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Oscillation of a Nonlinear Difference Equation of Power Type | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Oscillation of a Nonlinear Difference Equation of Power Type

Bakery AA*

Department of Mathematics, Faculty of Science and Arts, University of Jeddah, Saudi Arabia

*Corresponding Author:
Bakery AA
Department of Mathematics
Faculty of Science and Arts
P.O. Box 355, Khulais 21921
University of Jeddah (U j), Saudi Arabia
Tel: 966126952000
E-mail: [email protected]

Received: October 27, 2015; Accepted: November 09, 2015; Published: November 13, 2015

Citation: Bakery AA (2015) Oscillation of a Nonlinear Difference Equation of Power Type. J Appl Computat Math 4:270. doi:10.4172/2168-9679.1000270

Copyright: © 2015 Bakery AA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We give in this work the sufficient conditions on the positive solutions of the difference equation 8 1 1 n n n n x x x x α − + = + , n=0,1,…., where α ≥ 0 and s > 0 with arbitrary positive initials x-1; x0 to be bounded and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. We have given a non-oscillatory positive solution which converges to the equilibrium point.

Keywords

Difference equation; Boundedness; Global asymptotic stability; Oscillation; Period two solution

Introduction

In this work we study the positive solutions of the difference equation

Equation (1)

where f is a continuously differentiable function, α ≥ 0 and s > 0 with arbitrary positive initials x-1,x0. Now we recall some basic definitions and results which will be used in the sequel.

Remark 3.1: A point Equationis an equilibrium point of equation (1), if and only if,

Equation (2)

If we replace xn and xn-1 in (1) by the variables u and v respectively, then we have

Equation

Equation

The linearized equation related with equation (1) about the equilibrium point Equationis

Equation

Its characteristic equation is

Equation (3)

Definition 3.2: An equilibrium [1] point x of the difference equation (1) is called locally stable if for every ?>0 there exists δ > 0 such that, if x-1, x0? (0;∞) withEquation for all n≥-1.

Definition 3.3: An equilibrium [2] point Equationof the difference equation (1) is called a global attractor if for every x-1,x0?(0;∞), we have Equation.

Definition 3.4: An equilibrium [3] point Equation of the equation (1) is called globally asymp-totically stable if it is locally stable and a global attractor.

Definition 3.5: An equilibrium [3] point Equation of the difference equation (1) is called unstable if it is not locally stable.

Definition 3.6: A sequence [4] Equation is said to be periodic with period p if xn+p = xn for all n ≥ -1.

Definition 3.7: A sequence [4] Equation is said to be periodic with prime period p if p is the smallest positive integer having this property.

Theorem 3.8: (i) Equation is locally asymptotically stable, if and only if [5,6],

Equation.

(ii) Equation is unstable and called a saddle point, if and only if,

Equation

(iii) Equation is called a non-hyperbolic point, if and only if,

Equation

or

Equation

For s = 0, the difference equation (1) will reduce to

Equation (4)

Amleh et al. [7] gave the following results:

(1) If α > 1, the equilibrium point Equation of (4) is globally asymptotically stable.

(2) If α = 1, then every positive solution converges to a solution of prime period-two.

(3) Every positive solution is bounded, if and only if, α ≥ 1.

(4) The equilibrium point is an unstable saddle point, if α ? (0,1).

As of late, there has been incredible enthusiasm for examining nonlinear and rational difference equations, see, for instance the references therein [8-16].

Throughout the article we denote by! the class

Equation

and l denote the class of bounded sequences of real numbers.

Global Behavior of Solutions and Boundedness

Firstly we determine the classification of the equilibrium points for equation (1) and its uniqueness. By applying theorem (3.8) in the special difference equation (1), we obtain the following result.

Lemma 4.1: (i) x is locally asymptotically stable, if and only if,

Equation

(ii) Equation is unstable, if and only if,

Equation

(iii) Equation is non-hyperbolic point, if and only if,

Equation

Proof: The proof is easy, so omitted.

Lemma 4.2: (1) If s = 1 and α ? (0,1), then there exist an unique equilibrium point Equation of equation (1).

(2) If s ? (0,1), then there exist an unique equilibrium point Equation of equation (1).

(3) For s > 1, if

(i) Equation then (1) has two equilibrium points Equation.

(ii) Equation then there exist an unique equilibrium point Equation of equation (1).

(iii) Equation , then there is no equilibrium point of equation (1).

Proof: (1) For s = 1 and α ? (0,1), then from the definition of the function g, we get Equation which gives Equation.

(2) Let s ? (0; 1), from the definition of the function g, we have Equation. The function g is decreasing on Equation and increasing on Equation hence g has a unique root Equation.

