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

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation x n+1 = (α+βx n +γx n−1 )/(A+Bx n +Cx n−1 ), n = 0, 1, 2, . . ., where the parametersA ≥ 0 and B,C, α, β, γ are positive real numbers and the initial conditions x −1 , x 0 are nonnegative real numbers such thatA+Bx 0 +Cx −1 > 0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).


Introduction
Rational difference equations, particularly bilinear ones, that is, attracted the attention of many researchers recently.For example, see the articles [1][2][3][4][5][6][7][8][9].As it turns out, many models, such as population models in mathematical biology, are members of the family of rational difference equations.The behavior of solutions of rational difference equations can provide prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one.Hence, the study of this family of equations is important from both a theoretical point of view and the point of view of applications.
For the general theory of difference equations, one can refer to Agarwal [10], Elaydi [11], Kelley and peterson [12], and the monograph of Kocic and Ladas [13].Many rational difference equations were studied extensively in [14,15] and the references cited therein.
Asystematic study of the second order rational difference equation of the form where the parameters , , , , ,  and the initial conditions  −1 ,  0 are nonnegative real numbers, was considered in the Kulenović and Ladas monograph [15].They came up with the idea of setting one or more parameters in (2) to zero and studying the resulting equation with fewer parameters.This approach gives rise to 49 different cases which exhibit variety dynamics.They presented the known results such as [16][17][18][19][20][21][22][23].
Next, Kulenović and Ladas [15] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of the form (2). 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 [24][25][26][27][28], there were some cases of the 49 different cases that have not been investigated till 2007.Amleh et al. in [29,30] give an up-to-date account on recent developments related to (2) up to 2007.Furthermore, they reposed several open problems and conjectures related to this equation.
where the parameters , , , , , , ,  and the initial conditions  −2 ,  −1 ,  0 are nonnegative real numbers.In their book, Camouzis and Ladas [14] have posed a series of open problems and conjectures related to (3).In addition, they reposed open problems and conjectures on these remaining equations of (2) that have resisted analysis so far.
Our approach handles the aforementioned case as well as other cases.Furthermore, the results of this paper improve and extend the asymptotic results in [15,Chapter 11].Indeed, our results provide affirmative answer to the following conjecture proposed by Camouzis and Ladas in [14,Conjecture 5.201.1].
Conjecture 1.This shows that, for the equilibrium  of (2), Local Asymptotic Stability ⇒ Global Attractikity.(4) It is worth mentioning that the aforementioned conjecture appeared previously in the Kulenović and Ladas monograph [15,Conjecture 11.4.2].
To this end and using the transformation equation ( 2) reduces to where are positive real numbers and the initial conditions  −1 ,  0 are nonnegative real numbers.The periodic character of positive solutions of (6) has been investigated by the authors in [40].They showed that the period-two solution is locally asymptotically stable if it exists.
Our results, together with the established results in [15,40], give a complete picture of the nature of solutions of the second order rational difference equation of the form (2). We believe that our results are 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.
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.It is worth mentioning that there are few global attractivity results in the literature that can be applied to rational difference equations of the form (2). Next we establish our main results in Sections 3-5.We determine the local stability character of (2) in Section 3. Section 4 examines the existence of intervals which attract all solutions of (2).In Section 5, we investigate the global asymptotic stability of the hyperbolic equilibrium solution of (6).Next, in Section 6 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 7 with suggestion for future research.

Preliminaries
For the sake of self-containment and convenience, we recall the following definitions and results from [15].
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 has a unique solution {  } ∞ =−1 .A constant sequence,   =  for all  where  ∈ , is called an equilibrium solution of (8) if Definition 2. Let  be an equilibrium solution of (8).
(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 denote the partial derivatives of (, V) evaluated at the equilibrium  of (8).Then the equation is called the linearized equation associated with (8) about the equilibrium solution .

