On the Dynamics of a Higher-Order Rational Difference Equation

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Volume 2011, Article ID 419789,8pages doi:10.1155/2011/419789

Research Article

On the Dynamics of a Higher-Order Rational Difference Equation

A. M. Ahmed

Department of Natural Sciences, Arriyadh Community College, King Saud University, Malaz, P.O. Box 28095, Riyadh 11437, Saudi Arabia

Correspondence should be addressed to A. M. Ahmed,[email protected] Received 23 May 2011; Revised 2 August 2011; Accepted 6 August 2011 Academic Editor: Cengiz C¸ inar

Copyrightq2011 A. M. Ahmed. 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 paper is to investigate the global asymptotic stability and the periodic character for the rational diﬀerence equationxn1αxn−1/βγΠ^k_ilx^p_n−2iⁱ , n0,1,2, . . ., where the parameters α, β, γ, pl, pl1, . . . , pk are nonnegative real numbers, andl, k are nonnegative integers such that l≤k.

1. Introduction

Diﬀerence equations have always played an important role in the construction and analysis of mathematical models of biology, ecology, physics, economic process, and so forth.

The study of nonlinear rational diﬀerence equations of higher order is of paramount importance, since we still know so little about such equations.

Amleh et al.1 investigated the third-order rational diﬀerence equation x_n1 abx_n−1

ABx_n−2, n0,1,2, . . . , 1.1

wherea, b, A, Bare nonnegative real numbers and the initial conditions are nonnegative real numbers.

Ahmed2 studied the global asymptotic behavior and the periodic character of solutions of the third-order rational diﬀerence equation

xn1 bx_n−1

ABx^p_nx^q_n−2, n0,1,2, . . . , 1.2

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where the parametersb, A, B, p, q are nonnegative real numbers, and the initial conditions x₋₂, x₋₁, x0are arbitrary nonnegative real numbers.

For other related results, see3 and also4–15 .

In this paper, the global asymptotic behavior and the periodic character of solutions of the rational diﬀerence equation

x_n1 αx_n−1

βγΠ^k_ilx^p_n−2iⁱ , n0,1,2, . . . , 1.3 where the parametersα, β, γ, p_l, p_l1, . . . , p_kare nonnegative real numbers,l, kare nonnegative integers such thatl≤k, and the initial conditionsx_−2k, x_−2k1, . . . , x0are arbitrary nonnegative real numbers such that

βγ^k

il

x_n−2i^pⁱ >0, ∀n≥0, 1.4

will be investigated.

LetIbe an interval of real numbers, and letf :I^2k1 → Ibe a continuously differen- tiable function. Consider the diﬀerence equation

xn1fxn, xn−1, . . . , xn−2k, n0,1,2, . . . , 1.5 withx−2k, x−2k1, . . . , x0 ∈ I. Letxbe the equilibrium point of1.5. The linearized equation of1.5aboutxis

y_n1c1y_nc2y_n−1· · ·c_2k1y_n−2k, n0,1,2, . . . , 1.6

where c1 ∂f

∂x_nx, x, . . . , x, c2 ∂f

∂x_n−1x, x, . . . , x, . . . , c_2k1 ∂f

∂x_n−2kx, x, . . . , x.

1.7

The characteristic equation of1.5is

λ^2k1−c1λ^2k−c2λ^2k−1− · · · −c2k1 0. 1.8 Definition 1.1. Letxbe an equilibrium point of1.5.

iThe equilibrium pointxof1.5is called locally stable if, for every >0, there exists δ >0 such that, for allx−2k, x−2k1, . . . , x0 ∈Iwith|x−2k−x||x−2k1−x|· · ·|x0−x|<

δ, we have|x_n−x|< for alln≥ −2k.

iiThe equilibrium point xof 1.5is called locally asymptotically stable if it is locally stable, and if there existsγ > 0 such that for allx−2k, x−2k1, . . . , x0 ∈Iwith|x−2k− x||x_−2k1−x|· · ·|x0−x|< γ, we have lim_n_{→ ∞}x_nx.

