Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 973714, 23 pages
http://dx.doi.org/10.1155/2009/973714
Research Article

Boundedness, Attractivity, and Stability of a Rational Difference Equation with Two Periodic Coefficients

School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

Received 24 August 2008; Accepted 11 January 2009

Academic Editor: Yong Zhou

Copyright © 2009 G. Papaschinopoulos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the boundedness, the attractivity, and the stability of the positive solutions of the rational difference equation , , where , are positive sequences of period 2.

1. Introduction

In [1], Camouzis et al. studied the global character of the positive solutions of the difference equation: where are positive parameters and the initial values are positive real numbers.

The mathematical modeling of a physical, physiological, or economical problem very often leads to difference equations (for partial review of the theory of difference equations and their applications see [212]). Moreover, a lot of difference equations with periodic coefficients have been applied in mathematical models in biology (see [1315]). In addition, between others in [1619], we can see some more difference equations with periodic coefficients that have been studied.

In this paper, we investigate the difference equation where , are positive sequences of period and the initial values are positive numbers.

Our goal in this paper is to extend some results obtained in [1]. More precisely, we study the existence of a unique positive periodic solution of (1.2) of prime period . In the sequel, we investigate the boundedness, the persistence, and the convergence of the positive solutions to the unique periodic solution of (1.2). Finally, we study the stability of the positive periodic solution and the zero solution of (1.2).

If we set , , it is easy to prove that (1.2) is equivalent to the following system of difference equations: where , are positive constants and the initial values , are positive numbers. So in order to study (1.2) we investigate system (1.3).

2. Existence of the Unique Positive Equilibrium of System (1.3)

In the following proposition, we study the existence of the unique positive equilibrium of system (1.3).

Proposition 2.1. Consider system (1.3) where , are positive constants and the initial values , are positive numbers. Suppose that are satisfied. Then system (1.3) possesses a unique positive equilibrium.

Proof. Let be a positive equilibrium of system (1.3) thenEquations (2.2) imply that is a solution of the equation
Suppose that Let , and be the solutions of (2.3). Then from (2.1), (2.3), and (2.4) we take and so (2.3) has unique positive solution . Then from (2.2) and (2.4) we haveand so system (1.3) has a unique positive equilibrium.
Now suppose that If , and are the solutions of (2.3), then from (2.3) and (2.7) we take and so (2.3) has a negative solution, but also (2.3) has a solution in the interval , sinceMoreover, (2.3) has a solution in the interval , since therefore, we get (2.6) and so system (1.3) has a unique positive equilibrium.
Finally, suppose that If , and are the solutions of (2.3), then from (2.3) and (2.11), we take We have and since , it is obvious that (2.3) has a solution in the interval . From (2.3), we get
If equation has complex roots, then it is obvious that is the unique solution of (2.3). Therefore, we get (2.6), and so system (1.3) has a unique positive equilibrium.
Now, suppose that the roots of are real numbers.
Suppose that , then it is obvious thatand so we have that (2.3) has a unique solution
If , then it holds that which implies that (2.3) has a unique solution
Therefore, we can take (2.6) and so system (1.3) has a unique positive equilibrium. This completes the proof of the proposition.

3. Boundedness and Persistence of the Solutions of System (1.3)

In the following propositions we study the boundedness and the persistence of the positive solutions of system (1.3). In the sequel we will use the following result which has proved in [20].

Theorem 3.1. Assume that all roots of the polynomial where have absolute value less than 1, and let be a nonnegative solution of the inequality Then, the following statements are true.
(i) If is a nonnegative bounded sequence, then is also bounded. (ii) If , then

Proposition 3.2. One considers the system of difference equations (1.3) where , are positive constants and the initial values , are positive numbers. Then the following statements are true.
(i) If then every solution of (1.3) is bounded. (ii) Ifthen every solution of (1.3) is bounded and persists.