Theorem 4 (linearized stability). (a) If both roots of the quadratic equation
lie in the open unit disk || < 1, then the equilibrium  of ( 8) is locally asymptotically stable.(b) If at least one of the roots of ( 16) has absolute value greater than one, then the equilibrium  of ( 8) is unstable.
(c) A necessary and sufficient condition for both roots of ( 16) 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 (16) to have absolute value greater than one is In this case  is a repeller.(e) A necessary and sufficient condition for one root of ( 16) 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 (16) to have absolute value equal to one is or In this case the equilibrium  is called a nonhyperbolic point.
The following result from [40] will be useful in the sequel.
(b) When equation ( 6) has prime period-two solution, . . ., , , , , . . ., (31) if and only if condition where  and  are the positive and distinct solutions of the quadratic equation

Local Stability
In this section, we address the local stability of the equilibrium  of (6) when all parameters are positive.In particular, we give explicit conditions on the parameter values of (6) for the equilibrium  to be locally asymptotically stable.Equation ( 6) has a unique positive equilibrium given by The linearized equation associated with (6) about the equilibrium solution is given by Therefore, its characteristic equation is By applying linearized stability (Theorem 4(c)) we have the following result.
Theorem 10.(a) Assume that then the positive equilibrium of ( 6) is locally asymptotically stable.
(b) Assume that  +  < 1; (38) then the positive equilibrium of ( 6) is locally asymptotically stable if and only if Proof.By employing linearized stability (Theorem 4(c)) we see that condition (17) is equivalent to the following three inequalities: or that is, from which (39) follows.
The proof is complete.

Invariant Intervals
In this section, we investigate the invariant intervals for (6) in order to obtain convergence results for the solutions of (6).
Let {  } ∞ =−1 be a positive solution of ( 6).Then we have the following identities: We consider the cases where  < ,  > , and  = .
By Remark 11, Lemma 12, and Table 1, we obtain the following key result.
Theorem 13.Assume that  <  and  ≤ ; then we have two cases to be considered.
(2) If  −  > 0, then we have three subcases to be considered.
Here we have two cases to be considered.
Our interest now is to show that the pairs ( −1 ,   ) cannot stay forever in  4  .Also, we need to show that the pairs First assume that ( −1 ,   ) ∈  4  ; then  −1 ,   < ( − )/( − ).In view of Table 1, Case 1, the solution increases in both arguments.Using the increasing character of , we obtain (57) As such the limit of the solution lies in the interval (0, /), which is impossible because  ≥ / by Lemma 12, part 1.
Hence every positive solution of (6) in the region  4  also eventually enters and remains in the interval Without loss of generality, we may assume that  2+1 < ( − )/( − ), whereas  2 > / for all .But, The first inequality holds true because  2−1 < ( − )/( − ).Furthermore, the second inequality follows from the facts that the graph of () = ( + )/( + ) looks like the one depicted in Figure 4, and Thus the odd terms converge and so do the even terms.This implies the existence of a period-two solution which is a contradiction since, by Theorem 9, (6) does not possess a period-two solution.
Without loss of generality, we may assume that  2+1 > ( − )/( − ), whereas  2 < / for all .But, The first inequality holds true because  2−1 > ( − )/(− ).Furthermore, the second inequality follows from the facts that the graph of () = ( + )/( + ) looks like the one depicted in Figure 6, and Thus the odd terms converge and so do the even terms.This implies the existence of a period-two solution which is a contradiction since, by Theorem 9, (6) does not possess a period-two solution.
The proof is complete.
By Remark 11, Lemma 14, and Table 1, we obtain the following key result.
Theorem 15.Assume that  <  and  > ; then we have two cases to be considered.Proof.Assume that  <  and  > .Identity (52) shows that   > / for all  ≥ 0. In this case, the plane  −1 −   divides into the regions depicted in Figure 8.
From the directed graph, we can see that if a solution is not eventually in [( − )/( − ), 1], it converges to a periodic solution with period 2 or 3.
Our interest now is to show that the pairs ( −1 ,   ) cannot stay forever in  2  ∪  3  .Also, we need to show that the pairs ( −1 ,   ) cannot stay forever in  2  ∪  3  ∪  4  .
From the directed graph, we can see that if a solution is not eventually in [1, ( − )/( − )], it converges to a periodic solution with period 2 or 3.
Our interest now is to show that the pairs ( −1 ,   ) cannot stay forever in  2  ∪  3  .Also, we need to show that the pairs ( −1 ,   ) cannot stay forever in  2  ∪  3  ∪  4  .
With that in mind, Thus the odd terms converge and so do the even terms.This implies the existence of a period-two solution which is a contradiction since, by Theorem 9, (6) does not possess a period-two solution.
The proof is complete.

