10_447_zayed.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 2, 2010, 254-268 ISSN 1307-5543 – www.ejpam.com Dynamics of the Nonlinear Rational Difference Equation xn+1 = Axn+ Bxn−k+ pxn+xn−k q+xn−k E. M. E. Zayed Mathematics Department, Faculty of Science, Taif University, El-Taif, EL-Hawiyah, Kingdom of Saudi Arabia Previously: Mathematics Department, Faculty of Science,Zagazig University, Zagazig, Egypt Abstract. In this article, we study the global stability and the asymptotic properties of the nonnegative solutions of the nonlinear difference equation xn+1 = Axn + Bxn−k + � pxn + xn−k � / � q+ xn−k � , n= 0,1,2, ..... where the parameters A, B, p,q and the initial conditions x−k,..., x−1, x0 are arbitrary nonnegative real numbers, while k is a positive integer number. Some numerical examples will be given to illustrate our results. 2000 Mathematics Subject Classifications: 39A10,39A11,39A99,34C99 Key Words and Phrases: Difference equations, prime period two solution, locally asymptotically sta- ble, global attractor, Global stability. 1. Introduction The qualitative study of difference equations is a fertile research area and increasingly attracts many mathematicians. This topic draws its importance from the fact that many real life phenomena are modeled using difference equations. Examples from economy, biology, etc. can be found in [2,16,19,29] . It is known that nonlinear difference equations are capable of producing a complicated behavior regardless its order. This can be easily seen from the family xn+1 = gµ � xn � , µ > 0, n≥ 0. This behavior is ranging according to the value of µ, from the existence of a bounded number of periodic solutions to chaos. There has been a great interest in studying the global attractivity, the boundedness char- acter and the periodicity nature of nonlinear difference equations. For example, in the articles [1,7-14,21–31] closely related global convergence results were obtained which can be applied to nonlinear difference equations in proving that every solution of these equations converges Email address: emezayed�hotmail. om http://www.ejpam.com 254 c© 2010 EJPAM All rights reserved. E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 255 to a period two solution. For other closely related results, (see [3-7,10,17,18]) and the refer- ences cited therein. The study of these equations is challenging and rewarding and is still in its infancy. We believe that the nonlinear rational difference equations are of paramount im- portance in their own right. Furthermore the results about such equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations. Our goal in this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation xn+1 = Axn+ Bxn−k + pxn+ xn−k q+ xn−k , n = 0,1,2, .. . . . (1) where the parameters A, B, p,q and the initial conditions x−k,. . . x−1, x0 are arbitrary nonneg- ative real numbers, while k is a positive integer number. The global stability of Eq.(1) for A= B = 0 has been investigated in [29]. Kulenvic et al.[22] studied Eq.(1) when A= B = 0 and k = 1. Our interest now is to study the behavior of solutions of Eq.(1) in the general case where A 6= 0, B 6= 0 and k is a positive integer number. For the related work see [32-45]. The study of these equations is challenging and rewarding and is still in its infancy. We believe that the nonlinear rational difference equations are of paramount importance in their own right. Furthermore the results about such equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations. Let us now recall some well know results [15] which will be useful in the sequel. Definition 1. A difference equation of order (k+ 1) is of the form xn+1 = F(xn, xn−k), n= 0,1,2, ..... (2) where F is a continuous function which maps some set J k+1 into J where J is a set of real numbers. An