(3) For s > 1, since g(0) = -1 and g(1) = -α, we have also g'(x) = 0 if and only if Equation which is a maximum point of g(x). Since

Equation

We have three cases, the first one is

Equation

Hence there is two equilibrium points Equation.

The second case is

Equation

Hence there is one equilibrium point Equation.

The third case is

Equation

Hence there is no equilibrium point. This ends the proof.

Secondly we give the sufficient condition for the positive solutions of equation (1) to be bounded and its equilibrium point to be global asymptotically stable.

Theorem 4.3: (i) If s = 1 and α ≥ 1, then ω ? l = θ.

(ii) If s > 1 and α = 1, then ω\l ≠ θ.

(iii) If s > 1 and α > 1, then ω\l ≠ θ.

(iv) If s > 1 and α ? (0,1), then ω\l ≠ θ.

Proof: (i) Let (xn) be a bounded solution of the difference equation Equation

Since xn+1 ≥ αxn ≥ xn for each n ≥ -1. Hence xn is convergent, which gives a xn contradiction. So all solutions of equation (1) are unbounded.

(ii) For s > 1 and α = 1, we have

Equation

(iii) For s > 1, α > 1 and by using condition (ii), we obtain

Equation

(iv) For s > 1 and 0 < α < 1, we have

Equation

Theorem 4.4: If s = 1, then ω = l.

Proof: We have two cases:

(i) Let s = 1 and α ? (1/2;1), then by using equation (1) we get

Equation

for all n ≥ -1, hence for odd indices we obtain

Equation

By induction, we deduce

Equation

And similarly, we get for even indices

Equation

This completes the proof.

Lemma 4.5: Given equation (1), for all x-1 and x0 ? (0,∞) there exists Equationsuch that if n ≥ n0, then xn ≥ 1.

Proof: Let x-1 and x0 ? (0,∞) and there exists Equation such that if n ≥ n0 with xn < 1. By using equation (1), we have

Equation

Hence there exists Equation for every n ≥ n0 -1, then xn < 1, so continuing in the same manner we get x-1 < 1. This gives a contradiction.

Theorem 4.6: If s ? (0,1), then ω = l.

Proof: We have three cases:

(i) For α ? (0,1) and s ? (0,1), since from lemma (4.5), there exists Equation such that if n ≥ n0, then xn ≥ 1. By using equation (1) and theorem (4.4), we obtain

Equation.

And since Equation is bounded, so the proof follows.

(ii) For α>1 and s ? (0,1), since from lemma (4.5), there exists Equation such that if n ≥ n0, then xn ≥ 1. By using equation (1) and theorem (4.4), we have

Equation

And since Equation is bounded, so the proof follows.

(iii) For α = 1 and s ? (0,1), since from lemma (4.5), there exists Equation such that if n ≥ n0, then xn ≥ 1. By using equation (1) and theorem (4.4), we have

Equation

And since Equation is bounded, so the proof ends.

Theorem 4.7: For s > 1; the unique positive equilibrium x of equation (1) is not locally asymptotically stable.

Proof: Since Equation of equation (1) is locally asymptotically stable, if and only if,

Equation

For s > 1, Equation the previous inequality gives a contradiction.

This ends the proof.

Theorem 4.8: For s = 1, the unique positive equilibrium Equation of equation (1) is locally asymptotically stable, if and only if, Equation

Proof: For s = 1, we have Equation then Equation is locally asymptotically stable, if and only if, Equation

Theorem 4.9: For s = 1 and Equation, the unique positive equilibrium Equation of equation (1) is global attractor.

Proof: Since (xn) is bounded, let (xn) be a divergent sequence with x-1 = x0 = 1, then without loss of generality there exists two subsequence (x2n) and (x2n+1) with Equation then from equation (1) we have

Equation

By taking the limit as n→∞, we obtain α = 0, this is a contradiction which completes the proof.

By using theorems (2.8) and (2.9), we get the following result.

Theorem 4.10: For s = 1 and Equation the unique positive equilibrium Equation of equation (1) is global asymptotically stable. This finishes the proof.

Theorem 4.11: For s?(0,1) and Equation the unique positive equilibrium Equation of equation (1) is global asymptotically stable.

Proof: By lemma (2.1), Equation is locally asymptotically stable. So, we have to show that for all Equation there exists an unique positive equilibrium Equation with Equation. Let Equation. By theorem (2.6), Equation. Thus, we obtain 0 < l = lim inf xn, L = lim sup xn < ∞.

