On the Dynamics of a Higher-Order Difference Equation

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2012, Article ID 263053, 8 pagesdoi:10.1155/2012/263053

Research ArticleOn the Dynamics of a Higher-OrderDifference Equation

H. El-Metwally1, 2

1 Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344,Rabigh 21911, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to H. El-Metwally, [email protected]

Received 26 September 2011; Accepted 5 January 2012

Academic Editor: Taher S. Hassan

Copyright q 2012 H. El-Metwally. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper deals with the investigation of the following more general rational difference equation:yn+1 = αyn/(β + γ

∑ki=0 y

p

n−(2i+1)∏k

i=0yn−(2i+1)), n = 0, 1, 2, . . . ,where α, β, γ, p ∈ (0,∞) with the initialconditions x0, x−1, . . . , x−2k, x−2k−1 ∈ (0,∞). We investigate the existence of the equilibrium pointsof the considered equation and then study their local and global stability. Also, some results relatedto the oscillation and the permanence of the considered equation have been presented.

1. Introduction

In this paper we investigate the global stability character and the oscillatory of the solutionsof the following difference equation:

yn+1 =αyn

β + γ∑k

i=0 yp

n−(2i+1)∏k

i=0yn−(2i+1), n = 0, 1, 2, . . . , (1.1)

where α, β, γ, p ∈ (0,∞) with the initial conditions x0, x−1, . . . , x−2k, x−2k−1 ∈ (0,∞). Alsowe study the permanence of (1.1). The importance of permanence for biological systems wasthoroughly reviewed by Huston and Schmidtt [1].

In general, there are a lot of interest in studying the global attractivity, boundednesscharacter, and periodicity of the solutions of nonlinear difference equations. In particularthere are many papers that deal with the rational difference equations and that is becausemany researchers believe that the results about this type of difference equations are of

2 Discrete Dynamics in Nature and Society

paramount importance in their own right, and furthermore they believe that these resultsoffer prototype towards the development of the basic theory of the global behavior ofsolutions of nonlinear difference equations of order greater than one.

Kulenovic and Ladas [2] presented some known results and derived several new oneson the global behavior of the difference equation xn+1 = (α+βxn+γxn−1)/(A+Bxn+Cxn−1) andof its special cases. Elabbasy et al. [3–5] established the solutions form and then investigatedthe global stability and periodicity character of the obtained solutions of the followingdifference equations:

xn+1 = axn − bxn

cxn − dxn−1, xn+1 =

αxn−kβ + γ

∏ki=0xn−i

, xn+1 =dxn−lxn−kcxn−s − b

+ a. (1.2)

El-Metwally [6] gave some results about the global behavior of the solutions of the followingmore general rational difference equations

xn+1 =axl0

n−k0xl1n−k1 . . . x

lin−ki + bxs0

n−r0xs1n−r1 . . . x

sjn−rj

cxl0n−k0x

l1n−k1 . . . x

lin−ki + dxs0

n−r0xs1n−r1 . . . x

sjn−rj

, yn+1 =α0yn + α1yn−1 + · · · + αtyn−tβ0yn + β1yn−1 + · · · + βtyn−t

.

(1.3)

Cinar [7–9] obtained the solutions form of the difference equations xn+1 = xn−1/(1 + xnxn−1),xn+1 = xn−1/(−1 + xnxn−1) and xn+1 = axn−1/(1 + bxnxn−1). Also, Cinar et al. [10] studied theexistence and the convergence for the solutions of the difference equation xn+1 = xn−3/(−1 +xnxn−1xn−2xn−3). Simsek et al. [11] obtained the solution of the difference equation xn+1 =xn−3/(1 + xn−1). In [12] Yalcinkaya got the solution form of the difference equation xn+1 =xn−(2k+1)/(1+xn−kxn−(2k+1)). In [13] Stevic studied the difference equation xn+1 = xn−1/(1+xn).Other related results on rational difference equations can be found in [14–19].

Let I be some interval of real numbers and let

f : Ik+1 −→ I (1.4)

be a continuously differentiable function. Then for every set of initial conditions x−k,x−k+1, . . . , x0 ∈ I, the difference equation

xn+1 = f(xn, xn−1, . . . , xn−k), n = 0, 1, . . . (1.5)

has a unique solution {xn}∞n=−k.

Definition 1.1 (permanence). The difference equation (1.5) is said to be permanent if thereexist numbers m and M with 0 < m ≤ M < ∞ such that for any initial conditionsx−k, x−k+1, . . . , x−1, x0 ∈ (0,∞) there exists a positive integer N which depends on the initialconditions such that m ≤ xn ≤ M for all n ≥ N.

Definition 1.2 (periodicity). A sequence {xn}∞n=−k is said to be periodic with period p if xn+p =xn for all n ≥ −k.

