## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2011, Number 1 (2011), Article ID 671967, 24 pages.

### Asymptotic Behavior of Solutions of Delayed Difference Equations

**Full-text: Open access**

#### Abstract

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 671967, 24 pages.

**Dates**

First available in Project Euclid: 12 August 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1313171421

**Digital Object Identifier**

doi:10.1155/2011/671967

**Mathematical Reviews number (MathSciNet)**

MR2817276

**Zentralblatt MATH identifier**

1217.39019

#### Citation

Diblík, J.; Hlavičková, I. Asymptotic Behavior of Solutions of Delayed Difference Equations. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 671967, 24 pages. doi:10.1155/2011/671967. https://projecteuclid.org/euclid.aaa/1313171421

#### References

- J. Diblík, “Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique,”
*Computers and Mathematics with Applications*, vol. 45, no. 6–9, pp. 1041–1057, 2003, Advances in difference equations, I. - J. Diblík, “Discrete retract principle for systems of discrete equations,”
*Computers and Mathematics with Applications*, vol. 42, no. 3–5, pp. 515–528, 2001, Advances in difference equations, II. - J. Diblík, “Anti-Lyapunov method for systems of discrete equations,”
*Nonlinear Analysis*, vol. 57, no. 7–8, pp. 1043–1057, 2004. - J. Diblík, M. Migda, and E. Schmeidel, “Bounded solutions of nonlinear discrete equations,”
*Nonlinear Analysis*, vol. 65, no. 4, pp. 845–853, 2006. - J. Diblík, I. R\accent23užičková, and M. R\accent23užičková, “A general version of the retract method for discrete equations,”
*Acta Mathematica Sinica*, vol. 23, no. 2, pp. 341–348, 2007.Zentralblatt MATH: 1120.39005

Mathematical Reviews (MathSciNet): MR2286928

Digital Object Identifier: doi:10.1007/s10114-005-0729-8 - J. Diblík and I. R\accent23užičková, “Compulsory asymptotic behavior of solutions of two-dimensional systems of difference equations,” in
*Proceedings of the 9th International Conference on Difference Equations and Discrete Dynamical Systems*, pp. 35–49, World Scientific Publishing, University of Southern Cali-fornia, Los Angeles, Calif, USA, 2005.Mathematical Reviews (MathSciNet): MR2188693 - J. Diblík and I. Hlavičková, “Asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation,” in
*Advances in Discrete Dynamical Systems*, vol. 53 of*Advanced Studies in Pure Mathematics*, pp. 23–32, Mathematical Society of Japan, Tokyo, Japan, 2009. - G. Ladas, C. G. Philos, and Y. G. Sficas, “Sharp conditions for the oscillation of delay difference equations,”
*Journal of Applied Mathematics and Simulation*, vol. 2, no. 2, pp. 101–111, 1989. - I. Györi and G. Ladas,
*Oscillation Theory of Delay Differential Equations*, Clarendon Press, Alderley, UK, 1991.Mathematical Reviews (MathSciNet): MR1168471 - J. Baštinec and J. Diblík, “Subdominant positive solutions of the discrete equation $\Delta u(k+n)=-p(k)u(k)$,”
*Abstract and Applied Analysis*, no. 6, pp. 461–470, 2004. - J. Baštinec and J. Diblík, “Remark on positive solutions of discrete equation $\Delta u(k+n)=-p(k)u(k)$,”
*Nonlinear Analysis*, vol. 63, no. 5-7, pp. e2145–e2151, 2004. - J. Baštinec, J. Diblík, and B. Zhang, “Existence of bounded solutions of discrete delayed equations,” in
*Proceedings of the Sixth International Conference on Difference Equations and Applications*, pp. 359–366, CRC, Boca Raton, Fla, USA, 2004. - R. P. Agarwal, M. Bohner, and W.-T. Li,
*Nonoscillation and Oscillation: Theory for Functional Differential Equations*, Marcel Dekker, New York, NY, USA, 2004. - L. Berezansky and E. Braverman, “On existence of positive solutions for linear difference equations with several delays,”
*Advances in Dynamical Systems and Applications*, vol. 1, no. 1, pp. 29–47, 2006. - G. E. Chatzarakis and I. P. Stavroulakis, “Oscillations of first order linear delay difference equations,”
*The Australian Journal of Mathematical Analysis and Applications*, vol. 3, no. 1, article 14, 2006. - D. Chengjun and S. Qiankun, “Boundedness and stability for discrete-time delayed neural network with complex-valued linear threshold neurons,”
*Discrete Dynamics in Nature and Society*, vol. 2010, article 368379, 2010.Zentralblatt MATH: 1200.39006

