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Journal of the Egyptian Mathematical Society (2015) 000, 1–4
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
Original Article
Attractivity of the recursive sequence xn+1 = (A − Bxn−2 )/(C + Dxn−1 ) A.M. Ahmed a, N.A. Eshtewy b,∗ a b
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt Department of Mathematics, Faculty of Science Arish, Suez Canal University, 41522, Ismailia, Egypt
Received 13 June 2015; revised 31 July 2015; accepted 12 August 2015 Available online xxx
Keywords Difference equation; Attractivity; Basin of attraction
Abstract In this paper, we investigate the global attractivity of the difference equation A − Bxn−2 xn+1 = , n = 0, 1, . . . , C + Dxn−1 where A, B are nonnegative real numbers, C, D are positive real numbers and C + Dxn−1 = 0 for all n ≥ 0. 2010 Mathematics Subject Classification: 39A10; 39A11 Copyright 2015, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Difference equations appear naturally as discrete analogues and numerical solutions of differential equations and delay differential equations having applications in biology, ecology, physics, etc. 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. The study ∗
Corresponding author. Tel.: +20 1067179166. E-mail addresses:
[email protected] (A.M. Ahmed),
[email protected] (N.A. Eshtewy). Peer review under responsibility of Egyptian Mathematical Society.
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of nonlinear rational difference equations of higher order is of paramount importance, since we still know so little about such equations. R. Abo-Zeid [1] investigated the attractivity of two nonlinear third order difference equations xn+1 =
A − Bxn−1 , ±C + Dxn−2
n = 0, 1, . . . ,
where A, B are nonnegative real numbers, C, D are positive real numbers and C + Dxn−2 = 0 for all n ≥ 0. El-Owaidy et al. [2] investigated the global attractivity of the difference equation xn+1 =
α − βxn−1 , γ + xn
n = 0, 1, . . . ,
where α, β, γ are non-negative real numbers and γ + xn = 0 for all n ≥ 0.
S1110-256X(15)00068-1 Copyright 2015, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.joems.2015.08.004
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A.M. Ahmed, N.A. Eshtewy
M. A. El-Moneam [3] studied the global behavior of the higher order nonlinear rational difference equation xn+1 = Axn + Bxn−k + Cxn−l + Dxn−σ
y2 =
bxn−k , + |dxn−k − exn−l |
n = 0, 1, . . . ,
xn+1
n = 0, 1, . . . ,
n = 0, 1, . . . ,
(1.1)
where A, B are non negative real numbers, C, D are positive real numbers and C + Dxn−1 = 0 for all n ≥ 0. Theorem 1.1 ([6]). Consider the third-degree polynomial equation λ3 + a2 λ2 + a1 λ + a0 = 0,
(1.2)
|a2 + a0 | < 1 + a1 , |a2 − 3a0 | < 3 − a1 and a20 + a1 − a0 a2 < 1. The change of variables xn = difference equation
where p =
AD C2
yi q λ+ = 0. 1 + yi 1 + yi
gi (λ) = λ3 +
yi q λ+ , i = 1, 2. 1 + yi 1 + yi
(2.2)
Theorem 2.1. (1) The sufficient condition for the equilibrium point y1 to be locally asymptotically stable is q ≤ 1.
(2) If q > 13 + 34 p + 49 , then y1 is unstable. (3) y2 is saddle equilibrium point. Proof.
where a1 , a0 and a2 are real numbers. Then a necessary and sufficient condition for all roots of Eq. (1.2) to lie inside the open disk |λ| < 1 is
p − qyn−2 , 1 + yn−1
Its associated characteristic equation is
Suppose that
A − Bxn−2 = , C + Dxn−1
yn+1 =
yi q zn−1 + zn−2 = 0, n = 0, 1, 2, . . . . 1 + yi 1 + yi
zn+1 +
λ3 +
where α, β, γ ≥ 0. For other related results, see [5,6]. In this paper we study the global attractivity of the difference equations xn+1
1 (−(1 + q) − (1 + q)2 + 4p). 2
The linearized equation associated with Eq. (2.1) about yi is
where the coefficients A, B, C, D, b, d, e ∈ (0, ∞), while k, l and σ are positive integers. The initial conditions x−σ , . . . , x−l , . . . , x−k , . . . , x−1 , x0 are arbitrary positive real numbers such that k < l < σ . A. E. Hamza et al. [4] investigated the global asymptotic stability of the recursive sequence α − βxn = , γ + xn−1
and
C y D n
reduces Eq. (1.1) to the
(1) If q ≤ 1, then by using Theorem 1.1 with a0 = yi , 1+yi
q , 1+yi
a1 =
a2 = 0. We can easily show that y1 is locally asymptotically stable.
