GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION

Journal of Applied Mathematics & Informatics.
2014.
Sep,
32(5_6):
843-848

- Received : February 15, 2014
- Accepted : June 19, 2014
- Published : September 30, 2014

Download

PDF

e-PUB

PubReader

PPT

Export by style

Share

Article

Metrics

Cited by

TagCloud

In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation
where {
p_{n}
} is a two periodic sequence of nonnegative real numbers and the initial conditions
x
_{-1}
,
x
_{0}
are arbitrary positive real numbers.
AMS Mathematics Subject Classification: 39A10.
where {
p_{n}
} is a two periodic sequence of nonnegative real numbers and the initial conditions
x
_{−1}
,
x
_{0}
are arbitrary positive numbers.
As far as we can examine, this is the first work devote to the investigation of the type Eq.(1.1).
Now, we assume
p
_{2n}
=
α
and
p
_{2n+1}
=
β
in Eq.(1.1). Then we have
and
The autonomous case of Eq.(1.1) is
where
p
> 0 and the initial conditions
x
_{−1}
,
x
_{0}
are arbitrary positive numbers. We now consider the local asymptotic stability of the unique equilibrium
of Eq.(1.4).
The linearized equation for Eq. (1.4) about the positive equilibrium
is
The following theorem is given in
[1]
.
Theorem A.
Consider Eq
. (1.4)
and assume that x
_{−1}
,
x
_{0}
,
p
∈ (0,∞).
Then the unique positive equilibrium
of Eq
. (1.4)
is globally asymptotically stable
.
Theorem 2.1.
Suppose that α
> 1
and β
> 1
with α
≠
β
,
then every positive solution of Eq
.(1.1)
is bounded
.
Proof
. It is clear from Eq. (1.2) and (1.3) that
Then, from (1.2) and (2.1) we obtain
and from (1.3) and (2.1) we obtain
From (2.3), (2.4) using induction we get
The result now follows.
Proposition 3.1.
Consider Eq
. (1.1)
when the case α
≠
β and assume that α
,
β
∈ (0,∞).
Then there exist prime two periodic solutions of Eq
. (1.1).
Proof.
In order Eq. (1.1) to possess a periodic solution {
x_{n}
} of prime period 2, we must find positive numbers
x
_{−1}
,
x
_{0}
such that
Let
x
_{−1}
=
x
,
x
_{0}
=
y
, then from (3.1) we obtain the system of equations
We prove that (3.2) has a solution
From the first relation of (3.2) we have
From (3.3) and the second relation of (3.2) we obtain
Now we consider the function
Then from (3.4) we get
Hence Eq. (3.4) has a solution
Then if
we have that the solution
of Eq. (1.1) with initial values
is a periodic solution of period two.
Theorem 3.2.
Consider Eq
. (1.1)
when the case α
≠
β and assume that α
,
β
∈ (0,∞).
Suppose that
Then the two periodic solutions of Eq
. (1.1)
are locally asymptotically stable
.
Proof
. From equations (1:3), (1:4) and Proposition 3:1 there exist
such that
We set
x
_{2n−1}
=
u_{n}
,
x
_{2n}
=
v_{n}
in equations (1.3), (1.4) and so we have
Then
is the positive equilibrium of Eq. (3.8), and the linearised system of Eq. (3.8) about
is the system
The characteristic equation of
B
is
Using Eq. (3.6), from Eq. (3.7), since
we have
and we obtain
Then, from (3.10) and Theorem 1.3.7 of Kocic and Ladas in
[4]
, all the roots of Eq. (3.9) are modulus less than 1. Therefore, from Proposition 3.1, system (3.8) is asymptotically stable. The proof is complete.
Theorem 3.3.
Consider Eq
. (1.1)
when the case α
≠
β
.
Assume that α
> 1,
β
> 1.
Then every positive solution of Eq
. (1.1)
converges to a two-periodic solution of Eq
. (1.1).
Proof
. Since
α
> 1,
β
> 1, we know by Theorem 2.1 that every positive solution of Eq. (1.1) is bounded, it follows that there are finite
exist. Then it is easy to see from Eq. (1.2) and (1.3) that
and
Thus, we have
and
This implies that
and
Then, we get
and
Now, we shall prove that
s
=
S
and
l
=
L
. It is clear that if
l
=
L
, then by (3.11) it must be
s
=
S
. Similarly, if
s
=
S
, then by (3.12) it must be
l
=
L
.
Hence we assume that
s
<
S
and
l
<
L
. From (3.11) and (3.12) we have
then we obtain a contradiction. So, we get
s
=
S
and
l
=
L
Moreover, it is obvious that since
α
≠
β
, then from Eq. (1.2) and Eq. (1.3)
Then it is clear that every positive solution of Eq. (1.1) converges to a twoperiodic solution of Eq. (1.1). The proof is complete.
Finally, using Proposition 3.1, Theorems 3.1 and 3.2, we have the following Theorem.
Theorem 3.4.
Consider Eq
. (1.1)
when the case α
≠
β
.
Assume that α
> 1,
β
> 1
and that
(3.6)
holds
.
Then two-period solutions of Eq
. (1.1)
are globally asymptotically stable
.
Özkan Öcalan received M. Sc. from Afyon Kocatepe University and Ph. D. at Ankara University. Since 1995 he has been working at Afyon Kocatepe University. His research interests include oscillation and stability of difference and differential equations.
Afyon Kocatepe University, Faculty of Science and Arts, Department of Mathematics, ANS Campus, 03200, Afyonkarahisar, Turkey.
e-mail:ozkan@aku.edu.tr

