Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 930410, 11 pages doi:10.1155/2012/930410 Research Article Global Attractivity of a Higher-Order Difference Equation R. Abo-Zeid Department of Mathematics, College of Science & Arts-Rabigh, King Abdulaziz University, Rabigh 21911, Saudi Arabia Correspondence should be addressed to R. Abo-Zeid, abuzead73@yahoo.com Received 7 May 2012; Accepted 8 July 2012 Academic Editor: M. De la Sen Copyright q 2012 R. Abo-Zeid. 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. The aim of this work is to investigate the global stability, periodic nature, oscillation, and the boundedness of all admissible solutions of the difference equation xn1 Axn−2r−1 /B − C kil xn−2i , n 0, 1, 2, . . . where A, B, C are positive real numbers and l, r, k are nonnegative integers, such that l ≤ k. 1. Introduction and Preliminaries Although some difference equations look very simple, it is extremely difficult to understand thoroughly the global behaviors of their solutions. One can refer to 1, 2. The study of nonlinear rational difference equations of higher order is of paramount importance, since we still know so little about such equations. It is worthwhile to point out that although several approaches have been developed for finding the global character of difference equations 2– 4, relatively a large number of difference equations have not been thoroughly understood yet 5–8. Aloqeili in 9 discussed the stability properties and semicycle behavior of the solutions of the difference equation: xn1 xn−1 , a − xn xn−1 n 0, 1, 2, . . . , with real initial conditions and positive real number a. 1.1 2 Discrete Dynamics in Nature and Society In 10, the authors investigated the global asymptotic stability of the difference equation: xn1 Axn−1 , B C kil xn−2i n 0, 1, 2, . . . , 1.2 where A, B, C are nonnegative real numbers and l, k are nonnegative integers, such that l ≤ k. Also in 11, they discussed the existence of unbounded solutions under certain conditions of the difference equation: k xn1 A il xn−2i−1 , k−1 B C il xn−2i n 0, 1, 2, . . . , 1.3 where A, B, C are nonnegative real numbers and l, k are nonnegative integers, l < k In 12, the global asymptotic stability of the difference equation: xn1 Axn−2r−1 , B Cxn−2l xn−2k n 0, 1, 2, . . . 1.4 was discussed, where A, B, C are nonnegative real numbers and r, l, k are nonnegative integers such that l ≤ k and r ≤ k. In 13, the global stability and periodic nature of the solutions of the difference equations: xn1 xn−2 , ±1 xn xn−1 xn−2 n 0, 1, 2, . . . 