Lemma 17. Assume that
Then we have two cases to be considered.
Proof.The proof is similar to the proof of Lemmas 12 and 14 and will be omitted.(105) Then we have two cases to be considered.
Here we have two cases to be considered.
(a) If  −  ≤ 0.  The proof is similar to that of Theorem 15 and will be omitted.
The proof is similar to that of Theorem 13, part 2, and will be omitted.
The proof is complete.
Table 3 gives the signs of / and / in all possible nondegenerate cases when  = .
By Identity (107) and Table 3, we obtain the following result.p/q p/q p/q p/q  6) possesses the following invariant intervals.
Notice that when  =  and  =  then the function (, ) = 1 for all values of  and , while if  =  and  <  then by Table 3 ( The proof is complete.

Global Stability of Hyperbolic Equilibrium Solution
The results about the global stability for the positive equilibrium of ( 6) are given in the following theorem.
then the positive equilibrium of ( 6) is globally asymptotically stable.
(ii) Assume that then the positive equilibrium of ( 6) is globally asymptotically stable if and only if  is  = .This system is equivalent to which implies  = .
Now the result is a consequence of Theorem 7.
Case 2:  > .It follows from Table 4 that each of (0, 1 is  = .Now the result is a consequence of Theorem 7.
The proof is complete.

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.
Since  = 0.1 satisfies condition (39), by Theorem 10, the equilibrium is locally asymptotically stable.Indeed,  ≤  and  −  ≤ 0; Theorem 13 implies that every positive solution of (121) eventually enters and remains in the interval [/, 1].Furthermore, the equilibrium  is globally asymptotically stable by Theorem 20.The dynamics of (121) are shown in Figure 15.

Conclusion
In this paper, we have established the global stability of the hyperbolic equilibrium solutions of the second order rational difference equation  +1 =  +   +  −1  +   +  −1 ,  = 0, 1, 2, . . ., where the parameters , , , , ,  are positive real numbers and the initial conditions  −1 ,  0 are nonnegative real numbers.Particularly, we showed that Local Asymptotic Stability ⇒ Global Attractivity.(127) However, it is natural to ask about the global stability of the nonhyperbolic equilibrium solutions for the equation of the form mentioned above.In particular, one may want to investigate the necessary and sufficient conditions for the equation to have nonhyperbolic solution and completely examine the existence of intervals which attract all solutions of the equation in order to obtain a convergence result for the the nonhyperbolic equilibrium solutions of the equation of the form mentioned above.
We consider the aforementioned result as a step forward in investigating bigger classes of difference equations which afford the LGAS property; that is, local stability of an equilibrium implies its global stability.
(iii)  is called a global attractor if, for every  0 ,  −1 ∈ , we have lim → ∞   = .(12)(iv)iscalled globally asymptotically stable if it is locally stable and a global attractor.(v)is called unstable if it is not stable.(vi)

Table
by Table 1, Case 1, the function ( −1 ,   ) is increasing in  −1 and decreasing in   .Using the decreasing character of  in   , and the increasing character in  −1 , we obtain 4 V ; then ( − )/( − ) ≤  −1 ,   ≤ 1.In this case, by Table 1, Case 5, the function ( −1 ,   ) is increasing in  −1 and decreasing in   .Using the decreasing character of  in   , and the increasing character in  −1 , we obtain

Table 2 ,
Cases 2, 3, and 4, the function ( −1 ,   ) is increasing in   and decreasing in  −1 for all values of   and  −1 .Using the decreasing character of  in  −1 , and the increasing character in   , we obtain , Case 1, the function (, ) is increasing in both arguments for all values of  and .Using the increasing character of  we have, Identity (107) shows that if  =  and  >  then   > 1 for all  ≥ 0. Furthermore, by Table3, Case 2, the function (, ) is decreasing in both arguments for all values of  and .Using the decreasing character of  we have,

)
Proof.We have established the local stability of the equilibrium solution in Theorem 10.To complete the proof it remains to show that the equilibrium  is a global attractor.