Global attractivity of the recursive sequence xn+1=(α−βxn−1)/(γ+xn)

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  • xn1 a bxn1=c xn

    Aboutaleb et al. [1] studied the rational recursive sequence

    xn1 a bxnc xn1 ; n 0; 1; . . . ;El-Owaidy et al. [2] studied the rational recursive sequences

    xn1 axn1b xn ;

    where the coecients a, b > 0 and obtained sucient conditions for the globalattractivity of the zero equilibrium points with basin that depend on certain

    conditions posed on the coecients.Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

    Abstract

    Our aim in this paper is to investigate the global attractivity of the recursive sequence

    xn1 a bxn1=c xn under specied conditions. We show that the positiveequilibrium point of the equation is a global attractor with a basin that depends on

    certain conditions posed on the coecients.

    2003 Elsevier Inc. All rights reserved.

    Keywords: Dierence equations; Attractivity; Asymptotic stability

    1. Introduction* Co

    E-m

    0096-3

    doi:10.H.M. El-Owaidy, A.M. Ahmed *, Z. ElsadyGlobal attractivity of the recursive sequenceApplied Mathematics and Computation 151 (2004) 827833

    www.elsevier.com/locate/amcrresponding author.

    ail address: ahmedelkb@yahoo.com (A.M. Ahmed).

    003/$ - see front matter 2003 Elsevier Inc. All rights reserved.1016/S0096-3003(03)00539-3

  • sequence

    where

    a; b; c > 0 and c > b: 1:2

    x F x ; x ; n 0; 1; . . . 1:3

    ii(i) Ttwe have jxn xj < for all nP 1,i(ii) The equilibrium point x of Eq. (1.3) is called locally asymptotically stable if

    it is locally stable, and if there exist c > 0 such that for all x1, x0 2 I withjx1 xj jx0 xj < c, we have limn!1 xn x.(iii) T

    xhe equilibrium point x of Eq. (1.3) is called locally stable if for every > 0here exists d > 0 such that for all x1, x0 2 I with jx1 xj jx0 xj < d,Denition 1.2. Let I be some interval of real numbers and let x be an equi-librium point of Eq. (1.3).point that satises the condition:

    x F x;x:n1 n n1

    is theWe show that the positive equilibrium point of Eq. (1.1) is a global attractor

    with a basin that depends on certain conditions posed on the coecients.

    The study of these equations is quite challenging and rewarding and is still in

    its infancy.We believe that the non-linear rational dierence equations are paramount

    importance in their own right, and furthermore that results about such equa-

    tions oer prototypes for the development of the basic theory of the global

    behaviour of non-linear dierence equations.

    We need the following denitions:

    Denition 1.1. The equilibrium point x of the equationc xnxn1 a bxn1

    ; n 0; 1; . . . ; 1:1where the coecients a, b and c are non-negative real numbers and obtainedsucient conditions for the global attractivity of the positive equilibrium point.

    Li and Sun [5] extended the above results to the following rational recursive

    sequence:

    xn1 a bxnc xnk ; n 0; 1; . . .

    Other related results refer to [3,4].

    Our aim in this paper is to investigate the global attractivity of the recursive

    828 H.M. El-Owaidy et al. / Appl. Math. Comput. 151 (2004) 827833he equilibrium point x of Eq. (1.3) is called a global attractor if for all x1,0 2 I , we have limn!1 xn x.

  • i(v) The equilibrium point x of Eq. (1.3) is called unstable if x is not locally sta-ble.

    (vi

    (iii) B

    (iv) B

    i(v) Ohp2 4q > 0 and jpj > j1 qj:In this case x, is a saddle point.

    2. Main resultsCo

    two ene root of Eq. (1.5) has absolute value greater than one while the other rootas absolute value less than one if and only ifjqj > 1 and jpj < j1 qj:In this case, x is a repeller.oth roots of Eq. (1.5) have absolute value greater than one if and only ifjpj < 1 q < 2:In this case x is locally asymptotically stable.oth roots of Eq. (1.5) have absolute value less than one if and only ifk pk q 0: 1:5We need the following theorem:

    Theorem A [4]

    ii(i) If both roots of Eq. (1.5) have absolute value less than one, then the equilib-rium point x of Eq. (1.3) is locally asymptotically stable.

    i(ii) If at least one of the roots of Eq. (1.5) has absolute value greater than one,then x is unstable.zn1 pzn qzn1 0; n 0; 1 . . . 1:4be the linearized equation associated with Eq. (1.3) about the equilibrium pointx. Then its characteristic equation is

    2jxN xjP r:Clearly, a repeller is an unstable equilibrium.