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iiiThe equilibrium point x of 1.5 is called global attractor if, for every x−2k, x_−2k1, . . . , x0∈I, we have lim_n_{→ ∞}x_nx.

ivThe equilibrium pointxof1.5is called globally asymptotically stable if it is locally stable and global attractor.

vThe equilibrium pointxof1.5is called unstable if it is not stable.

viThe equilibrium pointxof1.5is called source or repeller if there existsr >0 such that, for allx_−2k, x_−2k1, . . . , x0∈Iwith 0<|x_−2k−x||x_−2k1−x|· · ·|x0−x|< r, there existsN≥1 such that|xN−x| ≥r. Clearly, a repeller is an unstable equilibrium.

Theorem Alinearized stability theorem. The following statements are true.

1If all roots of 1.8have modulus less than one, then the equilibrium pointxof 1.5is locally asymptotically stable.

2If at least one of the roots of 1.8has modulus greater than one, then the equilibrium point xof1.5is unstable.

The equilibrium pointxof 1.5is called a “saddle point” if 1.8has roots both inside and outside the unit disk.

2. The Special Cases αβγ Σ

^k_il

p

i

0

In this section, we examine the character of solutions of 1.3 when one or more of the parameters in1.3are zero.

There are four such equations; namely,

x_n10, n0,1,2, . . . , 2.1

xn1 α

βxn−1, n0,1,2, . . . , 2.2

x_n1 α

βγx_n−1, n0,1,2, . . . , 2.3

xn1 αxn−1

γΠ^k_ilx^p_n−2iⁱ , n0,1,2, . . . . 2.4

Equation2.1is trivial,2.2and2.3are linear, and2.4is a non-linear diﬀerence equation; the change of variablesx_ne^yⁿreduces it to a linear diﬀerence equation.

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3. A General Oscillation Result

The change of variablesx_n β/γ^1/Σ^k^il^pⁱy_nreduces1.3to the diﬀerence equation

yn1 ryn−1

1 Π^k_ily^p_n−2iⁱ , n0,1,2, . . . , 3.1 whererα/β >0.

Note thaty₁ 0 is always an equilibrium point. Whenr > 1,3.1also possesses the unique positive equilibriumy₂ r−1^1/Σ^k^il^pⁱ.

Theorem B see 8 . Assume that F ∈ C0,∞^2k1 → 0,∞ is nonincreasing in the odd arguments, and nondecreasing in the even arguments. Letxbe an equilibrium point of the diﬀerence equation

xn1Fxn, xn−1, . . . , xn−2k, n0,1,2, . . . , 3.2

and let{xn}^∞_n−2kbe a solution of 3.2such that either

x−2k, x−2k2, . . . , x0≥x, x−2k1, x−2k3, . . . , x−1< x, 3.3

or

x−2k, x−2k2, . . . , x0< x, x−2k1, x−2k3, . . . , x−1≥x. 3.4

Then{xn}^∞_n−2koscillates aboutxwith semicycles of length one.

Corollary 3.1. Assume thatr >1; let{y_n}^∞_n−2kbe a solution of 3.1such that either y−2k, y−2k2, . . . , y0≥y₂ r−1^1/Σ^k^il^pⁱ,

y_−2k1, y_−2k3, . . . , y₋₁< y₂ r−1^1/Σ^k^il^pⁱ, 3.5

or

y_−2k, y_−2k2, . . . , y0< y₂ r−1^1/Σ^k^il^pⁱ

y_−2k1, y_−2k3, . . . , y₋₁≥y₂ r−1^1/Σ^k^il^pⁱ. 3.6

Then{y_n}^∞_n−2koscillates about the positive equilibrium pointy₂ r−1^1/Σ^k^il^pⁱ with semicycles of length one.

Proof. The proof follows immediately from Theorem B.

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4. The Dynamics of 3.1

In this section, we investigate the dynamics of3.1with nonnegative initial conditions.

Theorem 4.1. For3.1, we have the following results.

iAssume thatr <1, then the zero equilibrium point is locally asymptotically stable.

iiAssume thatr >1, then the zero equilibrium point is saddle point.

iiiThe positive equilibrium pointy₂is unstable.

Proof. The linearized equation associated with3.1abouty₁0 has the form

zn1−rzn−10, n0,1,2, . . . , 4.1

so, the characteristic equation of3.1abouty₁ 0 is

λ^2k1−rλ^2k−10, 4.2

then the proof ofiandiifollows immediately from Theorem A.