(2) If q > 13 + 43 p + 49 , then g1 (λ) has a zero λ1 in (−∞, −1), which implies that the equilibrium point y1 is unstable. q )= (3) It is clear that g2 (λ) has a zero λ1 ∈ (0, 1), and g2 (− 1+y 2 q 2 [y + 1 − q ]. 2 (1+y )3 2
n = 0, 1, . . . .
and q =
(1.3)
B . C
It is clear that g2 (λ) is an increasing function. Since q q ) > 0, then λ1 < − 1+y ⇒ |λ2 λ3 | > 1 ⇒ |λ2 | = g2 (− 1+y 2 2 |λ3 | > 1, which implies that y2 is unstable equilibrium point (saddle).
2. The recursive sequence yn+1 = (p − qyn−2 )/(1 + yn−1 )
Lemma 1. Assume that q ≤ 1. Then the interval [0, qp ] is an invariant interval for Eq. (2.1).
In this section we study the global attractivity of the difference equation
Proof. Let {yn }∞ n=−2 be a solution of Eq. (2.1) with y−2 , y−1 , y0 ∈ [0, qp ]. , U1 is decreasing in x Consider the function U1 (x, y) = p−qy 1+x and y on (−1, ∞) × (−∞, qp ). Hence 0 = U1 ( qp , qp ) ≤ y1 = U1 (y−1 , y−2 ) < U1 (0, 0) = p < qp , by induction we obtain 0 ≤ yn ≤ qp ∀ n ≥ 1. Assume that there exists k ≥ 2 such that the following conditions hold . p ≥ kq2 and 1 ≥ kp q
yn+1 =
p − qyn−2 , 1 + yn−1
n = 0, 1, . . . ,
(2.1)
where p and q are positive real numbers. The equilibrium points of Eq. (2.1) are the zeros of the function f (y) = y2 + (1 + q)y − p. That is
Lemma 2. Assume that condition (2.3) hold for some k ≥ 2. Let {yn } be a solution of Eq. (2.1)
1 y1 = (−(1 + q) + (1 + q)2 + 4p) 2
If yn , yn+1 , yn+2 ∈ [−(k − 1) qp , qp ] for some n ≥ −2, then yn+i ∈ [0, qp ] ∀ i ≥ 3.
Please cite this article as: A.M. Ahmed, N.A. Eshtewy, Attractivity of the recursive sequence xn+1 = (A − Bxn−2 )/(C + Dxn−1 ), Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.08.004
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Attractivity of the recursive sequence xn+1 = (A − Bxn−2 )/(C + Dxn−1 ) Proof. (i) If yn , yn+1 , yn+2 ∈ [−(k − 1) qp , qp ] for some n ≥ −2, then p p , ≤ yn+3 = U1 (yn+1 , yn ) 0 = U1 q q p p p ≤ . ≤ U1 −(k − 1) , −(k − 1) q q q 0 = U1 ≤ U1
p p , q q
≤ yn+4 = U1 (yn+2 , yn+1 ) p p p ≤ . −(k − 1) , −(k − 1) q q q
We get the inequality
Therefore by Lemma 1, we have 0 ≤ yn+i ≤ qp , ∀ i ≥ 3. Lemma 3. Assume that condition (2.3) hold for some k ≥ 2 and q ≤ 1. Let {yn } be a solution of Eq (2.1). If p mkp for some n ≥ −2, yn , yn+1 , yn+2 ∈ −(k − 1) , q q p m ∈ Z+ then yn+i ∈ 0, ∀i ≥ 6. q Proof. We prove the theorem by induction. For m = 1, let ] for some n ≥ −2, then yn , yn+1 , yn+2 ∈ [−(k − 1) qp , kp q −(k − 1)pq p − kp −(k − 1)p ≤ ≤ ≤ yn+3 q q + kp 1 + kp q
One may show that yn+4 and yn+5 ∈ [−(k − 1) qp , qp ]. Then by Lemma 2 we have 0 ≤ yn+i ≤ qp , ∀ i ≥ 6. Skp ], q
Now, assume that yn , yn+1 , yn+2 ∈ [−(k − 1) qp ,
for some
