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ON THE DYNAMICS OF
ON THE DYNAMICS OF
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 599-607
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : September 20, 2013
  • Accepted : March 22, 2014
  • Published : September 30, 2014
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A. M. AHMED

Abstract
In this paper, we investigate the behavior of solutions of the difference equation where k ∈ {1,2}, a ≥ 0, and x-j > 0, j = 0, 1, ..., k . AMS Mathematics Subject Classification : 39A10, 39A21, 39A30.
Keywords
1. Introduction
Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. More importantly, difference equations also appear in the study of discretization methods for differential equations. Several results in the theory of difference equations have been obtained as more or less natural discrete analogues of corresponding results of differential equations.
Recently there has been a lot of interest in studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. For example,
Abu-Saris et al. [1] investigated the asymptotic stability of the difference equation
PPT Slide
Lager Image
For other related results( [2 - 16] ).
In this paper, we investigate the behavior of solutions of the difference equation
PPT Slide
Lager Image
where k ∈ {1,2}, a ≥ 0, and x-j > 0, j = 0, 1, ..., k .
We need the following definitions.
Definition 1.1. Let I be an interval of real numbers and let
PPT Slide
Lager Image
be a continuously differentiable function. Consider the difference equation
PPT Slide
Lager Image
with x−k , x k+1 , ..., x 0 I . Let
PPT Slide
Lager Image
be the equilibrium point of Eq.(1.2). The linearized equation of Eq.(1.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
where
PPT Slide
Lager Image
The characteristic equation of Eq.(1.3) is
PPT Slide
Lager Image
(i) The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is locally stable if for every ϵ > 0, there exists δ > 0 such that for all x−k , x k+1 , ..., x −1 , x 0 I with
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
(ii) The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is locally asymptotically stable if
PPT Slide
Lager Image
is locally stable and there exists γ > 0, such that for all x−k , x k+1 , ..., x −1 , x 0 I with
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
(iii) The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is global attractor if for all x−k , x k+1 , ..., x −1 , x 0 I we have
PPT Slide
Lager Image
(iv) The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is globally asymptotically stable if
PPT Slide
Lager Image
is locally stable, and
PPT Slide
Lager Image
is also a global attractor of Eq.(1.2).
(v) The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is unstable if
PPT Slide
Lager Image
is not locally stable.
Definition 1.2. A positive semicycle of
PPT Slide
Lager Image
of Eq.(1.2) consists of a ‘string’ of terms { xl , x l+1 , ..., xm }, all greater than or equal o
PPT Slide
Lager Image
, with l ≥ - k and m ≤∞ and such that either l = - k or l > - k and
PPT Slide
Lager Image
and either m = ∞ or m < ∞ and
PPT Slide
Lager Image
A negative semicycle of
PPT Slide
Lager Image
of Eq.(1.2) consists of a ‘string’ of terms { xl , x l+1 , ..., xm }, all less than
PPT Slide
Lager Image
, with l ≥ - k and m ≤ ∞ and such that either l = - k or l > - k and
PPT Slide
Lager Image
and either m = ∞ or m < ∞ and
PPT Slide
Lager Image
.
Definition 1.3. A solution
PPT Slide
Lager Image
of Eq.(1.2) is called nonoscillatory if there exists N ≥ - k such that either
PPT Slide
Lager Image
and it is called oscillatory if it is not nonoscillatory.
We need the following theorems.
Theorem 1.4 ( [16] ). (i) If all roots of Eq.(1.4) have absolute value less than one, then the equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is locally asymptotically stable.
(ii) If at least one of the roots of Eq.(1.4) has absolute value greater than one, then
PPT Slide
Lager Image
is unstable.
The equilibrium point
PPT Slide
Lager Image
of Eq.(1.2) is called a saddle point if Eq.(1.4) has roots both inside and outside the unit disk.
Theorem 1.5 ( [16] ). Assume that p 1 , p 2 , ..., pk ∈ ℝ and k ∈ {1, 2, ...}. Then
PPT Slide
Lager Image
is a sufficient condition for the asymptotic stability of the difference equation
PPT Slide
Lager Image
2. Behavior of solutions of Eq.(1.1) when k = 1 and a = 0.
In this section we give the closed form of solutions of Eq.(1.1) when k = 1 and a = 0.
In this case the difference equation (1.1) becomes
PPT Slide
Lager Image
with positive initial conditions x -1 and x 0 .