The linearized equation of3.1abouty₂ r−1^1/Σ^k^il^pⁱ is

z_n1−z_n−1^k

il

p_i

1− 1 r

z_n−2i0, n0,1,2, . . . , 4.3

so, the characteristic equation of3.1abouty₂ r−1^1/Σ^k^il^pⁱ is

λ^2k1−λ^2k−1^k

il

pi

1−1

r

λ^2k−2i0. 4.4

Set

fλ λ^2k1−λ^2k−1^k

il

pi

1−1

r

λ^2k−2i, 4.5

thenf−1 Σ^k_ilp_ir −1/r > 0, and lim_λ_{→ −∞}fλ −∞sofλhas at least a root in

−∞,−1. Then the proof ofiiifollows.

Theorem 4.2. Assumer <1. Then the zero equilibrium point of3.1is globally asymptotically stable.

Proof. We know byTheorem 4.1that the equilibrium pointy₁ 0 is locally asymptotically stable of3.1, and so it suﬃces to show thaty₁0 is a global atractor of3.1as follows:

0≤y_n1 ry_n−1

1 Π^k_ily^p_n−2iⁱ ≤ry_n−1, 4.6

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sincer <1, then

nlim→ ∞yn0. 4.7

The next theorem shows that3.1has a prime-period two solutions whenr1.

Theorem 4.3. For3.1, we have the following results.

aEquation3.1possesses the prime-period two solutions

. . . , φ,0, φ,0, φ, . . . 4.8

withφ >0, whenr1.

bAssume thatr 1, then every solution of3.1converges to a periodnot neces- sarily primetwo solutions4.8withφ≥0.

Proof. aLet

. . . , φ, ψ, φ, ψ, . . . 4.9

be period two solutions of3.1. Then

φ rφ

1ψ^Σ^k^il^pⁱ, ψ rψ

1φ^Σ^k^il^pⁱ. 4.10

Ifφ /0 andψ /0, thenφψ r−1^1/Σ^k^il^pⁱ, which is impossible. Hence,ψ0 which implies thatr−1φ0, sor1.

bAssume thatr1, and let{y_n}^∞_n−2kbe a solution of3.1, then

yn1−yn−1 −yn−1Π^k_ily_n−2i^pⁱ

1 Π^k_ily^p_n−2iⁱ ≤0. 4.11

So the even terms of this solution decrease to a limitsayΦ≥0, and the odd terms decrease to a limitsayΨ≥0. Thus,

Φ Φ

1 Ψ^Σ^k^il^pⁱ, Ψ Ψ

1 Φ^Σ^k^il^pⁱ, 4.12

which implies that

ΦΨ^Σ^k^il^pⁱ 0, ΨΦ^Σ^k^il^pⁱ 0. 4.13

This completes the proof.

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The next theorem shows that whenr >1,3.1possesses unbounded solutions.

Theorem 4.4. Assumer > 1. Then3.1possesses unbounded solutions. In particular, every solution of3.1which oscillates about the equilibriumy₂ r−1^1/Σ^k^il^pⁱwith semicycles of length one is unbounded.

Proof. we will prove that every solution{yn}^∞_n−2kof3.1which oscillates with semicycles of length one is unboundedsee corollary3.1.

Assume that{y_n}^∞_n−2kis a solution of3.1such that

y2n1< y₂ r−1^1/Σ^k^il^pⁱ, y2n> y₂ r−1^1/Σ^k^il^pⁱ, n≥ −k. 4.14

Then

y_2n2 ry_2n

1 Π^k_ily^p_2n1−2iⁱ > ry_2n

1y^Σ₂^k^il^pⁱ y_2n,

y2n3 ry2n1

1 Π^k_ily_2n2−2i^pⁱ < ry2n1

1y^Σ₂^k^il^pⁱ y2n1.

4.15

From which it follows that

nlim→ ∞y_2n∞, lim

n→ ∞y_2n1 0, 4.16

which completes the proof.

Acknowledgments

This paper was supported by King Saud University, Deanship of Scientific Research. The author would like to thank Deanship of Scientific Research, King Saud University, Riyadh, Saudi Arabia, for funding and supporting this research.

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