ON THE RATIONAL RECURSIVE SEQUENCE X N+1 = γx N K + (AX N + BX N K ) / (CX N DX N K ) Communicated by Mohammad Asadzadeh. 1.

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1 Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp ON THE RATIONAL RECURSIVE SEQUENCE X N+1 γx N K + (AX N + BX N K ) / (CX N DX N K ) E.M.E. ZAYED AND M.A. EL-MONEAM* Communicated by Mohammad Asadzadeh Abstract. Our main objective is to study some qualitative behavior of the solutions of the difference equation x n+1 γx n k + (ax n + bx n k ) / (cx n dx n k ), n 0, 1, 2,..., where the initial conditions x k,..., x 1, x 0 are arbitrary positive real numbers the coefficients γ, a, b, c d are positive constants, while k is a positive integer number. 1. Introduction Our goal is to investigate some qualitative behavior of the solutions of the difference equation, (1.1) x n+1 γx n k + ax n + bx n k cx n dx n k, n 0, 1, 2,..., where the initial conditions x k,..., x 1, x 0 are arbitrary positive real numbers the coefficients γ, a, b, c d are positive constants, while k is a positive integer number. The case where any of α, a, c, d is allowed to be zero gives different special cases of the equation (1) which are studied by many authors (see for example, 3, 6, 8, 12, 17, 29]). For MSC(2000): Primary: 39A10, 39A11, 39A99; Secondary: 34C99. Keywords: Difference equations, prime period two solution, locally asymptotically stable, global attractor, convergence. Received: 15 July 2008, Accepted: 07 March Corresponding author c 2010 Iranian Mathematical Society. 103

2 104 Zayed El-Moneam related work, see also 1, 2, 4, 5, 7, 9-11, 13-16, 18-28, 30 40]. The study of these equations is challenging rewarding still in its infancy. We believe that the nonlinear rational difference equations are of paramount importance in their own rights. Furthermore, the results about such equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations. Note that Eq. (1.1) can be considered as a generalization of that obtained in 8, 33]. Definition 1.1. A difference equation of order (k + 1) is of the form (1.2) x n+1 F (x n, x n 1,..., x n k ), n 0, 1, 2,..., where F is a continuous function which maps some set J k+1 into J J is a set of real numbers. An equilibrium point x of this equation is a point that satisfies the condition x F ( x, x,..., x). That is, the constant sequence {x n } n k with x n x, for all n k, is a solution of that equation. Definition 1.2. Let x (0, ) be an equilibrium point of the difference equation (1.2). Then, we have: (i) An equilibrium point x of the difference equation (1.2) is called locally stable if for every ε > 0 there exists δ > 0 such that, if x k,..., x 1, x 0 (0, ) with x k x x 1 x + x 0 x < δ, then x n x < ε, for all n k. (ii) An equilibrium point x of the difference equation (1.2) is called locally asymptotically stable if it is locally stable there exists γ > 0 such that, if x k,..., x 1, x 0 (0, ) with x k x x 1 x + x 0 x < γ, then lim n x n x. (iii) An equilibrium point x of the difference equation (1.2) is called a global attractor if x k,..., x 1, x 0 (0, ), then lim x n x. n (iv) An equilibrium point x of the equation (1.2) is called globally asymptotically stable if it is locally stable a global attractor. (v) An equilibrium point x, of the difference equation (2) is called unstable if it is not locally stable.

3 On the rational recursive sequence 105 Definition 1.3. A sequence {x n } n k is said to be periodic with period p if x n+p x n, for all n k. A sequence {x n } n k is said to be periodic with prime period p if p is the smallest positive integer having this property. The linearized equation of the difference equation (1.2) about the equilibrium point x is the linear difference equation, (1.3) y n+1 k i0 F ( x, x,..., x) x n i y n i. Now, assume that the characteristic equation associated with (1.3) is (1.4) p (λ) p 0 λ k + p 1 λ k p k 1 λ + p k 0, where, p i F ( x, x,..., x) / x n i. Theorem 1.1. (See 18]). Assume that p i R, i 1, 2,..., k {0, 1, 2,...}. Then, (1.5) k p i < 1 i1 is a sufficient condition for the asymptotic stability of the difference equation, (1.6) x n+k + p 1 x n+k p k x n 0, n 0, 1, 2,... Theorem 1.2. (The linearized stability theorem; see15,18,19]). Suppose F is a continuously differentiable function defined on an open neighbourhood of the equilibrium x. Then, the following statements are true. (i) If all roots of the characteristic equation (1.4) of the linearized equation (1.3) have absolute value less than one, then the equilibrium point x is locally asymptotically stable. (ii) If at least one root of Eq.(1.4) has absolute value greater than one, then the equilibrium point x is unstable. The following Theorem will be useful for the proof of our main results.