(S+1)kp ], q
then
−((S + 1)k − 1)pq p − (S + 1)kp −(k − 1)p ≤ ≤ ≤ yn+3 q q + (S + 1)kp 1 + (S+1)kp q =
p − qyn p − kp + p p ≤ ≤ . p 1 + yn+1 q q
By using the same steps we obtain yn+4 , yn+5 ∈ [−(k − 1) qp , qp ], then by Lemma 2 we have 0 ≤ yn+i ≤ qp , ∀ i ≥ 6. This complete the proof. Theorem 2.2. Suppose that q ≤ 1 and there exists k ≥ 2, m ≥ 1 such that conditions (2.3) hold. Then the positive equilibrium point y1 of Eq (2.1) is a global attractor with a basin p mkp 3 S = −(k − 1) , q q
p − q(λ − ) p − q( + ) ≤λ≤≤ . 1++ 1+λ− Hence we have p − q p − qλ ≤λ≤≤ . 1+ 1+λ This implies p − q ≤ λ + λ and + λ ≤ p − qλ, then p − q ≤ λ + λ ≤ λ + p − qλ − . That is λ(q − 1) ≤ (q − 1). This is contradiction and therefore, = λ = y1 . Theorem 2.3. Assume the initial conditions y−2 , y−1 , y0 ∈ [0, qp ]. If they are not both equal to y1 , then the following statements are true (1) (2) (3) (4)
{yn } cannot have three consecutive terms equal to y1 . Every negative semicycle of {yn } has at most three terms. Every positive semicycle of {yn } has at most four terms. {yn } is strictly oscillatory.
Proof.
p − qyn p − kp + p p ≤ ≤ kq ≤ . p 1 + yn+1 q q
Assume that if yn , yn+1 , yn+2 ∈ [−(k − 1) qp , n ≥ −2, S ∈ Z+ , then 0 ≤ yn+i ≤ qp , ∀i ≥ 6.
Proof. Suppose that q < 1 and let {yn }∞ n=−2 be a solution of Eq. (2.1) with y−2 , y−1 , y0 ∈ S. Then by Lemma 2 and Lemma 3 we have yn ∈ [0, qp ], n ≥ 6. Set λ = lim inf yn and = lim sup yn . Let > 0 such that < min{(p/q) − , λ}. There exists n0 ∈ N such that λ − < yn < + ∀ n ≥ n0 . Hence p − q(λ − ) p − q( + ) < yn+1 < , ∀ n ≥ n0 + 2. 1++ 1+λ−
As we know −(k − 1) qp ≤ 0 ≤ yn+3 ≤ qp , then we can write p p , ≤ yn+5 = U1 (yn+3 , yn+2 ) 0 = U1 q q p p p ≤ . ≤ U1 −(k − 1) , −(k − 1) q q q
=
3
(1) If yl−1 = yl = yl+1 = y1 for some l ∈ N, then yl−2 = y1 , which implies that yl−1 = yl−2 = yl−3 = · · · = y0 = y−1 = y−2 = y1 which is impossible. (2) Assume a negative semicycle starts with three terms then yl−2 , yl−1 , yl < y1 , and yl+1 = yl−2 , yl−1 , yl , f (yl−1 , yl−2 ) > f (y1 , y1 ) = y1 . (3) Assume a positive semicycle starts with three terms yl−2 , yl−1 , yl , then yl−2 , yl−1 , yl ≥ y1 , provided that yl−2 , yl−1 , yl = y1 at the same time. yl+1 = f (yl−1 , yl−2 ) ≤ f (y1 , y1 ) = y1 and yl+1 = y1 iff yl−1 = yl−2 = y1 which implies that yl > y1 . In all cases yl+2 = f (yl , yl−1 ) < f (y1 , y1 ) = y1 . (4) From (1)–(3), we obtain {yn } is strictly oscillatory. 3. The recursive sequence yn+1 = (−qyn−2 )/(1 + yn−1 ) In this section we study the global behavior of the difference equation yn+1 =
−qyn−2 , 1 + yn−1
n = 0, 1, . . . .
(3.1)
Eq. (3.1) has two equilibrium points y1 = 0 and y2 = −1 − q.