1.5 were discussed, where the initial conditions x−2 , x−1 , x0 are real numbers. In 14, we discussed the oscillation, boundedness, and the global behavior of all admissible solutions of the difference equation: xn1 Axn−1 , B − Cxn xn−2 n 0, 1, 2, . . . , 1.6 where A, B, C are positive real numbers. In this paper, we study the global asymptotic stability of the difference equation xn1 Axn−2r−1 , B − C kil xn−2i n 0, 1, 2, . . . , 1.7 where A, B, C are nonnegative real numbers and l, r, k are nonnegative integers, such that l ≤ k. Discrete Dynamics in Nature and Society 3 Consider the difference equation: xn1 fxn , xn−1 , . . . , xn−k , n 0, 1, 2 . . . , 1.8 where f : Rk1 → R. An equilibrium point for 1.8 is a point x ∈ R such that x fx, x, . . . , x. 1 An equilibrium point x for 1.8 is called locally stable if for every > 0, ∃δ > 0 such that every solution {xn } with initial conditions x−k , x−k1 , . . . , x0 ∈x − δ, x δ is such that xn ∈x − , x , ∀n ∈ N. Otherwise x is said to be unstable. 2 The equilibrium point x of 1.8 is called locally asymptotically stable if it is locally stable and there exists γ > 0 such that for any initial conditions x−k , x−k1 , . . . , x0 ∈ x − γ, x γ, the corresponding solution {xn } tends to x. 3 An equilibrium point x for 1.8 is called global attractor if every solution {xn } converges to x as n → ∞. 4 The equilibrium point x for 1.8 is called globally asymptotically stable if it is locally asymptotically stable and global attractor. The linearized equation associated with 1.8 is yn1 k ∂f x, . . . , xyn−i , ∂x n−i i0 n 0, 1, 2, . . . . 1.9 The characteristic equation associated with 1.9 is λk1 − k ∂f x, . . . , xλk−i 0. ∂x n−i i0 1.10 Theorem 1.1 see 2. Assume that f is a C1 function and let x be an equilibrium point of 1.8. Then the following statements are true. 1 If all roots of 1.10 lie in the open disk |λ| < 1, then x is locally asymptotically stable. 2 If at least one root of 1.10 has absolute value greater than one, then x is unstable. 2. Linearized Stability Analysis Consider the difference equation: xn1 Axn−2r−1 , B − C kil xn−2i n 0, 1, 2, . . . , 2.1 where A, B, C are nonnegative real numbers and l, r, k are nonnegative integers, such that l ≤ k. 4 Discrete Dynamics in Nature and Society The change of variables xn yn1 k−l1 B/Cyn reduces 1.7 to the difference equation: γyn−2r−1 , 1 − kil yn−2i n 0, 1, 2, . . . , 2.2 where γ A/B. Now we determine the equilibrium points of 2.2 and discuss their local asymptotic behavior. It is clear that the values of the equilibrium points depend on whether k − l is even or odd. When k − l is odd, we have the equilibrium points y 0 and y ± k−l1 1 − γ if γ < 1 and y 0 only if γ ≥ 1. When k − l is even, we have the equilibrium points y 0 and y k−l1 1 − γ. Now assume that K max{2k, 2r 1}. The linearized equation associated with 2.2 about y is zn1 − γ 1−y γy k−l1 z − k−l1 n−2r−1 1 − y k−l1 k 2 zn−2i 0, n 0, 1, 2, . . . . il 2.3 The characteristic equation associated with this equation is λK1 − γy k−l1 γ 1−y λK−2r−1 − k−l1 1−y k−l1 2 k λK−2i 0. il 2.4 We summarize the results of this section in the following two theorems. Theorem 2.1. Assume that K 2k > 2r 1. Then the following statements are true. 1 The zero equilibrium point is locally asymptotically stable if γ < 1 and unstable (saddle point) if γ > 1. 