4 106 Zayed El-Moneam Theorem 1.3. (See15, p.18]). Let F : a, b] k+1 a, b] be a continuous function, where k is a positive integer, a, b] is an interval of real numbers consider the difference equation (1.2). Suppose that F satisfies the following conditions: (i) For every integer i with 1 i k+1, the function F (z 1, z 2,..., z k+1 ) is weakly monotonic in z i, for fixed z 1, z 2,..., z i 1, z i+1,..., z k+1. (ii) If (m, M) is a solution of the system (1.7) m F (m 1, m 2,..., m k+1 ) M F (M 1, M 2,..., M k+1 ), then m M, where for each i 1, 2,..., k + 1, we set m if F is nondecreasing in z i m i M if F is nonincreasing in z i M if F is nondecreasing in z i M i m if F is nonincreasing in z i. Then, there exists exactly one equilibrium point x of the difference equation (1.2), every solution of (1.2) converges to x. 2. Periodic solutions Theorem 2.1. If k is an even positive integer c d, then Eq. (1.1) has no positive solution of prime period two. Proof. Assume that there exists a distinctive positive solution..., P, Q, P, Q,... of prime period two of Eq. (1.1). If k is even, then x n x n k. It follows from Eq. (1.1) that P γq + aq + bq cq dq Q γp + ap + bp cp dp, provided that c d. Hence, we have (P Q) (γ + 1) 0. Thus, P Q, which is a contradiction. The proof of Theorem 2.1 is now complete.

5 On the rational recursive sequence 107 Theorem 2.2. If k is an odd positive integer, γ > 1, b > a, then the difference equation (1.1) has no positive solution of prime period two. Proof. Assume that there exists a distinctive positive solution..., P, Q, P, Q,... of prime period two of Eq. (1.1). If k is odd, then x n+1 x n k. It follows from the difference equation (1.1) that P γp + Consequently, we obtain: aq + bp cq dp Q γq + ap + bq cp dq. (2.1) cp Q dp 2 γcp Q γdp 2 + aq + bp, (2.2) cp Q dq 2 γcp Q γdq 2 + ap + bq. By subtracting, we deduce: (2.3) P + Q b a d (γ 1), while by adding we obtain: (2.4) P Q a (b a) d (c + d) (γ 1) 2. Since γ > 1 b > a, then P Q is negative. But P Q are both positive, we have a contradiction. Therefore, the proof of Theorem 2.2 is complete. Theorem 2.3. If k is an odd positive integer, then the necessary sufficient condition for the difference equation (1.1) to have a positive prime period two solution is that the inequality (2.5) (c + d) (a b) > 4ad

6 108 Zayed El-Moneam is valid, provided that a > b 0 < γ < 1. Proof. First, suppose that there exists a positive prime period two solution..., P, Q, P, Q,... of the difference equation (1.1). If k is odd, then x n+1 x n k. We shall prove that the condition (2.5) holds. It follows from the difference equation (1.1) that aq + bp P γp + cq dp Consequently, we have, Q γq + ap + bq cp dq. (2.6) cp Q dp 2 γcp Q γdp 2 + aq + bp, (2.7) cp Q dq 2 γcp Q γdq 2 + ap + bq. By subtracting (2.6) from (2.7), we deduce: (2.8) P + Q a b d (1 γ). while, by adding (2.6) (2.7), we have (2.9) P Q a (a b) d (c + d) (1 γ) 2, where a > b 0 < γ < 1. Assume that P Q are two positive distinct real roots of the quadratic equation, (2.10) t 2 ( P + Q) t + P Q 0. Thus, we deduce: (2.11) ( ) a b 2 > d (1 γ) 4a (a b) d (c + d) (1 γ) 2. From (2.11), we obtain the condition (2.5). Thus, the necessary condition is satisfied. Conversely, suppose that the condition (2.5) is valid. Then, we deduce immediately from (2.5) that the inequality (2.11) holds.

7 On the rational recursive sequence 109 Consequently, there exist two positive distinct real numbers P Q such that (2.12) P (a b) + β 2d (1 γ) Q (a b) β 2d (1 γ), where β (a b) 2 4ad (a b) / (c + d). Thus, P Q represent two positive distinct real roots of the quadratic equation (2.10). Now, we prove that P Q form a positive prime period two solution of the difference equation (1.1). To this end, we assume that x k P, x k+1 Q,..., x 1 P, x 0 Q. We wish to show that x 1 P x 2 Q. To this end, we deduce from the difference equation (1.1) that (2.13) x 1 γx k + ax 0 + bx k aq + bp γp + cx 0 dx k cq dp. Thus, we deduce from (2.12) (2.13) that (2.14) aq + bp x 1 P (1 γ) P cq dp aq + bp c (1 γ) P Q + d (1 γ) P 2 cq dp ] ] ] a (a b) β +b (a b)+β c (1 γ) +d (1 γ) d(c+d)(1 γ) ] ] 2 c d. (a b) β (a b)+β ] 2 (a b)+β