Please cite this article as: A.M. Ahmed, N.A. Eshtewy, Attractivity of the recursive sequence xn+1 = (A − Bxn−2 )/(C + Dxn−1 ), Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.08.004
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A.M. Ahmed, N.A. Eshtewy
The linearized equation associated with Eq. (3.1) about yi , i = 1, 2 is qyi q zn−1 + zn−2 = 0, i = 1, 2, zn+1 − (1 + yi )2 1 + yi n = 0, 1, . . . .
0.
(3.2)
The characteristic equation associated with Eq. (3.2) is λ3 −
qyi q λ+ = 0, i = 1, 2. 2 (1 + yi ) 1 + yi
Let gi (λ) = λ3 −
qyi q λ+ , i = 1, 2. (1 + yi )2 1 + yi
Theorem 3.1. (1) The equilibrium point y1 = 0 is locally asymptotically stable iff q < 1, saddle point iff q > 1, and non hyperbolic if q = 1. (2) The equilibrium point y2 is unstable. Proof. (1) The linearized equation associated with Eq. (3.2) about y1 = 0 is zn+1 + qzn−2 = 0, n = 0, 1, . . . . Its associated characteristic equation is λ3 + q = 0. Easily one can show that y1 = 0 is locally asymptotically stable iff q < 1, saddle point iff q > 1, and non hyperbolic if q = 1. (2) The characteristic equation about y2 = −1 − q is λ3 +
1 (1 + q)λ − 1 = 0, q
g2 (λ) has a root λ1 ∈ (0, 1), then |λ2 λ3 | > 1, which implies y2 = −1 − q is unstable (saddle point). Theorem 3.2. Assume that q < 12 . Then the interval [−q, q] is an invariant interval of Eq. (3.1) Proof. Suppose that q < 12 and let |y−i | < q, i = 0, 1, 2. we show that yn ∈ [−q, q], n = 1, 2, . . . for Eq. (3.1). 0 < 1 − q < 1 + y−i < q + 1. Hence −qy−2 |y1 | = 1+y
= q|y−2 | < q|y−2 | < |y−2 | < q, |1 + y | 1−q −1 −1
by induction we obtain |yn+1 | <
q|yn−2 | < |yn−2 | < q, n ≥ 1−q
Theorem 3.3. Assume that q < 12 and let {yn }∞ n=−2 be a solution of Eq. (3.1) with y−2 , y−1 , y0 ∈ [−q, q]. Then {yn }∞ n=−2 oscillate with semicycles of length at most three. Proof. Let {yn }∞ n=−2 be a solution of Eq. (3.1) with y−2 , y−1 , y0 ∈ [−q, q]. Then by Theorem 3.2., we get 1 + yn > 0, n ≥ −2, that is sgn(yn+1 ) = −sgn(yn−2 ), n ≥ 0. ∞ This implies that the subsequences {y3n−2 }∞ n=0 , {y3n−1 }n=0 and oscillate with semicycles of length one. Therefore, the {y3n }∞ n=0 oscillates with semicycles of length at most solution {yn }∞ n=−2 three. Theorem 3.4. If q < 12 , then the equilibrium point y1 = 0 is a global attractor with basin [−q, q]3 . Proof. According to Theorem 3.2, we have n+1 n+1 q q |y−2 |, |y3n+2 | < |y−1 |, and |y3n+1 | < 1−q 1−q n+1 q |y0 |, |y3n+3 | < 1−q since q < 12 , we have lim yn = 0. n−→∞
Acknowledgment The authors are grateful to the referees for numerous comments that improved the quality of the paper. References [1] R. Abo-Zied, Attractivity of two nonlinear third order difference equations, J. Egypt. Math. Soc. 21 (3) (2013) 241–247. [2] H.M. El-Owaidy, A.M. Ahmed, Z. Elsady, Global attractivity of the α − βxn−1 recursive sequence xn+1 = , Appl. Math. Comput. 151 γ + xn (2004) 827–833. [3] M.A. El-Moneam, On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett. 3 (2) (2014) 121–129. [4] M.T. Aboutaleb, M.A. Elsayed, A.E. Hamza, Stability of the recursive sequence xn+1 = (α − βxn )/(γ + xn−1 ), J. Math Anal. Appl. 261 (2001) 126–133. [5] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993. [6] M.R.S. Kulenovi´c, G. Ladas, Dynamics of Second-Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002.
Please cite this article as: A.M. Ahmed, N.A. Eshtewy, Attractivity of the recursive sequence xn+1 = (A − Bxn−2 )/(C + Dxn−1 ), Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2015.08.004
