if for every , we have (iv) is called globally asymptotically stable if it is locally stable and a global attractor.(v) is called* unstable* if it is not stable.(vi) is called a* source*, or a* repeller*, if there exists such that, for all , with , there exists such that Clearly a source is an unstable equilibrium.

*Definition 2. *Letdenote the partial derivatives of evaluated at the equilibrium of (2). Then the equationis called the linearized equation associated with (2) about the equilibrium solution .

Theorem 3 (linearized stability). *(a) If both roots of the quadratic equation**lie in the open unit disk , then the equilibrium of (2) is locally asymptotically stable.**(b) If at least one of the roots of (13) has absolute value greater than one, then the equilibrium of (2) is unstable.**(c) A necessary and sufficient condition for both roots of (13) to lie in the open unit disk is**In this case the locally asymptotically stable equilibrium is also called a sink.**(d) A necessary and sufficient condition for both roots of (13) to have absolute value greater than one is**In this case is a repeller.**(e) A necessary and sufficient condition for one root of (13) to have absolute value greater than one and for the other to have absolute value less than one is**In this case the unstable equilibrium is called a saddle point.**(f) A necessary and sufficient condition for a root of (13) to have absolute value equal to one is**or**In this case the equilibrium is called a nonhyperbolic point.*

Theorem 4. *Consider the difference equation (2). Let be some interval of real numbers and assume that**is a continuous function satisfying the following properties: *(a) * is nonincreasing in for each , and is nondecreasing in for each ;*(b) *the difference equation (2) has no solutions of prime period two in ;**then (2) has a unique equilibrium and every solution of (2) converges to .*

The following result from [12] will become handy in the sequel.

Theorem 5. *Let be a set and let**be a function which decreases in and increases in . Then for every solution of the equation**the subsequences and of even and odd terms of the solution do exactly one of the following:*(i)*they are both monotonically increasing;*(ii)*they are both monotonically decreasing;*(iii)*eventually, one of them is monotonically increasing and the other is monotonically decreasing.*

*The following result was established in [2] and will prove to be useful in our investigation.*

*Theorem 6. (a) When (4) has no nonnegative prime period-two solution.*

(b) When (4) has prime period-two solution, if and only if conditionwhere and are the positive and distinct solutions of the quadratic equation

*The following two results were established in part 1 of this investigation [1] and will prove to be useful in our investigation.*

*Theorem 7. Assume that and ; then one has two cases to be considered.(1)If , then is invariant.(2)If , then we have two subcases to be considered:(a) if , then every positive solution of (4) eventually enters and remains in the interval ;(b) if , then every positive solution of (4) eventually enters and remains in the interval .*

*Theorem 8. Assume that and ; then one has two cases to be considered. (1)If , then every positive solution of (4) eventually enters and remains in the interval .(2)If , then every positive solution of (4) eventually enters and remains in the interval .*

*3. Existence of Nonhyperbolic Equilibrium Solution*

*3. Existence of Nonhyperbolic Equilibrium Solution**In this section, we give explicit conditions on the parameter values of (4) for the equilibrium to be nonhyperbolic.*

*Equation (4) has a unique positive equilibrium given byThe linearized equation associated with (4) about the equilibrium solution is given byTherefore, its characteristic equation is*

*By applying Theorem 3(f) we have the following result.*

*Theorem 9. Assume that then the positive equilibrium of (4) is nonhyperbolic if and only if*

*Proof. *By employing Theorem 3(f), conditions (17) and (18) are equivalent to the following two inequalities:respectively. Notice that Part (1) of (33) implies , which is impossible to be satisfied since , while (32) is equivalent to the following two inequalities:orEquation (34) implies , which contradicts (27), while (35) is equivalent to From which we haveClearly the equilibrium is the positive solution of the quadratic equationNow setand (37) holds if and only ifThat is, from which (31) follows.

The proof is complete.

*4. Global Stability Analysis*

*4. Global Stability Analysis**In this section, we give necessary and sufficient conditions for the nonhyperbolic solution of (4) to be globally attractive.*

*The characteristic polynomial associated with (4) about the positive equilibrium is given byBy the Stable Manifold Theorem, there is a manifold of solutions that converge to the equilibrium solution.*

*Now, since , condition (31) impliesIndeed, since we have*

*With that in mind we examine the existence of intervals which attract all solutions of (1) in the next section.*

*4.1. Invariant Intervals*

*4.1. Invariant Intervals**Table 1 gives the signs of and of (4) in all possible nondegenerate cases when .*