2 When k − l is even, the equilibrium point y k−l1 1 − γ is unstable if γ < 1 and unstable (saddle point) if γ > 1. 3 When k − l is odd, the equilibrium points y ± k−l1 1 − γ are unstable. Proof. 1 The linearized equation 2.3 about y 0 is zn1 − γzn−2r−1 0, n 0, 1, 2, . . . . 2.5 The characteristic equation associated with this equation is λ2k1 − γλ2k−2r−1 0. √ So λ 0, λ ± 2r2 γ. Therefore the result follows. 2.6 Discrete Dynamics in Nature and Society 5 2 Suppose that k − l is even. The linearized equation 2.3 about y zn1 − zn−2r−1 − k 1 zn−2i 0, 1−γ γ il k−l1 n 0, 1, 2, . . . . 1 − γ is 2.7 The associated characteristic equation 2.4 becomes k 1 λ2k−2i 0. 1−γ γ il λ2k1 − λ2k−2r−1 − 2.8 Let fλ λ2k1 − λ2k−2r−1 − k 1 λ2k−2i 0. 1−γ γ il 2.9 We can see that fλ has a real root in 1, ∞ if γ < 1 and when γ > 1, fλ has a root in 1, ∞ and some roots with |λ| < 1. Therefore the result follows. 3 Whenk − l is odd, fλ has a root in 1, ∞ and some roots with |λ| < 1, if γ < 1. Therefore y ± 1 − γ are unstable. Theorem 2.2. Assume that K 2r 1 > 2k. Then the following statements are true. 1 The zero equilibrium point is locally asymptotically stable if γ < 1 and a source if γ > 1. 2 If k − l is even, then the equilibrium point y k−l1 1 − γ is unstable (saddle point). 3 If k − l is odd, then the equilibrium points y ± k−l1 1 − γ are unstable (saddle points). Proof. It is sufficient to consider the linearized equation about y: zn1 − γy k−l1 γ 1−y z − k−l1 n−2r−1 1 − yk−l1 k 2 zn−2i 0, n 0, 1, 2, . . . il 2.10 and its associated characteristic equation: λ2r2 − γ 1 − y k−l1 γy k−l1 − 1 − y k−l1 2 k λ2r1−2i 0. il 2.11 3. Oscillation Let t be the largest nonnegative integer such that 0 < 2t 1 ≤ K and let s be the largest nonnegative integer such that 0 ≤ 2s ≤ K. 6 Discrete Dynamics in Nature and Society Theorem 3.1. Assume that γ < 1. Then the interval − k−l1 1 − γ, k−l1 1 − γ is an invariant interval for 2.2. Proof. The proofis by induction. Suppose that y−i ∈ − k−l1 1 − γ, k−l1 1 − γ, i 0, 1, . . . , K. Hence |y−i | < k−l1 1 − γ, i 0, 1, . . . , K. This implies that | kil y−2i | < 1 − γ. Then γ y−2r−1 γ y−2r−1 y1 ≤ < y−2r−1 , 1 − kil y−2i 1 − kil y−2i γ y−2r γ y−2r y2 ≤ < y−2r . k k 1 − il y−2i1 1 − il y−2i1 If for a certain n0 ∈ N we have yn0 −K , yn0 −K1 , . . . , yn0 ∈ − k−l1 1 − γ, k−l1 γ yn− 2r−1 γ yn0 −2r−1 ≤ < yn0 −2r−1 < 0 1 k k 1 − il y−2i 1 − il y−2i yn 3.1 1 − γ, then k−l1 1 − γ. 3.2 This completes the proof. ∞ Corollary of 2.2 such that either y−K , y−K1 , . . . , y−1 , y0 ∈ 3.2. Assume that {yn }n−K be a solution ∞ k−l1 k−l1 1 − γ or− 1 − γ, 0. Then {yn }∞ 0, n−K is positive (or negative). Moreover, {yn }n−K converges to the zero equilibrium point. Theorem 3.3. Let {yn }∞ n−K be a nontrivial solution of 2.2 such