Discrete Dynamics in Nature and Society 3

Definition 1.3 (semicycles). A positive semicycle of a sequence {xn}∞n=−k consists of a “string”of terms {xl, xl+1, . . . , xm} all greater than or equal to the equilibrium point x, with l ≥ −kand m ≤ ∞ such that either l = −k or l > −k and xl−1 < x; and, either m = ∞ or m < ∞and xm+1 < x. A negative semicycle of a sequence {xn}∞n=−k consists of a ”string” of terms{xl, xl+1, . . . , xm} all less than the equilibrium point x, with l ≥ −k and m ≤ ∞ such that:either l = −k or l > −k and xl−1 ≥ x; and, either m = ∞ or m < ∞ and xm+1 ≥ x.

Definition 1.4 (oscillation). A sequence {xn}∞n=−k is called nonoscillatory about the point x ifthere is exists N ≥ −k such that either xn > x for all n ≥ N or xn < x for all n ≥ N. Otherwise{xn}∞n=−k is called oscillatory about x.

Recall that the linearized equation of (1.5) about the equilibrium x is the lineardifference equation

yn+1 =k∑

i=0

∂f(x, x, . . . , x)∂xn−i

yn−i. (1.6)

2. Dynamics of (1.1)

The change of variables yn = (β/γ)1/(p+k+1) xn reduces (1.1) to the following differenceequation

xn+1 =rxn

1 +∑k

i=0 xp

n−(2i+1)∏k

i=0xn−(2i+1), n = 0, 1, 2, . . . , (2.1)

where r = α/β.In this section we study the local stability character and the global stability of the

equilibrium points of the solutions of (2.1). Also we give some results about the oscillationand the permanence of (2.1).

Recall that the equilibrium point of (2.1) are given by

x =rx

1 + (k + 1)xp+k+1. (2.2)

Then (2.1) has the equilibrium points x = 0 and whenever r > 1, (2.1) possesses the uniqueequilibrium point x = ((r − 1)/(k + 1))1/(p+k+1).

The following theorem deals with the local stability of the equilibrium point x = 0 of(2.1).

Theorem 2.1. The following statements are true:(i) if r < 1, then the equilibrium point x = 0 of (2.1) is locally asymptotically stable,(ii) if r > 1, then the equilibrium point x = 0 of (2.1) is a saddle point.

Proof. The linearized equation of (2.1) about x = 0 is un+1 − run = 0. Then the associatedeigenvalues are λ = 0 and λ = r. Then the proof is complete.


Theorem 2.2. Assume that r < 1, then the equilibrium point x = 0 of (2.1) is globally asymptoticallystable.

Proof. Let {xn}∞n=−2k+1 be a solution of (2.1). It was shown by Theorem 2.1 that the equilibriumpoint x = 0 of (2.1) is locally asymptotically stable. So, it is suffices to show that

limxnn→∞

= 0. (2.3)

Now it follows from (2.1) that

xn+1 =rxn

1 + xp+1n−1xn−3 · · ·xn−2k+1 + xn−1x

p+1n−3 · · ·xn−2k+1 + · · · + xn−1xn−3 · · ·xp+1

n−2k+1

≤ rxn

< xn.

(2.4)

Then the sequence {xn}∞n=0 is decreasing and this completes the proof.

Theorem 2.3. Assume that r > 1. Then every solution of (2.1) is either oscillatory or tends to theequilibrium point x = ((r − 1)/(k + 1))1/(p+k+1).

Proof. Let {xn}∞n=−2k+1 be a solution of (2.1). Without loss of generality assume that {xn}∞n=−2k+1is a nonoscillatory solution of (2.1), then it suffices to show that limn→∞xn = x. Assume thatxn ≥ x for n ≥ n0 (the case where xn ≤ x for n ≥ n0 is similar and will be omitted). It followsfrom (2.1) that

xn+1 =rxn

1 + xp+1n−1xn−3· · ·xn−2k−1 + xn−1x

p+1n−3· · ·xn−2k−1 + · · · + xn−1xn−3 · · ·xp+1

n−2k−1

≤ xn

(r

1 + (k + 1)xp+k+1

)

= xn.

(2.5)

Hence {xn} is monotonic for n ≥ n0 + 2k + 1, therefore it has a limit. Let limn→∞xn = μ,and for the sake of contradiction, assume that μ > x. Then by taking the limit of both sideof (2.1), we obtain μ = rμ/(1 + (k + 1)μp+k+1), which contradicts the hypothesis that x =((r − 1)/(k + 1))1/(p+k+1) is the only positive solution of (2.2).

Theorem 2.4. Assume that {xn}∞n=−2k+1 is a solution of (2.1) which is strictly oscillatory about thepositive equilibrium point x = ((r − 1)/(k + 1))1/(p+k+1) of (2.1). Then the extreme point in anysemicycle occurs in one of the first 2(k + 1) terms of the semicycle.


