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GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION
GLOBAL DYNAMICS OF A NON-AUTONOMOUS RATIONAL DIFFERENCE EQUATION
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 843-848
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 15, 2014
  • Accepted : June 19, 2014
  • Published : September 30, 2014
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ÖZKAN ÖCALAN

Abstract
In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation where { pn } 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.
Keywords
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
where { pn } 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
PPT Slide
Lager Image
and
PPT Slide
Lager Image
The autonomous case of Eq.(1.1) is
PPT Slide
Lager Image
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
PPT Slide
Lager Image
of Eq.(1.4).
The linearized equation for Eq. (1.4) about the positive equilibrium
PPT Slide
Lager Image
is
PPT Slide
Lager Image
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
PPT Slide
Lager Image
of Eq . (1.4) is globally asymptotically stable .
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.
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
PPT Slide
Lager Image
Then, from (1.2) and (2.1) we obtain
PPT Slide
Lager Image
and from (1.3) and (2.1) we obtain
PPT Slide
Lager Image
From (2.3), (2.4) using induction we get
PPT Slide
Lager Image
PPT Slide
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The result now follows.
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.
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 { xn } of prime period 2, we must find positive numbers x −1 , x 0 such that
PPT Slide
Lager Image
Let x −1 = x , x 0 = y , then from (3.1) we obtain the system of equations
PPT Slide
Lager Image
We prove that (3.2) has a solution
PPT Slide
Lager Image
From the first relation of (3.2) we have
PPT Slide
Lager Image
From (3.3) and the second relation of (3.2) we obtain
PPT Slide
Lager Image
Now we consider the function
PPT Slide
Lager Image
Then from (3.4) we get
PPT Slide
Lager Image
Hence Eq. (3.4) has a solution
PPT Slide
Lager Image
Then if
PPT Slide
Lager Image
we have that the solution
PPT Slide
Lager Image
of Eq. (1.1) with initial values
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
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such that
PPT Slide
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We set x 2n−1 = un , x 2n = vn in equations (1.3), (1.4) and so we have
PPT Slide
Lager Image
Then
PPT Slide
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is the positive equilibrium of Eq. (3.8), and the linearised system of Eq. (3.8) about
PPT Slide
Lager Image
is the system
PPT Slide
Lager Image
The characteristic equation of B is
PPT Slide
Lager Image
Using Eq. (3.6), from Eq. (3.7), since
PPT Slide
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we have
PPT Slide
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and we obtain
PPT Slide
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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
PPT Slide
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exist. Then it is easy to see from Eq. (1.2) and (1.3) that
PPT Slide
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and
PPT Slide
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Thus, we have
PPT Slide
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and
PPT Slide
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This implies that
PPT Slide
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and
PPT Slide
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Then, we get
PPT Slide
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and
PPT Slide
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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
PPT Slide
Lager Image
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)
PPT Slide
Lager Image
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 .
BIO
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
References
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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
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