Proof. Let be an arbitrary solution of (1.3).
(i) From (3.3), we get that one of the three following conditions holds:
We assume that (3.5) is satisfied. We prove that there exists a positive integer such that First, we show that if there exists a positive integer such that then In contradiction, we assume thatUsing relations (1.3), (3.5), and (3.11), we get that and so relations (1.3) and (3.3) imply that which contradicts (3.9). So and working inductively, we get (3.10).
If then from the analogous relations (3.9) and (3.10), we getNow, suppose that we prove that there exists a positive integer such that From (3.3), there exists a positive integer such that If , then (3.16) is true for .
Now, suppose that Then from (1.3), (3.5), and (3.18), we get and so from (1.3), (3.3), and (3.5), we have that If , then (3.16) is true for .
Now, suppose that Using (1.3), (3.3), (3.5), (3.20) and arguing as to prove (3.19) we getWorking inductively, we get that From (3.22) for , we get which contradicts (3.17). So which means that (3.16) holds for .
Arguing as for , we can prove that there exist positive integers such that From (3.16) and (3.23), we get that there exists a positive integer such that Finally, from (1.3) and (3.24), we get and so (3.8) is true for .
Similarly, we can prove that if (3.6) holds, then there exists a positive integer such that
Finally, suppose that (3.7) hold. From (1.3) and (3.7), we have and so, From (3.27), we get and so the subsequences either are bounded from below by and decreasing or bounded from above by and increasing. Hence, is bounded and persists. Similarly, we can prove that is bounded and persists. This completes the proof of part (i) of the proposition.
(ii) In statement (i), we have already proved that if (3.7) hold, then every solution of (1.3) is bounded and persists. So, from (3.4), it remains to show that if either orholds, then the solution persists. From (3.3), (3.8), (3.25), (3.29), and (3.30), we get that We consider the positive number such that Moreover, if then it is easy to see that for the functions (3.33), is increasing with respect to for any , and is increasing with respect to for any .
Therefore, from (1.3), (3.31), and (3.32) we have and working inductively, we take Therefore, persists and using statement (i), then is bounded and persists. This completes the proof of the proposition.

Proposition 3.3. One considers the system of difference equations (1.3) where , are positive constants, and the initial values , are positive numbers. Then, the following statements are true.
(i) If then every solution of (1.3) persists. (ii) If then every solution of (1.3) is bounded and persists.

Proof. Let be an arbitrary solution of (1.3).
(i)From (3.36), we have or
Arguing as in the proof of statement (i) of Proposition 3.2, we can easily prove that if (3.38) holds, then there exists a positive integer such that and if (3.39) holds, then there exists a positive integer such that
(ii)From Proposition 3.2, we have that if (3.7) holds, then every solution of (1.3) is bounded and persists. So, from (3.37), it remains to show that if eitheror holds, then the solution is bounded and persists.
From (3.36), (3.40), (3.41), (3.42), and (3.43), we get that Suppose that
From (1.3) and (3.44), we haveWe have for the functions (3.33) that is decreasing with respect to for any , and is decreasing with respect to for any , . Therefore, relations (1.3), (3.44), and (3.46) imply that and so from (1.3) and (3.46), Working inductively, we can prove thatThen from (3.42), (3.43), (3.45), and Theorem 3.1, is bounded. Similarly, we take that is bounded. Therefore, from (3.44), the solution is bounded and persists.
Now, suppose that We claim that is bounded. For the sake of contradiction, we assume that is not bounded. Then, there exists a subsequence such that Moreover, from (1.3) and (3.50), we get and so from (3.51), Moreover, from (1.3) and (3.50), and so from (3.54), Working inductively, we can prove that We claim that is a bounded sequence. Suppose on the contrary that there exists an unbounded subsequence of and without loss of generality, we may suppose that Arguing as above, we can easily prove that Also, since from (1.3), from (3.58), we have that and so eventually, From (1.3), (3.50), and (3.61), we have where Therefore, using (1.3) and (3.50), we get and since from (3.57) and (3.59), we have that as we can easily prove that eventually, which contradicts to (3.52).
Therefore, is a bounded sequence. From (1.3), (3.50), and (3.57), we get Similarly, from (1.3), (3.50) and (3.57) and (3.66) follows, Now, we prove that Otherwise, and without loss of generality, we may suppose that So, relations (1.3), (3.50), and (3.67) imply that and so Moreover, from (1.3), (3.44), and (3.50), we get eventually and so from (3.66), which contradicts to (3.70).
Hence, (3.68) is true.
Similarly, we can prove that Therefore, from (3.68) and (3.72), we have eventually where are positive real numbers.
Hence, from (1.3), (3.50), and (3.73) we have Then from (3.57), we can prove that eventually which contradicts to (3.52).
Therefore, is a bounded sequence. Moreover, from (1.3), (3.50), we take that is bounded. Therefore, the solution is bounded and persists. This completes the proof of the proposition.