Eq. (2.1) is linear which have the solution
PPT Slide
Lager Image
3. Behavior of solutions of Eq.(1.1) when k = 2 and a = 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when k = 2 and a = 0.
In this case the difference equation (1.1) becomes
PPT Slide
Lager Image
with positive initial conditions x -2 , x -1 and x 0 .
Eq.(3.1) has a unique equilibrium point
PPT Slide
Lager Image
Theorem 3.1. The equilibrium point
PPT Slide
Lager Image
of Eq.(3.1) is locally asymptotically
Proof . Since the linearized equation of Eq.(3.1) about the equilibrium point
PPT Slide
Lager Image
can be written in the following form
PPT Slide
Lager Image
then the proof follows immediately from Theorem B.
Theorem 3.2. The equilibrium point
PPT Slide
Lager Image
of Eq.(3.1) is globally asymptotically stable.
Proof . From Eq.(3.1) it is easy to show that x n+1 < x n−1 for all n ≥ 0 and so the even terms converge to a limit (say L 1 ≥ 0) and the odd terms converge to a limit (say L 2 ≥ 0). Then
PPT Slide
Lager Image
which implies that L 1 = L 2 = 0, and the proof is complete.
4. Behavior of solutions of Eq.(1.1) when k = 1 and a > 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when k = 1 and a > 0.
In this case the difference equation (1.1) becomes
PPT Slide
Lager Image
with positive initial conditions x −1 and x 0 .
The change of variables
PPT Slide
Lager Image
reduces Eq.(4.1) to the difference equation
PPT Slide
Lager Image
Eq.(4.2) has a unique positive equilibrium point
PPT Slide
Lager Image
Theorem 4.1. The equilibrium point
PPT Slide
Lager Image
of Eq.(4.2) is locally asymptotically stable.
Proof . The linearized equation of Eq.(4.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
and so, the characteristic equation of Eq.(4.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
which implies that | λ 1 | = | λ 2 | = 0 < 1. Hence, the proof is complete.
Theorem 4.2. The equilibrium point
PPT Slide
Lager Image
of Eq.(4.2) is globally asymptotically stable .
Proof . Since
PPT Slide
Lager Image
for all n ≥ 0, then we have yn ≥ 1 for all n ≥ 1.
Furthermore
PPT Slide
Lager Image
for all n ≥ 2. So the even terms
PPT Slide
Lager Image
converge to a limit (say L 1 ≥ 0) and the odd terms
PPT Slide
Lager Image
converge to a limit (say L 2 ≥ 0). Then
PPT Slide
Lager Image
which implies that L 1 = L 2 = 1. Thus, the proof is complete.
5. Behavior of solutions of Eq.(1.1) when k = 2 and a > 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when k = 2 and a > 0.
In this case the difference equation (1.1) becomes
PPT Slide
Lager Image
with positive initial conditions x −2 , x −1 and x 0 .
The change of variables
PPT Slide
Lager Image
reduces Eq.(5.1) to the difference equation
PPT Slide
Lager Image
Eq.(5.2) has two equilibrium points
PPT Slide
Lager Image
Theorem 5.1. The equilibrium point
PPT Slide
Lager Image
of Eq.(5.2) is unstable equilibrium point.
Proof . The linearized equation of Eq.(5.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
and so, the characteristic equation of Eq.(5.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
is clear that f ( λ ) has a root in the interval (1,∞), and so,
PPT Slide
Lager Image
is an unstable equilibrium point. This completes the proof.
Theorem 5.2. The equilibrium point
PPT Slide
Lager Image
of Eq.(5.2) is locally asymptotically stable.
Proof . The linearized equation of Eq.(5.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
and so, the characteristic equation of Eq.(5.2) about the equilibrium point
PPT Slide
Lager Image
is
PPT Slide
Lager Image
which implies that | λ 1 | = | λ 2 | = | λ 3 | = 0 < 1, from which the proof is complete.
Lemma 5.3. The following identities are true
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Proof . (i)
PPT Slide
Lager Image
for n ≥ 0.
(ii)
PPT Slide
Lager Image
for n ≥ 0.
(iii)
PPT Slide
Lager Image
for n ≥ 0.
( iv )
PPT Slide
Lager Image
( v )
PPT Slide
Lager Image
Then, the proof is complete.
Theorem 5.4. Let
PPT Slide
Lager Image
be a solution of Eq.(5.2), then the following statements are true
(i) If
PPT Slide
Lager Image
for some n 0 ∈ {-1, 0, 1, 2, ...}, then
PPT Slide
Lager Image
for all n n 0 + 2.