equilibrium point ex of this equation is a point that satisfies the condition ex = F (ex , ex). That is, the constant sequence � xn ∞ n=−k with xn = ex for all n ≥ −k is a solution of that equation. Definition 2. Let ex ∈ (0,∞) be an equilibrium point of the difference equation (2). Then (i) An equilibrium point ex of the difference equation (2) is called locally stable if for every ǫ > 0 there exists δ > 0 such that, if x−k, . . . , x−1, x0 ∈ (0,∞) with ��x−k − ex ��+ . . . +��x−1− ex ��+ ��x0 − ex �� < δ, then ��xn− ex ��< ǫ for all n≥ −k. (ii) An equilibrium point ex of the difference equation (2) is called locally asymptotically stable if it is locally stable and there exists γ > 0 such that, if x−k, . . . , x−1, x0 ∈ (0,∞) with��x−k − ex ��+ ...+ ��x−1− ex ��+ ��x0 − ex �� < γ, then lim n→∞ xn = ex . (iii) An equilibrium point ex of the difference equation (2) is called a global attractor if for every x−k, . . . , x−1, x0 ∈ (0,∞) we have lim n→∞ xn = ex . E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 256 (iv) An equilibrium point ex of the equation (2) is called globally asymptotically stable if it is locally stable and a global attractor. (v) An equilibrium point ex of the difference equation (2) is called unstable if it is not locally stable. Definition 3. A sequence � xn ∞ n=−k is said to be periodic with period p if xn+p = xn for all n ≥ −k. A sequence � xn ∞ n=−k is said to be periodic with prime period p if p is the smallest positive integer having this property. Definition 4. A positive semi-cycle of � xn ∞ n=−k consists of "a string" of terms � x l , x l+1, . . . xm all greater than or equal to ex , with l ≥ −k and m≤∞ such that either l = −k or + l > −k and x l−1 < ex , and either m=∞ or +m <∞and xm−1 < ex , A negative semi-cycle of � xn ∞ n=−k consists of "a string" of terms � x l , x l+1, . . . xm all less than ex , with l ≥ −k and m ≤∞ such that either l = −k or + l > −k and x l−1 ≥ ex , and either m=∞ or +m <∞and xm−1 ≥ ex , Definition 5. Eq.(2) is said to be permanent if there exist positive real numbers m and M such that for every solution � xn ∞ n=−k of Eq.(2) there exists a positive integer N ≥ −k which depends on the initial conditions, such that m ≤ xn ≤ M , for all n≥ N . The linearized equation of the difference equation (2) about the equilibrium point ex is the linear difference equation zn+1 = ∂ F (ex , ex) ∂ xn zn+ ∂ F (ex , ex) ∂ xn−k zn−k, (3) The characteristic equation associated with Eq.(3) is p (λ) = λk+1− p0λ k − p1 = 0, (4) where p0 = ∂ F (ex , ex) ∂ xn , p1 = ∂ F (ex , ex) ∂ xn−k . Theorem 1. ([15]). The linearized stability theorem. Suppose F is a continuously differentiable function defined on an open neighbourhood of the equilibrium ex . Then the following statements are true. E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 257 (i) If all the roots of the characteristic equation (4) of the linearized equation (3) have absolute value less than one, then the equilibrium point ex of Eq.(2) is locally asymptotically stable. (ii) If at least one root of Eq.(4) has absolute value greater than one, then the equilibrium point ex of Eq.(2) (iii) If all the roots of Eq.(4) have absolute value greater than one, then the equilibrium point ex of Eq.(2) is a source. 1.1. Equilibrium Points In this section, we examine the nonnegative equilibrium points ex of Eq.(1) and their local asymptotic behavior. The equilibrium points of Eq.(1) are the nonnegative solutions of the equation ex = (A+ B) ex + � p+ 1 � ex q+ ex . (5) So, ex = 0 is always an equilibrium point of Eq.(1). If 0< A+B < 1, p−q > − � 1+ q (A+ B) � and p > q then the positive equilibrium point is ex = � p− q � + � 1+ q (A+ B) � [1− (A+ B)] . (6) Lemma 1. If p > q and 0 < A+ B < 1, then the positive equilibrium point (6 satisfies the inequality ex > q p . Proof. From (6 we deduce that ex = p+ 1 1− (A+ B) − q > q+ 1 1− (A+ B) − q = 1+ q (A+ B) 1− (A+ B) = � 1+ q (A+ B) �� 1+ (A+ B) + (A+ B)2 + ...... � > 1> q p . The proof of Lemma 1 is now completed. 