Proof. Assume that {xn}∞n=−2k−1 is a strictly oscillatory solution of (2.1). Let N ≥ 2k + 2and let {xN, xN+1, . . . , xM} be a positive semicycle followed by the negative semicycle{xM, xM+1, . . . , xM}. Now it follows from (2.1) that

xN+2k+2 − xN =rxN+2k+1

1 + xp+1N+2kxN+2k−2· · ·xN + xN+2kx

p+1N+2k−2· · ·xN + · · · + xN+2k· · ·xp+1

N

− xN

≤ xN+2k+1

(r

1 + (k + 1)xp+k+1

)

− xN

= xN+2k+1 − xN

=rxN+2k

1 + xp+1N+2k−1xN+2k−3· · ·xN−1 + · · · + xN+2k−1· · ·xp+1

N−1− xN

≤ xN+2k

(r

1 + (k + 1)xp+k+1

)

− xN

= xN+2k − xN ≤ xN+2k−1 − xN ≤ · · · ≤ xN+1 − xN ≤ xN − xN = 0.

(2.6)

Then xN ≥ xN+2(k+1) for all N ≥ 2(k + 1).Similarly, we see from (2.1) that

xM+2k+2 − xM =rxM+2k+1

1 + xp+1M+2kxM+2k−2· · ·xM + xM+2kx

p+1M+2k−2· · ·xM + · · · + xM+2k· · ·xp+1

M

− xM

≥ xM+2k+1

(r

1 + (k + 1)xp+k+1

)

− xM

= xM+2k+1 − xM

=rxM+2k

1 + xp+1M+2k−1xM+2k−3· · ·xM−1 + · · · + xM+2k−1· · ·xp+1

M−1− xM

≥ xM+2k

(r

1 + (k + 1)xp+k+1

)

− xM

= xM+2k − xM ≤ xM+2k−1 − xM ≥ · · · ≥ xM+1 − xM ≥ xM − xM = 0.(2.7)

Therefore xM+2(k+1) ≥ xM for all M ≥ 2(k + 1). The proof is so complete.

Theorem 2.5. Equation (2.1) is permanent.

Proof. Let {xn}∞n=−2k+1 be a solution of (2.1). There are two cases to consider:(i) {xn}∞n=−2k+1 is a nonoscillatory solution of (2.1). Then it follows from Theorem 2.3

that

limxnn→∞

= x, (2.8)


that is there is a sufficiently large positive integer N such that |xn − x| < ε for all n ≥ N andfor some ε > 0. So, x − ε < xn < x + ε, this means that there are two positive real numbers, sayC and D, such that

C ≤ xn ≤ D. (2.9)

(ii) {xn}∞n=−2k+1 is strictly oscillatory about x = ((r − 1)/(k + 1))1/(p+k+1).Now let {xs+1, xs+2, . . . , xt} be a positive semicycle followed by the negative semicycle

{xt+1, xt+2, . . . , xu}. If xV and xW are the extreme values in these positive and negativesemicycle, respectively, with the smallest possible indices V and W , then by Theorem 2.4we see that V − s ≤ 2(k + 1) andW − u ≤ 2(k + 1). Now for any positive indices μ and Lwithμ < L, it follows from (2.1) for n = μ, μ + 1, . . . , L − 1 that

xL = xL−1

⎛

⎝ r

1 + xp+1L−2xL−4· · ·xL−2k−2 + xL−2x

p+1L−4· · ·xL−2k−2 + · · · + xL−2xL−4· · ·xp+1

L−2k−2

⎞

⎠

=r2xL−2

(1 + x

p+1L−3· · ·xL−2k−3 + · · · + xL−3· · ·xp+1

L−2k−3)(

1 + xp+1L−2· · ·xL−2k−2 + · · · + xL−2· · ·xp+1

L−2k−2)

...

= xL−ζrζζ∏

η=1

⎛

⎝ 1

1 +∑k

i=0 xp

L−(2i+1)−η∏k

i=0xL−(2i+1)−η

⎞

⎠

= xμrL−μ

L−1∏

η=μ

⎛

⎝ 1

1 +∑k

i=0 xp

η−(2i+1)∏k

i=0xη−(2i+1)

⎞

⎠.

(2.10)

Therefor for V = L and s = μwe obtain

xV = xsrV−s

V−1∏

η=s

⎛

⎝ 1

1 +∑k

i=0 xp

η−(2i+1)∏k

i=0xη−(2i+1)

⎞

⎠

≤ xr2k+1 = H.

(2.11)


Again whenever W = L and μ = t, we see that

xW = xtrW−t

W−1∏

η=t

⎛

⎝ 1

1 +∑k

i=0 xp

η−(2i+1)∏k

i=0xη−(2i+1)

⎞

⎠

≥ xrW−tW−1∏

η=t

(1

1 + (k + 1)Hp+k+1

)

= xrW−t(

11 + (k + 1)Hp+k+1

)W−t−1

≥ x

(1

1 + (k + 1)Hp+k+1

)2k+1

= G.

(2.12)

That is, G ≤ xn ≤ H. It follows from (i) and (ii) that

min{C,G} ≤ xn ≤ max{D,H}. (2.13)

Then the proof is complete.

References

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[2] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations With OpenProblems and Conjectures, Chapman & Hall, Boca Raton, Fla, USA, 2002.

[3] E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation xn+1 = axn −bxn/(cxn − dxn−1),” Advances in Difference Equations, vol. 2006, Article ID 82579, 10 pages, 2006.

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