8 110 Zayed El-Moneam Multiplying the denominator numerator of (2.14) by 4d 2 (1 γ) 2, we get: (2.15) x 1 P 4d 2 (1 γ) 2 ( a ] (a b) β + b 4d 2 (1 γ) 2 ( c 4d 2 (1 γ) (d 2 (1 γ) + ( 4d 2 (1 γ) 2 c (a b) β ] (a b)+β ] (a b) β (a b)+β (a b)+β ] d d ] ) 2 c (1 γ) ]) (a b)+β 2ad (1 γ) (a b) β]+2bd (1 γ) (a b)+β] 4cd (1 γ) 2cd (1 γ) (a b) β] 2d 2 (1 γ) (a b)+β] d (1 γ) (a b) + β] 2 + 2cd (1 γ) (a b) β] 2d 2 (1 γ) (a b) + β] 2a (a b) β] + 2b (a b) + β] 4c (c+d) 2 c (a b) β] d (a b) + β]] 2 (a b) (a+b) 2 (a b) β 4c 2 ( a 2 b 2) 4c (c+d) 2 c (a b) β] d (a b) + β]] ] + (a b) 2 + β 2 (c+d) 2 c (a b) β] d (a b) + β]] 2 ( a 2 b 2) 4c (c+d) ]) d(c+d)(1 γ) 2 ]) ] (c+d) ] + (a b) + β] 2 ] +(a b) 2 +2 (a b) β+β 2 ] + (a b) 2 + (a b) 2 4ad(a b) c+d 2 c (a b) β] d (a b) + β]] Thus, x 1 P. Similarly, we can show, x 2 γx 1 k + ax 1 + bx 1 k ap + bq γq + cx 1 dx 1 k cp dq Q. Using the mathematical induction, we have, x n P x n+1 Q, for all n k. 0. Thus, the difference equation (1.1) has a positive prime period two solution,..., P, Q, P, Q,... Hence, the proof is now complete.

9 On the rational recursive sequence Local stability of the equilibrium point Here, we study the local stability character of the solutions of the difference equation (1.1). The equilibrium points of the difference equation (1.1) are given by the relation (3.1) x γ x + a x + b x c x d x. If (1 γ) (c d) > 0, then the only positive equilibrium point x of the difference equation (1.1) is given by (3.2) x a + b (1 γ) (c d), where 0 < γ < 1 c > d. Let F : (0, ) k+1 (0, ) be a continuous function defined by (3.3) F (u 0, u 1 ) γu 1 + au 0 + bu 1 cu 0 du 1, provided that cu 0 du 1. Therefore, F (u 0, u 1 ) (ad + bc) u 1 u 0 (cu 0 du 1 ) 2 F (u 0, u 1 ) γ+ (ad + bc) u 0 u 1 (cu 0 du 1 ) 2. (3.4) Then, we see that F ( x, x) u 0 (1 γ) (ad + bc) ρ 0 (a + b) (c d) F ( x, x) u 1 γ + (1 γ) (ad + bc) (a + b) (c d) ρ 1. Then, the linearized equation of the difference equation (1.1) about x is: (3.5) y n+1 ρ 0 y n ρ 1 y n k 0. Theorem 3.1. Assume that 0 < γ < 1, c > d (3.6) 2 (1 γ) (ad + bc) + γ (a + b) (c d) < (a + b) (c d). Then, the equilibrium point x of the difference equation (1.1) is locally asymptotically stable.

10 112 Zayed El-Moneam Proof. From (3.5), we get: ρ 0 + ρ 1 (1 γ) (ad + bc) (a + b) (c d) + (1 γ) (ad + bc) γ + (a + b) (c d) (1 γ) (ad + bc) (1 γ) (ad + bc) + γ + (a + b) (c d) (a + b) (c d) 2 (1 γ) (ad + bc) + γ (a + b) (c d) (a + b) (c d) (3.7) 2 (1 γ) (ad + bc) + γ (a + b) (c d). (a + b) (c d) From (3.6) (3.7), we deduce that (3.8) ρ 0 + ρ 1 < 1. It is followed by Theorem 1.1 that Eq. (1.1) is locally asymptotically stable. Thus, the proof of Theorem 3.1 is now complete. 4. Global attractor of the equilibrium point Here, we investigate the global attractivety character of the solutions of the difference equation (1.1). Theorem 4.1. The equilibrium point x of the difference equation (1.1) is a global attractor if 0 < γ < 1. Proof. By using (3.4), we can see that the function F (u 0, u 1 ) which is defined by (3.3) is decreasing in u 0 increasing in u 1. Suppose that (m, M) is a solution of the system (4.1) m F (M, m) M F (m, M). Then, we get: we have, (4.2) am + bm m F (M, m) γm + cm dm, M F (m, M) γm + am + bm cm dm, am + bm m (cm dm) (1 γ) am + bm (1 γ). M (cm dm)