Also if
PPT Slide
Lager Image
then
PPT Slide
Lager Image
for all n ≥ 3.
(ii) If
PPT Slide
Lager Image
for some n 0 ∈ {-2,-1, 0, 1, 2, ...}, then
PPT Slide
Lager Image
for all n n 0 .
(iii) If (i) and (ii) are not satisfied, then
PPT Slide
Lager Image
oscillates about
PPT Slide
Lager Image
, with positive semicycles of length at most three, and negative semicycles of length at most two.
Proof . (i) Let
PPT Slide
Lager Image
for some n 0 ∈ {-1, 0, 1, 2, ...}, then from Eq.(5.3) we have
PPT Slide
Lager Image
for all n n 0 + 2.
If
PPT Slide
Lager Image
then from Eq.(5.3) we have
PPT Slide
Lager Image
which implies that
PPT Slide
Lager Image
for all n ≥ 3.
(ii) Let
PPT Slide
Lager Image
for some n 0 ∈ {-2,-1, 0, 1, 2, ...}, then from Eq.(5.3) we have
PPT Slide
Lager Image
for all n n 0 .
(iii) Suppose without loss of generality that there exists n 0 ∈ {-2,-1, 0, 1, 2, ...}, such that
PPT Slide
Lager Image
Then from Eq.(5.3) we have
PPT Slide
Lager Image
The proofs of the other possibilities are similar, and will be omitted.
Theorem 5.5. The equilibrium point
PPT Slide
Lager Image
of Eq.(5.2) is globally asymptotically stable.
Proof . We proved that
PPT Slide
Lager Image
of Eq.(5.2) is locally asymptotically stable, and so it suffices to show that lim n→∞ zn = 1. If there exists n 0 ∈ {-2,-1, 0, 1, 2, ...}g, such that
PPT Slide
Lager Image
then from Theorem 5.4 we have lim n→∞ zn = 1. Also, if
PPT Slide
Lager Image
then by Theorem 5.4 we have
PPT Slide
Lager Image
for all n ≥ -2, and from Eq.(5.4), we have z n+1 > z n-1 , for n ≥ 0. So the sequences
PPT Slide
Lager Image
are increasing and bounded, which implies that the even terms
PPT Slide
Lager Image
converge to a limit (say M 1 > 0) and the odd terms
PPT Slide
Lager Image
converge to a limit (say M 2 > 0). Then
PPT Slide
Lager Image
which implies that M 1 = M 2 = 1.
Now, Suppose that
PPT Slide
Lager Image
then from Eqs.(5.4) - (5.7) we have the following results
The sequence
PPT Slide
Lager Image
is decreasing and bounded, and so converges to a limit (say L 0 > 0).
The sequence
PPT Slide
Lager Image
is increasing and bounded, and so converges to a limit (say L 1 > 0).
The sequence
PPT Slide
Lager Image
is increasing and bounded, and so converges to a limit (say L 2 > 0).
The sequence
PPT Slide
Lager Image
is decreasing and bounded, and so converges to a limit (say L 3 > 0).
The sequence
PPT Slide
Lager Image
is increasing and bounded, and so converges to a limit (say L 4 > 0).
The sequence
PPT Slide
Lager Image
is decreasing and bounded, and so converges to a limit (say L 5 > 0).
The sequence
PPT Slide
Lager Image
is decreasing and bounded, and so converges to a limit (say L 6 > 0).
So we have from Eq.(5.2) that
PPT Slide
Lager Image
The solution of this system is either Li = -1, i = 0, 1, ...6, or Li = 0, i = 0, 1, ...6, or Li = 1, i = 0, 1, ...6. Since Li > 0, i = 0, 1, ...6 , we have lim n→∞ zn = 1.
The proofs for the other cases are as follows.
PPT Slide
Lager Image
or
PPT Slide
Lager Image
are similar to the proof of the last case, and will be omitted. Therefore the proof is complete.
BIO
A.M. Ahmed obtained his B.Sc. (1997), M.Sc. (2000) and Ph. D. (2004) from Al-Azhar University. He obtained the rank associate professor from Al-Azhar University in 2012. His research interests in difference equations and its applications.
Mathematics Department, Faculty of Science, Al-Azhar University , Nasr City(11884), Cairo, Egypt.
e-mail: ahmedelkb@yahoo.com
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