1.2. Linearization In this section, we derive the linearized equation of Eq.(1). To this end, we introduce a continuous function F : (0,∞)2→ (0,∞) which is defined by F(u0,u1) = Au0 + Bu1 + pu0 + u1 q+ u1 . (7) Therefore,    ∂ F(u0 ,u1) ∂ u0 = A+ p q+u1 , ∂ F(u0,u1) ∂ u1 = B+ q−pu0 (q+u1) 2 . (8) E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 258 From (6 and (8) we have    ∂ F(ex ,ex) ∂ u0 = A+ p[1−(A+B)] p+1 = ρ0, ∂ F(ex ,ex) ∂ u1 = B − [1−(A+B)][(p−q)+q(A+B)] p+1 = ρ1. (9) The linearized equation of Eq.(1) about the zero equilibrium point ex = 0 is zn+1 − � A+ p q � zn − � B+ 1 q � zn−k = 0, (10) and the linearized equation of Eq.(1) about the positive equilibrium point ex is zn+1 −ρ0 zn−ρ1 zn−k = 0, (11) where ρ0 and ρ1 are given by (9). Theorem 2. [20] Assume that ρ0,ρ1 ∈ R and k ∈ {1,2, ...}. Then ��ρ0 ��+ ��ρ1 ��< 1, (12) is a sufficient condition for the asymptotic stability of the difference equation (2). Suppose in addition that one of the following two cases holds: (i) k is an odd integer and ρ1 > 0. (ii) k is an even integer and ρ0ρ1 > 0. Then (12) is also a necessary condition for the asymptotic stability of Eq.(2). Theorem 3. [15] Consider the difference equation (2) where the function F ∈ C � Ik+1,R � and I is an open interval of real numbers. Let ex ∈ I be an equilibrium point of Eq.(2). Suppose also that (i) F is a nondecreasing function in each of its arguments. (ii) The function F satisfies the negative feedback property [F (x , x)− x] (x − ex)< 0 for all x ∈ I − {ex} . Then the equilibrium point ex of Eq.(2) is a global attractor for all solutions of Eq.(2) . E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 259 2. Semi-Cycle Analysis Theorem 4. Assume that F : (0,∞)2 → (0,∞) is a continuous function such that F(x , y) is increasing in x for fixed y, and F(x , y) is increasing in y for fixed x . Let ex be a positive equilibrium of Eq.(1). Then, except possibly for the first semi-cycle, every oscillatory solution of Eq.(1) has semi-cycle of length at least k. Proof. We just give the proof of the theorem 4 for k = 2. The proof of the theorem 4 for k ≥ 3, is similar and omitted here. Let � xn be a solution of Eq.(1) with at least three semi-cycles. Then, there exists N ≥ 0 such that either xN+1 ≥ xN−1 ≥ ex , or xN−1 ≥ xN+1 ≥ ex . We first assume that xN+1 ≥ xN−1 ≥ ex . Since the function F(x , y) given by (7) is increasing in x for fixed y and increasing in y for fixed x , then we get xN+2 = F(xN+1, xN−1) = AxN+1+ BxN−1 + pxN+1+ xN−1 q+ xN−1 ≥ Aex + BxN−1 + pex + xN−1 q+ xN−1 = F(ex , xN−1)≥ F(ex , ex) = ex , and xN+3 = F(xN+2, xN )> F(ex , xN )> F(ex , ex) = ex f or xN > ex . Similarly, we can prove the theorem if xN−1 ≥ xN+1 ≥ ex which is omitted. Now, the proof of Theorem 4 is completed. 3. Local Stability In this section, we investigate the local stability of the positive solutions of Eq.(1). By using Theorems 1 and 3, we have the following result. Theorem 5. The zero equilibrium point ex = 0 is locally asymptotically stable if p − q < − � 1+ q (A+ B) � . In particular, if p− q ≥ − � 1+ q (A+ B) � , then ex = 0 is unstable. Proof. First, suppose that p− q < − � 1+ q (A+ B) � . Then, from Eq.(10) we deduce that ����A+ p q ����+ ����B+ 1 q ���� = (A+ B) + p+ 1 q < (A+ B) + q [1− (A+ B)] q = 1. E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 260 Thus ex = 0 is locally asymptotically stable. In particular, assume p− q ≥ − � 1+ q (A+ B) � , then we have ����A+ p q ����+ ����B+ 1 q ���� = (A+ B) + p+ 1 q ≥ (A+ B) + q [1− (A+ B)] q = 1. Thus ex = 0 is unstable. The proof of Theorem 6 is now completed. Theorem 6. If � p− q � > − � 1+ q (A+ B) � , 0< A+ B < 1, p > q and B > [1− (A+ B)] �� p− q � + q (A+ B) � � p+ 1 �� q+ 1 � . Then, the positive equilibrium point ex is locally asymptotically stable. Furthermore, the condition (12) can be considered as a necessary and sufficient condition for the asymptotically stability of Eq.