Hence from equation (1), we get

Equation (5)

We have to show that L = l, otherwise L > l. From equation (5), we obtain

Equation (6)

By using the mean value theorem for the function f(x) = x(1-s) in (l,L), we find a constant c ? (l,L) and from inequality (6), we obtain

Equation (7)

From lemma (2.5) and for α ≥ 1, we have

Equation (8)

which gives L > l ≥ 2, therefore inequality (7) will be

2-3s ≥ α (1-s). (9)

Equation (9) with the values α = 1 and s = 1/3 gives a contradiction. Thus, we find L = l.

Periodicity of the Solutions of Equation

Firstly we study the convergence of the positive solution of equation (1) when s = 1 to a prime two periodic solution.

Theorem 5.1: If Equation, then equation (1) has a solution of prime period two.

Proof: Let Equation be a periodic solution of period two, we have

Equation (10)

then (x-1, x0) is a solution of the system

Equation (11)

Let Equation . It is obvious that if equation (10) holds, hence Equation has a solution of period two. The two equalities (11) is correspondent to

Equation.

Since Equation To find another root (x = a) of F(x), we must have Equation By simple calculations, we obtain Equation

From theorem (4.3), we omit the condition α > 1. Hence by taking x-1 = a and Equation we get a prime 2-periodic solution. This completes the proof.

Secondly we study the convergence of the positive solution of equation (1) when s ? (0,1) to a prime two periodic solution.

Theorem 5.2: If s ? (0,1), Equation, there exists a positive number ?1 such that

Equation(12)

and

Equation. (13)

Then, equation (1) has a solution of prime period two.

Proof: Let Equation be a periodic solution of period two, we have

Equation (14)

then x-1 and x0 is a solution of the system

Equation (15)

The system (15) is correspondent to the equation

Equation (16)

Thus, we have

Equation

More, from inequality (13) and equation (16), we have

Equation

Hence, the equation F(x) = 0 has a root b = 1 + ?0 other than Equation of equation (1), where 0 < ?0 < ?1, for all ? ? (1,1 + ?1). So, we have

Equation

Consider the function

F(x) = ? - α(1 + ?)s

For Equation we have f is increasing and from inequality (12), we get f(?0) < f(?1) < 0, which gives

Equation

By taking x-1 = b and Equation then we have a prime 2-periodic solution. This ends the proof.

Thirdly for s > 1, we study the convergence of the positive solution of equation (1) to a prime two periodic solution.

Theorem 5.3: If Equation there exists a positive number ?1 such that

Equation (17)

and

Equation (18)

Then, equation (1) has a solution of prime period two.

Proof: The same previous proof with Equation we have f is decreasing and from inequality (17), we get

f(?0) > f(?1) > 0,

which gives

Equation

By taking x-1 = b and Equation we have a prime 2-periodic solution. This completes the proof.

Convergence of the Solutions of Equation

Firstly we study the convergence of every positive solution of equation (1) to a prime two periodic solution. we begin with the following lemma.

Lemma 6.1: Suppose Equation and β>1. Then, the following conditions are contented.

(i) Equation if and only if, Equation

(ii) Equation if and only if, Equation

Proof: (i) Replace n by 2n + 1 in equation (1), we get

Equation

If Equation and vice versa.

(ii) Replace n by 2n in equation (1), we have

Equation

Let Equation and vice versa.

Theorem 6.2: If one of the following

(1) s = 1 and α < 1/3.

(2) s ? (0,1), 0 < α ≤ 1/s, inequalities (12) and (13).

is satisfied, then each positive solution of equation (1) converges to a prime two periodic solution.

Proof: Let (xn) be a positive solution of equation (1), then from theorem (4.4) or (4.6) we have (xn) is bounded and not convergent. By using Bolzano-Weierstrass theorem, there exists a subsequence convergent let it without loss of generality (x2n), hence from lemma (6.1) also (x2n + 1) is convergent and with theorem (5.1) or (5.2) the proof follows.

By the same manner as above we give the following result.

Theorem 6.3: If s > 1, α ≥ 1/s, inequalities (17) and (18) are hold, then there exists a positive solution of equation (1) converges to a prime two periodic solution.

Secondly we study the existence of a non-oscillatory solution of equation (1) and the convergence of every positive solution of equation (1) to the positive equilibrium Equation.

Theorem 6.4: There exist a non-oscillatory solution of equation (1).

Proof: By choosing x-1 and Equation, we obtain

Equation

By induction, we deduce Equationfor all n ≥ 0. If x-1 and Equation we have also Equation

Theorem 6.5: All non-oscillatory solutions of equation (1) converge to the positive equilibrium Equation.

Proof: Let (xn) be a non-oscillatory solutions of equation (1), then we get

Equation

Then (xn) is decreasing and bounded from below, then it is convergent. Also if

Equation

Then (xn) is increasing and bounded from above, hence it is convergent. This ends the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this article.

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