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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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

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

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

(1)

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

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

(2)

If we replace x_{n} and x_{n-1} in (1) by the variables u and v respectively, then we have

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

Its characteristic equation is

(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}, x_{0}? (0;∞) with for all n≥-1.

**Definition 3.3:** An equilibrium [2] point of the difference equation (1) is called a global attractor if for every x_{-1},x_{0}?(0;∞), we have .

**Definition 3.4: **An equilibrium [3] point 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 of the difference equation (1) is called unstable if it is not locally stable.

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

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

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

.

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

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

or

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

(4)

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

(1) If α > 1, the equilibrium point 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

and l_{∞} denote the class of bounded sequences of real numbers.

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,

(ii) is unstable, if and only if,

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

**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 of equation (1).

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

(3) For s > 1, if

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

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

(iii) , 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 which gives .

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

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

We have three cases, the first one is

Hence there is two equilibrium points .

The second case is

Hence there is one equilibrium point .

The third case is

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 (x_{n}) be a bounded solution of the difference equation

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

(ii) For s > 1 and α = 1, we have

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

(iv) For s > 1 and 0 < α < 1, we have

**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

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

By induction, we deduce

And similarly, we get for even indices

This completes the proof.

**Lemma 4.5: **Given equation (1), for all x_{-1} and x_{0} ? (0,∞) there exists such that if n ≥ n_{0}, then x_{n} ≥ 1.

**Proof: **Let x_{-1} and x_{0} ? (0,∞) and there exists such that if n ≥ n_{0} with x_{n} < 1. By using equation (1), we have

Hence there exists for every n ≥ n_{0} -1, then x_{n} < 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 such that if n ≥ n_{0}, then x_{n} ≥ 1. By using equation (1) and theorem (4.4), we obtain

.

And since is bounded, so the proof follows.

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

And since is bounded, so the proof follows.

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

And since 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 of equation (1) is locally asymptotically stable, if and only if,

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

This ends the proof.

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

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

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

**Proof: **Since (x_{n}) is bounded, let (x_{n}) be a divergent sequence with x_{-1} = x_{0} = 1, then without loss of generality there exists two subsequence (x_{2n}) and (x_{2n+1}) with then from equation (1) we have

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 the unique positive equilibrium of equation (1) is global asymptotically stable. This finishes the proof.

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

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

Hence from equation (1), we get

(5)

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

(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

(7)

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

(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.

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

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

(10)

then (x_{-1}, x_{0}) is a solution of the system

(11)

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

.

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

From theorem (4.3), we omit the condition α > 1. Hence by taking x_{-1} = a and 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), , there exists a positive number ?1 such that

(12)

and

. (13)

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

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

(14)

then x_{-1} and x_{0} is a solution of the system

(15)

The system (15) is correspondent to the equation

(16)

Thus, we have

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

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

Consider the function

F(x) = ? - α(1 + ?)^{s}

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

By taking x_{-1} = b and 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 there exists a positive number ?_{1} such that

(17)

and

(18)

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

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

f(?_{0}) > f(?_{1}) > 0,

which gives

By taking x_{-1} = b and we have a prime 2-periodic solution. This completes the proof.

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 and β>1. Then, the following conditions are contented.

(i) if and only if,

(ii) if and only if,

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

If and vice versa.

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

Let 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 (x_{n}) be a positive solution of equation (1), then from theorem (4.4) or (4.6) we have (x_{n}) is bounded and not convergent. By using Bolzano-Weierstrass theorem, there exists a subsequence convergent let it without loss of generality (x_{2n}), hence from lemma (6.1) also (x_{2n} + 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 .

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

**Proof:** By choosing x_{-1} and , we obtain

By induction, we deduce for all n ≥ 0. If x_{-1} and we have also

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

**Proof: **Let (x_{n}) be a non-oscillatory solutions of equation (1), then we get

Then (x_{n}) is decreasing and bounded from below, then it is convergent. Also if

Then (x_{n}) is increasing and bounded from above, hence it is convergent. This ends the proof.

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

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