4. Attractivity of the Positive Equilibrium of System (1.3)

In the following propositions, we study the convergency of the solutions of system (1.3) to its positive equilibrium.

Proposition 4.1. One considers the system of difference equations (1.3) where , are positive constants, and the initial values , are positive numbers. If either (3.29) or (3.30) hold, then every solution of (1.3) tents to the positive equilibrium of (1.3).

Proof. Let be an arbitrary solution of (1.3). From Proposition 3.2, there exist From (1.3), (3.31), and the monotony of functions (3.33), we have and hence The third inequality of (4.3), implies thatand so from the last inequality of (4.3), we have Hence, we get or The first inequality of (4.3), implies that and so from second inequality of (4.3), we get Using (4.3), we have Therefore, from (4.5) and (4.10), we get
Using (4.9) and (4.11), we have In Proposition 2.1, we proved that (2.3) has a unique positive solution . We can write where is defined in (2.3) and for any . Then from (4.7), (4.12), and (4.13), we have Therefore, from (4.10) and (4.14), which implies that
In addition, using (4.15), the first and the third inequalities of (4.3), we have and so (4.10) implies that This completes the proof of the proposition.

Proposition 4.2. One considers the system of difference equations (1.3) where , are positive constants, and the initial values , are positive numbers. If either (3.42) or (3.43) hold, then every solution of (1.3) tents to the positive equilibrium of (1.3).

Proof. Let be an arbitrary solution of (1.3). From Proposition 3.3, there exist , such that (4.1) are satisfied.
From (1.3), the monotony of functions (3.33) and (3.44), we have and hence which implies that Therefore, First suppose that (3.45) holds. Then from (3.42) or (3.43), and (3.45), we get which means thatUsing (4.20), it is obvious that So if (3.45) holds, the proof is completed.
Now, suppose that (3.50) hold. Then from (4.20), we have Moreover, from (4.24), it follows that In addition, from (3.50), the first and the second inequalities of (4.19), we getTherefore, from (4.25) and (4.26), we have We may assume that there exists a positive integer such that Moreover, from (1.3), (3.50), and (4.28), we get Since is decreasing with respect to , for any if or , then from (3.44), and (3.50), we get which contradicts to (4.27). So, Using the same argument, we can prove that and so Also, from (4.24), we have Therefore, where obviously . This completes the proof of the proposition.

Proposition 4.3. One considers the system of difference equations (1.3) where , are positive constants, and the initial values , are positive numbers. If relations (3.7) hold, then every solution of (1.3) tents to the positive equilibrium of (1.3).

Proof. Let be an arbitrary solution of (1.3). From the proof of Proposition 3.2, the subsequences , and are monotone and , are bounded and persist. So, there exist positive numbers , and such that and from (1.3) and (3.7), we get Then, we have and hence, Therefore, we take So, Therefore, if (resp., ), we have (resp., ) and so from (4.37), (resp., ). Hence, and from (4.37), Similarly, we can prove that . This completes the proof of the proposition.

5. Stability of System (1.3)

In this section we find conditions so that the positive equilibrium and the zero equilibrium of (1.3) are stable.

Proposition 5.1. Consider system (1.3) where , are positive constants and the initial values , are positive numbers. Then, the following statements are true.
(i) If then the unique positive equilibrium of (1.3) is globally asymptotically stable. (ii) If then the zero equilibrium of (1.3) is locally asymptotically stable.