1. Introduction

Recently, there has been an increasing interest in the study of the qualitative analyses of rational difference equations. For example, see
[1
−
8]
and the references cited therein.
This work studies the boundedness character and the global asymptotic stability for the positive solutions of the difference equation
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

2. Boundedness Character of Eq. (1.1)

In this section, we investigate the boundedness character of Eq. (1.1). So, we have the following result.
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

3. Stability and Periodicity for Eq. (1.1)

In this section, we investigate the periodicity and stability character of positive solutions of Eq. (1.1). Now, we have the following result.
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

BIO

Devault R.
,
Ladas G.
,
Schultz S.W.
(1998)
On the recursive sequence xn+1= (A/xn) + (1/xn-2)
Proc. Amer. Math. Soc.
126
(11)
3257 -
3261
** DOI : 10.1090/S0002-9939-98-04626-7**

Grove E.A.
,
Ladas G.
2005
Periodicities in nonlinear difference equations
Chapman and Hall/Crc

Kocić V.
,
Ladas G.
1993
Global behavior of nonlinear difference equations of higher order with applications
Kluwer Academic Publishers
Dordrecht

Kulenović M.R.S.
,
Ladas G.
,
Overdeep C.B.
(2003)
On the dynamics of xn+1= pn+ (xn-1/xn)
J. Difference Equ. Appl.
9
(11)
1053 -
1056
** DOI : 10.1080/1023619031000154644**

Kulenović M.R.S.
,
Ladas G.
,
Overdeep C.B.
(2004)
On the dynamics of xn+1= pn+(xn-1/xn) with a period-two coefficient
J. Difference Equ. Appl.
10
(10)
905 -
914
** DOI : 10.1080/10236190410001731434**

Öcalan Ö.
(2014)
Dynamics of the difference equation xn+1= pn+(xn-k)/(xn) with a Period-two Coefficient
Appl. Math. Comput.
228
31 -
37
** DOI : 10.1016/j.amc.2013.11.020**

Papanicolaou V.G.
(1996)
On the asymptotic stability of a class of linear difference equations
Mathematics Magazine
69
34 -
43
** DOI : 10.2307/2691392**

Stević S.
(2003)
On the recursive sequence xn+1= αn+ (xn-1/xn)
Int. J. Math. Sci.
2
(2)
237 -
243

Citing 'GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION
'

@article{ E1MCA9_2014_v32n5_6_843}
,title={GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2014.843}, DOI={10.14317/jami.2014.843}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={ÖCALAN, ÖZKAN}
, year={2014}
, month={Sep}