that either C1 − k−l1 1 − γ < y−2t−1 , y−2t1 , . . . , y−1 < 0 < y−2s , y−2s2 , . . . , y0 < k−l1 1−γ or C2 − k−l1 1 − γ < y−2s , y−2s2 , . . . , y0 < 0 < y−2t−1 , y−2t1 , . . . , y−1 < k−l1 1 − γ is satisfied. Then {yn }∞ n−K oscillates about y 0 with semicycles of length one. Moreover y2r1n2j < or >y2r1n−12j and y2r1n2j1 > or < y2r1n−12j1 , j 1, 2, . . . , r 1 and n 0, 1, 2, . . .. Proof. Assume that condition C1 is satisfied. Then we have y1 γy−2r−1 /1 − kil y−2i > y−2r−1 , and y2 γy−2r /1 − kil y−2i1 < y−2r . By induction we get 0 > y2r1n2j1 > y2r1n−12j1 , and 0 < y2r1n2j < y2r1n−12j , n 0, 1, 2, . . .. If condition C2 is satisfied, the result is similar and will be omitted. Discrete Dynamics in Nature and Society 7 4. Global Behavior of 2.2 Theorem 4.1. The following statements are true. 1 If γ < 1, then the zero equilibrium point is a global attractor with basin − k−l1 1 − γ, k−l1 1 − γK1 . 2 If γ 1, then 2.2 has prime period two solutions of the form . . . , 0, ϕ, 0, ϕ, 0, . . ., where ϕ ∈ R. 3 If γ > 1, then there exist solutions which are neither bounded nor persist. , . . . , y−1 , y ∈ − k−l1 1 − γ, k−l1 1 − γ. Then using Proof. 1 Suppose that y−K , y−K1 0 Theorem 3.1, we have that yn ∈ − k−l1 1 − γ, k−l1 1 − γ, n ≥ 1. Moreover, we have |yn1 | < |yn−2r−1 |, n 0, 1, 2, . . .. That is, the subsequences {|y2r1nj |}∞ n−1 , j 1, 2, . . . , 2r 2 are decreasing. From 2.2 we have γ y2r1n−1j γ y2r1n−1j y2r1nj . ≤ 1 − kil y2r1nj−2i−1 1 − kil y2r1nj−2i−1 4.1 Now suppose that |y2r1nj | → Lj as n → ∞, j 1, 2, . . . , 2r 2. Then the last inequality implies that γLj , Lj ≤ 1 − kil Lj−2i−1 j 1, 2, . . . , 2r 2. If for a certain j ∈ {1, 2, . . . , 2r 2} we have Lj / 0, then |1 − 1γ ≥ k Lj−2i−1 ≥ 1 − γ. k il Lj−2i−1 | 4.2 ≤ γ. This implies that 4.3 il This is a contradiction as the the subsequences {|y2r1nj |}∞ n−1 , j 1, 2, . . . , 2r 2 are decreasing. Therefore, Lj 0, j 1, 2, . . . , 2r 2, and {yn }∞ n−K converges to zero. 2 Clear! with initialconditions, |y−i | < k−l1 γ − 1 > 3 Let {yn }∞ n−K be a solution of 2.2 k−l1 γ 1, i 2s, 2s − 2, . . . , 2, 0, and |y−i | >k−l1 γ 1 < k−l1 γ − 1, i 2t − 1, 2t − 3, . . . , 1. We consider only the case |y−i | < k−l1 γ − 1, i 2s, 2s − 2, . . . 2, 0, and |y−i | > k−l1 γ 1, i 2t − 1, 2t − 3, . . . , 1. It follows that | kil y−2i | |y−2k ||y−2k2 | · · · |y0 | < γ − 1. That is, −γ 1 < k il y−2i < γ − 1. 4.4 8 Discrete Dynamics in Nature and Society This implies that −γ 2 < 1 − k il y−2i < γ and so |1 − k il y−2i | γy−2r−1 γ y−2r−1 y1 > y−2r−1 > γ 1 − kil y−2i k−l1 < γ. Hence we have γ 1. 