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

βx n + γx n−1 )/(A + Bx n + Cx n−1 ), n = 0, 1, 2, . . ., where the parameters A ≥ 0 and B, C, α, β, γ are positive real numbers and the initial conditions x −1 , x 0 are nonnegative real numbers such that A + Bx 0 + Cx −1 > 0, 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.


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 ≥ 0 and , , , , are positive real numbers and the initial conditions −1 , 0 are nonnegative real numbers such that + 0 + −1 > 0. 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 −1 , 0 are nonnegative real numbers, was considered in the monograph of Kulenovic and Ladas [4]. They presented the known results up to 2002. 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.
Amleh et al. in [10,11] give an up-to-date account on recent developments related to (1) up to 2007. Furthermore, they reposed several open problems and conjectures related to this equation.
Our approach handles the aforementioned case as well as other cases. Furthermore, the results in this paper, together 2 Journal of Difference Equations 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 form We 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 −1 , 0 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.

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 0 , −1 ∈ , the difference equation of form (2) has a unique solution { } ∞ =−1 . A constant sequence, = for all where ∈ , is called an equilibrium solution of (2) if = ( , ) .
(i) is called locally stable if for every > 0, there exists > 0 such that, for all 0 , −1 ∈ , with | 0 − | + | −1 − | < , we have (ii) is called locally asymptotically stable if it is locally stable, and if there exists > 0, such that, for all (iv) is called globally asymptotically stable if it is locally stable and a global attractor.
(v) is called unstable if it is not stable.
denote the partial derivatives of ( , V) evaluated at the equilibrium of (2). Then the equation is 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 | | < 1, 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 | | < 1 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 2 + 4 > 0, 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 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: (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.
the subsequences { 2 } ∞ =0 and { 2 +1 } ∞ =−1 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.
(b) When (4) has prime period-two solution, . . . , , , , , . . . ; (24) if and only if condition where 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.
(2) If − > 0, then we have two subcases to be considered:

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 by The linearized equation associated with (4) about the equilibrium solution is given by Therefore, 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 −(1 + ) = , which is impossible to be satisfied since , , > 0, while (32) is equivalent to the following two inequalities: Equation (34) implies 1 + − = 2( + 1) , which contradicts (27), while (35) is equivalent to From which we have Clearly the equilibrium is the positive solution of the quadratic equation Now set and (37) holds if and only if That is, from which (31) follows. The proof is complete.

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 by By the Stable Manifold Theorem, there is a manifold of solutions that converge to the equilibrium solution.
With that in mind we examine the existence of intervals which attract all solutions of (1) in the next section. Table 1 gives the signs of ( , −1 )/ and ( , −1 )/ −1 of (4) in all possible nondegenerate cases when < .

Invariant Intervals.
The following two lemmas will be useful in investigating the attracting intervals of solutions of (4). Then By condition (31), inequality (46) is equivalent to Since + < 1 and > 1, the right-hand side of inequality (47) is equivalent to Thus, inequality (47) implies which contradicts condition (43).
Assume for the sake of contradiction that Then ≥ − + .

Proof. Condition (37) implies
Since + < 1, it is clear that < 1. To complete the proof we have two cases to be considered.
Our interest is to show that Assume for the sake of contradiction that By condition (31), the last inequality is equivalent to which is impossible.
The proof is complete.
By Theorem 7, Lemmas 10 and 11, we obtain the following key result.  Proof. We have two cases to be considered.
Since ≤ and condition (44) holds, then we have two subcases to be considered: The proof is complete.