    Let) The equilibrium point x of Eq. (1.3) is called a repeller if there exists r > 0such that if x1, x0 2 I and jx1 xj jx0 xj < r, then there existsN P 1 such that(iv) The equilibrium point x of Eq. (1.3) is called globally asymptotically stableif x is locally stable and x is also global attractor.

    H.M. El-Owaidy et al. / Appl. Math. Comput. 151 (2004) 827833 829nsider the dierence equation (1.1) under conditions (1.2). Eq. (1.1) has

    quilibrium points

  • Theor

    while 2

    a bxi b

    Lemm

    (1.1).

    where

    Proof

    Proof

    ately

    Lemm

    soluti. We can see that xm2, xm3 2 0; a=b. Then the proof follows immedi-from Lemma 2.2. Lemma 2.3. Assume that the conditions (2.4) hold for some k P 2. Let fxng be asolution of Eq. (1.1).

    If xm, xm1 2 k 1a=b; a=b for some mP 1, thenxmi 2 C;D 8 iP 6:cP ka=b and a P kb2: 2:4C ac ba c2 6 xm46

    ac D:

    So, the result follows by induction.

    Assume that there exists k P 2 such that the following conditions hold:. We can see that xm2, xm3 2 0; a=c. ThenC ac ba c2 and D

    ac: 2:3xmi 2 C;D 8 iP 4;a 2.2. Assume that condition (1.2) holds. Let fxng be a solution of Eq.If xm, xm1 2 0; a=b for some mP 1, thenThen the proof follows from Theorem A. zn1 c xi2zn c xi zn1 0; i 1; 2; n 0; 1; . . . 2:2Proof. The linearized equation associated with Eq. (1.1) about xi, i 1; 2, hasthe formem 2.1. The positive equilibrium point x1 is locally asymptotically stable,the negative equilibrium point x is unstable (saddle point).xi b c

    b c2 4a

    q

    2; i 1; 2: 2:1

    830 H.M. El-Owaidy et al. / Appl. Math. Comput. 151 (2004) 827833a 2.4. Assume that the conditions (2.4) hold for some k P 2. Let fxng be aon of Eq. (1.1).

  • jxm

    Proof

    If a

    m2

    In bo

    This

    Theor

    positi

    Then

    where C and D are dened in (2.3).Setk limn!1

    inf xn and K limn!1

    sup xn:

    Let > 0 be such that < minfa=b K; kg. Then there exists n0 2 N suchthatxm 2 C;D 8mP 6;

    by Lemmas 2.3 and 2.4, xm 2 0; a=b 8mP 2. By Lemma 2.2Proof. Let fxng be a solution of Eq. (1.1) with initial conditions x1; x0 2 S.a=b6M 6 2k1a=b

    S k 1a=b; a=b2

    [fMg M 2ka=b; a=b:em 2.5. If there exists kP 2 such that conditions (2.4) hold, then theve equilibrium point x1 of Eq. (1.1) is a global attractor with a basincompletes the proof. xmi 2 C;D 8 iP 7:

    th cases Lemma 2.3 yieldsxm2 2 0; a=b:If a bxm < 0, then

    x 2 a=b; 0:bc xm1 k 1a6 a bxm6 ka:bxm P 0, thenc xmi P a=b; i 0; 1and. We can see thatxmi 2 C;D 8 iP 7:

    xm1j6 2ka=b, thenIf xm 2 k 1a=b;1 and xm1 2 k 1a=b; a=b for some mP 1such thatH.M. El-Owaidy et al. / Appl. Math. Comput. 151 (2004) 827833 831k < xn < K 8nP n0:

  • Hence

    Then, the proof is complete.

    Theorem 2.6. Assume that the initial conditions x , x 2 0; a=b such that theyare no

    ii(i) fxng cannot have two consecutive terms equal to x1.i(ii) E

    Proof

    ii(i) I

    0

    N

    x(iii) F

    pRema

    Owail1 Px1, then we have xl2 < x1.rom (i) and (ii), we get fxng is strictly oscillatory. This completes theroof. If xl Px1 and xl1 > x1, then we have xl2 < x1, also, if xl > x1 and6 xl, xl1 < x1 which implies that xl2 > x1.ow Suppose that C is a positive semicycle starts with two terms xl, xl1.xl1 xl2 x0 x1 x1, which is a contradiction.i(ii) Assume that C is a negative semicycle starts with two terms xl, xl1, thenf xl xl1 x1 for some l 2 N, then xl1 x1, which implies that(iii) fxng is strictly oscillatory.

    very semicycle of fxng has at most two terms.1 0t equal to x1, then the following statements are true:which implies that

    a bK ck6 a bk cK:Therefore K6 k. Hence

    k K x1:a bK c K < xn1

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