4.5 Also | kil y−2i1 | > γ 1 implies that γ 2 < 1− kil y−2i1 < −γ and so |1− kil y−2i1 | > γ. Hence we have γy−2r γ y−2r y2 < y−2r < γ 1 − kil y−2i1 γ − 1. k−l1 4.6 By induction we get y2r1n2j1 > y2r1n−12j1 > y2r1n2j < y2r1n−12j < γ 1, k−l1 γ − 1, 4.7 k−l1 n ≥ 0 and j 0, 1, . . . , r. Now suppose that y2r1n−12j1 −→ L2j1 ∈ γ 1, ∞ , k−l1 y2r1n−12j −→ L2j ∈ 0, γ −1 , 4.8 k−l1 as n → ∞, j 0, 1, . . . , r. But as γ y2r1n−12j γ y2r1n−12j y2r1n2j ≤ , 1 − kil y2r1n2j−2i−1 1 − kil y2r1nj−2i−1 4.9 then γL2j , L2j ≤ 1 − kil L2j−2i−1 j 0, 1, . . . , r. 4.10 We claim that for each j 0, 1, . . . , r, L2j 0. For the sake of contradiction suppose that there exists j ∈ {0, 1, . . . , r} with L2j ∈ 0, k−l1 γ − 1. Then 4.10 gives k L2j−2i−1 ≤ γ. 1 − il 4.11 Discrete Dynamics in Nature and Society 9 10 8 6 4 2 0 −2 0 10 20 30 40 50 Figure 1: The difference equation yn1 0.6yn−1 /1 − yn yn−2 . 1 0.5 0 −0.5 −1 0 10 20 30 40 50 Figure 2: The difference equation yn1 0.7yn−3 /1 − yn yn−2 . This implies that −γ 1 ≤ k 4.12 L2j−2i−1 ≤ γ 1. il As L2j1 ∈ k−l1 γ 1, ∞, j 0, 1, . . . , r, we have a contradiction. Thus it is true that for each j 0, 1, . . . , r we have L2j 0 and so limn → ∞ y2n 0. We now claim that for each j 0, 1, . . . , r, L2j ∞. For the sake of contradiction, suppose that there exists j ∈ {0, 1, . . . , r} with L2j1 ∈ k−l1 γ 1, ∞. Then L2j1 γlimn → ∞ y2r1n−12j1 lim y2r1n−12j1 n→∞ 1 − kil limn → ∞ y2r1n2j−2i γlimn → ∞ y2r1n−12j1 γL2j1 ≥ γL2j1 . k 1 limn → ∞ y2r1n2j−2i 1 k L2j il il 4.13 10 Discrete Dynamics in Nature and Society ×105 Global attractivity of a higher 10 8 6 4 2 0 0 10 20 30 40 Figure 3: The difference equation yn1 2yn−1 /1 − yn yn−2 . 2000 1500 1000 500 0 0 10 20 30 40 Figure 4: The difference equation yn1 2yn−3 /1 − yn yn−2 . This is a contradiction. Therefore for each j 0, 1, . . . , r we have L2j1 ∞ and so limn → ∞ y2n1 ∞. The case when |y−i | > k−l1 γ 1, i 2s, 2s − 2, . . . , 2, 0 and |y−i | < k−l1 γ − 1, i 2t − 1, 2t − 3, . . . , 1 is similar and will be omitted. 5. Numerical Examples Example 5.1. Figure 1 shows that if r 0, l 0, k 1 K max{2k, 2r 1} 2 and γ 0.6, then the solution {yn }∞ n−2 with initial conditions y−2 −1, y−1 1.3, y0 1.1 converges to zero. Example 5.2. Figure 2 shows that if r 1, l 0, k 1 K max{2k, 2r 1} 3 and γ 0.7, then the solution {yn }∞ n−3 with initial conditions y−3 −1, y−2 −1.2, y−1 1.3, y0 1.1 converges to zero. Example 5.3. Figure 3 shows that if r 0, l 0, k 1 K max{2k, 2r 1} 2 and γ 2, then the solution {yn }∞ n−3 with initial conditions y−2 2, y−1 0.4, y0 2.1 is unbounded. Discrete Dynamics in Nature and Society 11 Example 5.4. Figure 4 shows that if r 1, l 0, k 1 K max{2k, 2r 1} 3 and γ 2, then the solution {yn }∞ n−3 with initial conditions y−3 0.5, y−2 2, y−1 0.4, y0 2.1 is unbounded. Acknowledgment This paper was funded by the Deanship of the Scientific Research DSR, King Abdulaziz University, Jeddah, under Grant no. 15-662-D1432. The author, therefore, acknowledge with thanks DSR technical and financial support. References 1 R. P. Agarwal, Difference Equations and Inequalities, vol. 155, Marcel Dekker, New York, NY, USA, 1st edition, 1992. 2 V. L. Kocić and G. 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