Global Stability of the Nonhyperbolic Equilibrium Solution.
The following lemma will be useful in investigating the global stability of the nonhyperbolic equilibrium solution of (4). Lemma 13. Assume > . Then +1 ≤ −1 .
Observe that ℎ( , −1 ) is continuously differentiable map defined on a compact interval. As such, either the maximum is attained at an interior stationary point or on the boundary. Figure 1  Furthermore, As such, there is one interior stationary point, namely, ((1 − − 2( / ))/ , / ). Recall that we only want stationary point in the region , so ((1 − − 2( / ))/ , / ) will be ignored, because / < ( − )/( − ). Now, we need to find the absolute maximum of the function ℎ( , −1 ) along the boundary of the rectangle .
The boundary of this rectangle is given by the following.
The value of this function at the end points is (2) Lower side: This is not in the range ( − )/( − ) ≤ −1 ≤ 1, so we will ignore it.
The value of this function at the end points is Then decreases and the maximum occurs at = ( − )/( − ). Furthermore, Now, collect up all the function values for ℎ( , −1 ): Clearly the maximum value of ℎ is 0 and it occurs at (( − )/( − ), 1).
The proof is complete.
The result about the global stability of the positive nonhyperbolic equilibrium solution of (4) is given in the following theorem.

Theorem 14. The positive nonhyperbolic equilibrium solution of (4) is globally asymptotically stable.
Proof. Let We have two cases to be considered.
Here we distinguish between two subcases.
With the understanding that function (74) is decreasing in and increasing in −1 in the interval [ / , 1] by Table 1 case 1, and since condition (31) holds, (4) does not possess a period-two solution by Theorem 6. Thus all conditions of Theorem 4 are satisfied and we conclude that is globally asymptotically stable. Lemma 13 shows that +1 ≤ −1 . Replacing by − 1 in the previous inequality then ≤ −2 . As such, the odd and even terms of any solution of (4) form two monotonic nondecreasing subsequences. Furthermore, both of these subsequences are bounded because is bounded. Hence by Monotone Convergence Theorem the subsequences { 2 } and { 2 +1 } of the solution converge to finite limits 1 and 2 . Set Using technique similar to the one used in proofing Case (a)(1), we have Hence, Thus, and by subtracting we have This is true if and only if However, if 1 = 2 = then by (84)  whereas, 1 + 2 = 1 − − = 2 is impossible in this case since by Theorem 6, (4) does not possess a period-two solution. The proof is complete.
Remark 15. The papers [22][23][24] give the proof of the existence of both stable and unstable manifolds for second order difference equations decreasing in first and increasing in second argument in nonhyperbolic case of stable type, that is, of the type when second characteristic value is in (−1, 1). In such a way one can avoid the use of center manifold for such equations.

Numerical Examples
In order to illustrate the results of the previous sections and to support our theoretical discussion, we consider several numerical examples generated by MATLAB. Example 1. Consider the following equation: Since = 0.01 satisfies condition (31), by Theorem 9, the equilibrium is nonhyperbolic. Indeed, ≤ and − ≤ 0, and Theorem 12 implies that every positive solution of (89) eventually enters and remains in the interval [ / , 1]. Furthermore, the equilibrium is globally asymptotically stable by Theorem 14. The dynamics of (89) is shown in Figure 3. Since = 0.0025 satisfies condition (31), by Theorem 9, the equilibrium is nonhyperbolic. Indeed, < and − > 0, and Theorem 12 implies that every positive solution of (90) eventually enters and remains in the interval [ / , 1]. Furthermore, the equilibrium is globally asymptotically stable by Theorem 14. The dynamics of (90) is shown in Figure 4. Since = 0.0025 satisfies condition (31), by Theorem 9, the equilibrium is nonhyperbolic. Indeed, > , and Theorem 12 implies that every positive solution of (91) eventually enters and remains in the interval [( − )/( − ), 1]. Furthermore, the equilibrium is globally asymptotically stable by Theorem 14. The dynamics of (91) is shown in Figure 5.

Conclusion
In this paper, we have established the global stability of the nonhyperbolic equilibrium solutions of the second order rational difference equation: where the parameters , , , , , are positive real numbers and the initial conditions −1 , 0 are nonnegative real numbers.
We believe that our result is an important stepping stone in understanding the behavior of solutions of rational difference equations which provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of higher order.