Proof. (i) Since is the unique positive positive equilibrium of (1.3), we have Then from (5.1) and (5.3), we get Without loss of generality we assume that . Then from (5.4), it results that which means that Moreover, from (5.4) and (5.6), we get In addition, from (5.3), we have and so Then the linearized system of (1.3) about the positive equilibrium is where The characteristic equation of is According to Remark 1.3.1 of [7], all the roots of (5.12) are of modulus less than 1 if and only if From (5.3), we get Then from (5.6), (5.7), and (5.14), inequality (5.13) is equivalent to Using (5.9), inequality (5.15) holds if (5.1) are satisfied. Using Propositions 4.1 and 4.3, we have that the unique positive equilibrium of (1.3) is globally asymptotically stable.
(ii) Arguing as above, we can prove that the linearized system of (1.3) about the zero equilibrium is where The characteristic equation of is Using [7, Remark 1.3.1], all the roots of (5.18) are of modulus less than 1 if and only if relation (5.2) holds. This completes the proof of the proposition.

6. Conclusion

In this paper, in order to investigate (1.2), we study the equivalent system (1.3). Summarizing the results of Sections 2, 3, 4, we get the following statements, concerning (1.2).

(i)If (2.1) hold, then (1.2) has a unique positive periodic solution of period 2.(ii)If either (3.4) or (3.37) holds, then every positive solution of (1.2) is bounded and persists and tends to the unique positive periodic solution.(iii)If (5.1) hold, then the unique periodic solution of (1.2) is globally asymptotically stable and if (5.2) holds, then the zero solution of (1.2) is locally asymptotically stable.

Open Problem
Consider the difference equation (1.2) where , are positive sequences of period 2, and the initial values , are positive numbers. Prove that
(i)if are satisfied, then every positive solution of (1.2) is bounded and persists;(ii)if relations (6.1) are satisfied, then every positive solution of (1.2) tends to the unique positive equilibrium of (1.2) as .

Acknowledgment

The authors would like to thank the referees for their helpful suggestions.

References

  1. E. Camouzis, E. Chatterjee, and G. Ladas, “On the dynamics of xn+1=(δxn2+xn3)/(A+xn3),” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 230–239, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, vol. 155 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992. View at Zentralblatt MATH · View at MathSciNet
  3. D. Benest and C. Froeschlé, Eds., Analysis and Modelling of Discrete Dynamical Systems, D. Benest and C. Froeschlé, Eds., vol. 1 of Advances in Discrete Mathematics and Applications, Gordon and Breach Science, Amsterdam, The Netherlands, 1998. View at Zentralblatt MATH · View at MathSciNet
  4. L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, New York, NY, USA, 1988. View at Zentralblatt MATH · View at MathSciNet
  5. S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1996. View at Zentralblatt MATH · View at MathSciNet
  6. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. View at Zentralblatt MATH · View at MathSciNet
  7. V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at Zentralblatt MATH · View at MathSciNet
  8. S. Stević, “On a discrete epidemic model,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 87519, 10 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Stević, “On global periodicity of a class of difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 23503, 10 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. Stević, “On the recursive sequence xn+1=A+xnp/xn1p,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 34517, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Stević, “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 56813, 20 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Stević, “Asymptotic periodicity of a higher-order difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 13737, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. M. Cushing, “Periodically forced nonlinear systems of difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 547–561, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859–872, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. M. Cushing and S. M. Henson, “A periodically forced Beverton-Holt equation,” Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1119–1120, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. R. S. Kulenović, G. Ladas, and C. B. Overdeep, “On the dynamics of xn+1=pn+xn1/xn with a period-two coefficient,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 905–914, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. G. Papaschinopoulos and C. J. Schinas, “On a (k+1)-th order difference equation with a coefficient of period k+1,” Journal of Difference Equations and Applications, vol. 11, no. 3, pp. 215–225, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. Papaschinopoulos and C. J. Schinas, “Periodic and asymptotic character of a difference equation with periodic coefficient,” Mathematical Sciences Research Journal, vol. 9, no. 2, pp. 32–40, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On a difference equation with 3-periodic coefficient,” Journal of Difference Equations and Applications, vol. 11, no. 15, pp. 1281–1287, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, vol. 32 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990. View at Zentralblatt MATH · View at MathSciNet