(1). Proof. Under these assumptions we deduce from (9) that ��ρ0 ��+ ��ρ1 �� = ����A+ p [1− (A+ B)] p+ 1 ����+ ����B− [1− (A+ B)] �� p− q � + q (A+ B) � p+ 1 ���� = A+ p [1− (A+ B)] p+ 1 + B − [1− (A+ B)] �� p− q � + q (A+ B) � p+ 1 < (A+ B) � p+ 1 � + � p+ 1 � [1− (A+ B)] p+ 1 = 1. This proves that the positive equilibrium point ex of Eq.(1) is locally asymptotically stable. Thus, the condition (12) is sufficient for the asymptotic stability of Eq.(1). In addition to that condition, we see that if k is an odd positive integer and ρ1 = B− [1− (A+ B)] �� p− q � + q (A+ B) � p+ 1 > 0, or if k is an even positive integer and ρ0ρ1 = � A+ p [1− (A+ B)] p+ 1 �� B− [1− (A+ B)] �� p− q � + q (A+ B) � p+ 1 � > 0, then the condition (12) is also necessary for the asymptotic stability of Eq.(1). According to Theorem 2, the proof of Theorem 7 is now completed. 4. Periodic Solutions In this section, we investigate the periodic character of the positive solutions of Eq.(1). E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 261 Theorem 7. If k is an even positive integer, then Eq.(1) has no positive solutions of prime period two for all A, B,p,q ∈ (0,∞). Proof. Assume for the sake of contradiction that there exists distinctive positive real num- bers Φ and Ψ, such that . . .Φ,Ψ,Φ,Ψ, . . . is a prime period two solution of Eq.(1). If k is even, then xn = xn−k. It follows from the difference equation (1) that Φ = (A+ B)Ψ+ � p+ 1 � Ψ q+Ψ and Ψ = (A+ B)Φ+ � p+ 1 � Φ q+Φ . Consequently, we obtain qΦ+ΦΨ = qAΨ+ AΨ2+ qBΨ+ BΨ2 + pΨ+Ψ, and qΨ+ΦΨ = qAΦ+ AΦ2+ qBΦ+ BΦ2 + pΦ+Φ. By subtracting, we deduce that (Φ−Ψ) � q (A+ B+ 1)+ (Φ+Ψ)(A+ B) + p+ 1 = 0. This implies Φ = Ψ. This contradicts the hypothesis Φ 6= Ψ. Thus, the proof of Theorem 7 is completed. Theorem 8. If k is an odd positive integer then for all A, B,p,q ∈ (0,∞) Eq.(1) has no prime period two solutions if A− B+ 1> 0. Proof. Assume for the sake of contradiction that there exists distinctive positive real num- bers Φ and Ψ, such that . . .Φ,Ψ,Φ,Ψ, . . . is a prime period two solution of Eq.(1). If k is odd, then yn+1 = yn−k. It follows from Eq.(1) that Φ = AΨ+ BΦ+ pΨ+Φ q+Φ and Ψ = AΦ+ BΨ+ pΦ+Ψ q+Ψ . Consequently, we obtain qΦ+Φ2 = qAΨ+ AΦΨ+ qBΦ+ BΦ2 + pΨ+Φ, (13) and qΨ+Ψ2 = qAΦ+ AΦΨ+ qBΨ+ BΨ2 + pΦ+Ψ. (14) By subtracting (13) from (14), we deduce that Φ+Ψ = �� p+ q � − � q (B− A) + 1 � B − 1 , (15) E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 262 while, by adding (13), (14) and using (15) we get ΦΨ = � qA+ p ��� p+ q � − � q (B− A) + 1 � (B− 1) [B− (A+ 1)] . (16) From (15) and (16) we have ΦΨ(Φ+Ψ) = − � qA+ p � (1+ A− B) ¨� p+ q � − � 1+ q (B− A) � B− 1 «2 < 0. (17) This contradicts the hypothesis that both Φ,Ψ are positive. Thus, the proof of Theorem 8 is now completed. 5. Boundedness Character In this section, we investigate the boundedness character of the positive solutions of Eq.(1). Theorem 9. Let � xn ∞ n=−k be a solution of Eq.(1). Then the following statements are true: (i) Suppose p < q and for some N ≥ 0, the initial conditions xN−k+1, . . . xN−1, xN ∈ � p q , 1 � , then xn ∈ � p q � A+ B+ p+ 1 q+ 1 � , q p (A+ B+ 1) � , for all n≥ N . (ii) Suppose p > q and for some N ≥ 0, the initial conditions xN−k+1, . . . xN−1, xN ∈ � 1, p q � , then xn ∈ � q p (A+ B+ 1) , p q � A+ B+ p+ 1 q+ 1 � � , for all n≥ N . Proof. First of all, if for some N ≥ 0 and p q ≤ xN ≤ 1 and p < q, then xn+1 = Axn+ Bxn−k + pxn + xn−k q+ xn−k ≤ Axn+ Bxn−k + qxn+ xn−k q+ xn−k ≤ A+ B + 1≤ q p (A+ B + 1) , and xn+1 = Axn+ Bxn−k + pxn + xn−k q+ xn−k ≥ p q � A+ B+ p+ 1 q+ 1 � . E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 263 Thus, the proof of part (i) is completed. Secondly, if for some N ≥ 0 and 1 ≤ xN ≤ p q and p > q, then xn+1 = Axn+ Bxn−k + pxn + xn−k q+ xn−k ≤ p q � A+ B+ p+ 1 q+ 1 � , and xn+1 = Axn+ Bxn−k + pxn + xn−k q+ xn−k ≥ Axn+ Bxn−k + qxn+ xn−k q+ xn−k ≥ A+ B+ 1 ≥ q p (A+ B+ 1) . Thus, the proof of part (ii) is completed. The proof of Theorem 9 is now finished. 