11 On the rational recursive sequence 113 From (4.2) we deduce that M m. It follows by Theorem 1.3 that x is a global attractor of the difference equation (1.1). Thus, the proof of Theorem 4.1 is now complete. References 1] M.T. Aboutaleb, M.A. El-Sayed A. E. Hamza, Stability of the recursive sequence x n+1 (α βx n)/(γ + x n 1), J. Math. Anal. Appl. 261 (2001) ] R. Agarwal, Difference Equations Inequalities. Theory, Methods Applications, Marcel Dekker Inc., New York, ] A.M. Amleh, E.A. Grove, G.Ladas D.A. Georgiou, On the recursive sequence x n+1 α + (x n 1/x n), J. Math. Anal. Appl. 233 (1999) ] C.W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol. 3 (1976) ] R. Devault, W. Kosmala, G. Ladas S.W. Schultz, Global behavior of y n+1 (p + y n k )/(qy n + y n k ), Nonlinear Analysis, 47 (2001) ] R. Devault, G. Ladas S.W. Schultz, On the recursive sequence x n+1 α + (x n/x n 1), Proc. Amer. Math. Soc. 126(11) (1998) ] R. Devault S.W. Schultz, On the dynamics of x n+1 (βx n+γx n 1)/(Bx n+ Dx n 2), Comm. Appl. Nonlinear Analysis 12 (2005) ] E.M. Elabbasy, H.El-Metwally E.M. Elsayed, On the difference equation x n+1 ax n bx n/ (cx n dx n 1), Advances in Difference Equations, Volume 2006, Article ID 82579, pages 1-10, doi: /2006/ ] H. El-Metwally, E.A. Grove G. Ladas, A global convergence result with applications to periodic solutions, J. Math. Anal. Appl. 245 (2000) ] H. El-Metwally, G. Ladas, E.A. Grove H.D. Voulov, On the global attractivity the periodic character of some difference equations, J. Difference Equations Appl. 7 (2001) ] H.A. El-Morshedy, New explicit global asymptotic stability criteria for higher order difference equations, J. Math. Anal. Appl. 336 (2007), ] H.M. EL- Owaidy, A.M. Ahmed M.S. Mousa, On asymptotic behavior of the difference equation x n+1 α + (x p n 1 /xp n), J. Appl. Math. & Computing 12 (2003) ] H.M. EL- Owaidy, A.M. Ahmed Z. Elsady, Global attractivity of the recursive sequence x n+1 (α βx n k )/(γ + x n), J. Appl. Math. & Computing 16 (2004) ] C.H. Gibbons, M.R.S. Kulenovic G. Ladas, On the recursive sequence x n+1 (α + βx n 1)/(γ + x n), Math. Sci. Res. Hot-Line 4 (2) (2000) ] E.A. Grove G. Ladas, Periodicities in nonlinear difference equations, Vol. 4, Chapman & Hall / CRC, ] G. Karakostas, Convergence of a difference equation via the full limiting sequences method, Diff. Equations Dynamical. System 1 (1993) ] G. Karakostas S. Stevic, On the recursive sequences x n+1 A + f(x n,..., x n k+1 )/x n 1, Comm. Appl. Nonlinear Analysis 11 (2004)

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13 On the rational recursive sequence ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 (α + βx n k ) / (γ x n), J. Appl. Math. & Computing, /s , in press. 37] E.M.E. Zayed M.A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., to appear. 38] E.M.E. Zayed M.A. El-Moneam, On the rational recursive two sequences x n+1 ax n k + bx n k / (cx n + δdx n k ), Acta Math. Vietnamica, to appear. 39] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 Ax n + (βx n + γx n k ) / (Cx n + Dx n k ), Comm. Appl. Nonlinear Analysis 16 (2009) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 Ax n + Bx n k + (βx n + γx n k ) / (Cx n + Dx n k ), Acta Appl. Math. doi /s y, in press. E. M. E. Zayed Mathematics Department, Faculty of Science, Taif University, El-Taif, EL-Hawiyah, P.O. Box 888, Kingdom of Saudi Arabia emezayed@hotmail.com M. A. El-Moneam Mathematics Department, Faculty of Science Arts, Jazan University, Farasan, Jazan, Kingdom of Saudi Arabia mabdelmeneam2004@yahoo.com

ON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE. E.M. Elsayed

Opuscula Mathematica Vol. 30 No. 4 2010 http://dx.doi.org/10.7494/opmath.2010.30.4.431 ON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE E.M. Elsayed Abstract. In this paper