6. Global Stability In this section, we investigate the global stability of the positive solutions of Eq.(1). Theorem 10. If p− q < − � 1+ q (A+ B) � , then, the zero equilibrium point ex = 0 of Eq.(1) is globally asymptotically stable. Proof. Under this condition, we have shown in Theorem 5 that ex = 0 is locally asymptoti- cally stable. It remains to prove that ex = 0 is a global attractor. To this end, we consider the function F(x , y) = Ax + B y + px + y q+ y . (18) We note that the function (18) is continuous and satisfying the following conditions: (i) F(x , y) is nondecreasing in x ∈ [ q p ,∞) for fixed y > −q. (ii) F(x , y) satisfies the inequality [F(x , x)− x][x − ex]< 0 for ex = 0. Let us now prove (ii) as follows: [F(x , x)− x] [x − 0] = � (A+ B) x + � p+ 1 � x q+ x − x � x . Since x ∈ [ q p ,∞), then p+1 q+x < p q and we have [F(x , x)− x][x − 0] < � (A+ B)q+ � p− q � q � x2 < � (A+ B)q− � 1+ q (A+ B) q � x2 = − x2 q < 0. E. Zayed / Eur. J. Pure Appl. Math, 3 (2010), 254-268 264 According to Theorem 4, the zero equilibrium point ex = 0 is a global attractor. The proof of Theorem 11 is now completed. Theorem 11. Assume that p − q > − � 1+ q (A+ B) � , p > q, 0 < A + B < 1 and B > [1−(A+B)][(p−q)+q(A+B)] p+1 , then, the positive equilibrium point ex of Eq.(1) is globally asymptot- ically stable. Proof. Under these assumptions, we have shown in Theorem 6 that the positive equilib- rium point ex of Eq.(1) is locally asymptotically stable. It remains to prove that the positive equilibrium point ex is a global attractor. To this end, we consider the function F � x , y � given by (18) which satisfies the following conditions: (i) F(x , y) is nondecreasing in x ∈ [ q p ,∞) for fixed y > −q. (ii) F(x , y) satisfies the inequality [F(x , x)− x][x − ex]< 0, where ex given by (6. Let us now prove the inequality (ii) using Lemma 1 as follows: [F(x , x)− x][x − ex] = � (A+ B) + � p+ 1 � q+ x − 1 �� x2− xex � < � (A+ B) + � p− q � q �� q2 p2 − q p ex � = − q p � (A+ B) + � p− q � q �� ex − q p � < 0. According to Theorem 3, the positive equilibrium point ex is a global attractor. The proof of Theorem 11 is now completed. 7. Numerical Examples In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These exam- ples represent different types of qualitative behavior of solutions to the nonlinear difference equation (1). REFERENCES 265 Example 1. Figure 1 shows that the solution of Eq.(1) is global stability if k = 1, x−1 = 1, x0 = 2, A= 0.25, B = 0.3, p = 2, q = 1, (p > q). 0 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 n−iteration so lu tio n of X (n + 1) = (A *X (n )+ B *X (n − 1) )+ (( p* X (n )+ X (n − 1) )/ (q + X (n − 1) )) plot of X(n+1)=(A*X(n)+B*X(n−1))+((p*X(n)+X(n−1))/(q+X(n−1))) Figure 1: xn+1 = 0.25xn + 0.3xn−1+ 2xn+xn−1 1+xn−1 Example 2. Figure 2 shows that the solution of Eq.(1) is global stability if k = 2, x−2 = 1, x−1 = 2, x0 = 3, A= 0.25, B = 0.3, p = 20, q = 5, (p > q). 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 n−iteration so lu tio n of X (n + 1) = (A *X (n )+ B *X (n − 2) )+ (( p* X (n )+ X (n − 2) )/ (q + X (n − 2) )) plot of X(n+1)=(A*X(n)+B*X(n−2))+((p*X(n)+X(n−2))/(q+X(n−2))) Figure 2: xn+1 = 0.25xn + 0.3xn−2 + 20xn+xn−2 5+xn−2 References [1] M. T. Aboutaleb, M. A. El-Sayed and A. E. Hamza, Stability of the recursive sequence xn+1 = (α− β xn)/(γ+ xn−1), J. Math. Anal. 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