Mathematical Reviews (MathSciNet): MR2677844

Digital Object Identifier: doi:10.1155/2010/368379 - J. Čermák, “Asymptotic bounds for linear difference systems,”
*Advances in Difference Equations*, vol. 2010, article 182696, 2010. - I. Györi and M. Pituk, “Asymptotic formulae for the solutions of a linear delay difference equation,”
*Journal of Mathematical Analysis and Applications*, vol. 195, no. 2, pp. 376–392, 1995.Mathematical Reviews (MathSciNet): MR1354549

Zentralblatt MATH: 0846.39003

Digital Object Identifier: doi:10.1006/jmaa.1995.1361 - P. Karajani and I. P. Stavroulakis, “Oscillation criteria for second-order delay and functional equations,”
*Studies of the University of Žilina. Mathematical Series*, vol. 18, no. 1, pp. 17–26, 2004. - L. K. Kikina and I. P. Stavroulakis, “A survey on the oscillation of solutions of first order delay dif-ference equations,”
*CUBO, A Mathematical Journal*, vol. 7, no. 2, pp. 223–236, 2005. - L. K. Kikina and I. P. Stavroulakis, “Oscillation criteria for second-order delay, difference, and functional equations,”
*International Journal of Differential Equations*, vol. 2010, article 598068, 2010.Zentralblatt MATH: 1207.34082

Mathematical Reviews (MathSciNet): MR2607727

Digital Object Identifier: doi:10.1155/2010/598068 - M. Kipnis and D. Komissarova, “Stability of a delay difference system,”
*Advances in Difference Equations*, vol. 2006, article 31409, 2006.Zentralblatt MATH: 1139.39015

Mathematical Reviews (MathSciNet): MR2238990

Digital Object Identifier: doi:10.1155/ADE/2006/31409 - E. Liz, “Local stability implies global stability in some one-dimensional discrete single-species models,”
*Discrete and Continuous Dynamical Systems. Series B*, vol. 7, no. 1, pp. 191–199, 2007.Zentralblatt MATH: 1187.39026

Mathematical Reviews (MathSciNet): MR2257458

Digital Object Identifier: doi:10.3934/dcdsb.2007.7.191 - R. Medina and M. Pituk, “Nonoscillatory solutions of a second-order difference equation of Poincaré type,”
*Applied Mathematics Letters*, vol. 22, no. 5, pp. 679–683, 2009.Zentralblatt MATH: 1169.39004

Mathematical Reviews (MathSciNet): MR2514890

Digital Object Identifier: doi:10.1016/j.aml.2008.04.015 - I. P. Stavroulakis, “Oscillation criteria for first order delay difference equations,”
*Mediterranean Journal of Mathematics*, vol. 1, no. 2, pp. 231–240, 2004.Zentralblatt MATH: 1072.39010

Mathematical Reviews (MathSciNet): MR2089091

Digital Object Identifier: doi:10.1007/s00009-004-0013-7 - J. Baštinec, J. Diblík, and Z. Šmarda, “Existence of positive solutions of discrete linear equations with a single delay,”
*Journal of Difference Equations and Applications*, vol. 16, no. 9, pp. 1047–1056, 2010.Zentralblatt MATH: 1207.39014

Mathematical Reviews (MathSciNet): MR2722821

Digital Object Identifier: doi:10.1080/10236190902718026

### More like this

- A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations

Temple, Blake and Young, Robin, Methods and Applications of Analysis, 2010 - New Results on Impulsive Functional Differential Equations with Infinite Delays

Yang, Jie and Xie, Bing, Abstract and Applied Analysis, 2013 - Solving Delay Differential Equations of Small and Vanishing Lag Using Multistep Block Method

Abdul Aziz, Nurul Huda, Abdul Majid, Zanariah, and Ismail, Fudziah, Journal of Applied Mathematics, 2014

- A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations

Temple, Blake and Young, Robin, Methods and Applications of Analysis, 2010 - New Results on Impulsive Functional Differential Equations with Infinite Delays

Yang, Jie and Xie, Bing, Abstract and Applied Analysis, 2013 - Solving Delay Differential Equations of Small and Vanishing Lag Using Multistep Block Method

Abdul Aziz, Nurul Huda, Abdul Majid, Zanariah, and Ismail, Fudziah, Journal of Applied Mathematics, 2014 - Liouville theorems for nonlinear parabolic equations of second order

Hile, G. N. and Mawata, C. P., Differential and Integral Equations, 1996 - Backward stochastic differential equations with time delayed generators—results and counterexamples

Delong, Łukasz and Imkeller, Peter, The Annals of Applied Probability, 2010 - Periodic solutions of periodic difference equations

Furumochi, Tetsuo and Muraoka, Masato, , 2009 - Nonlinear Schrödinger equations with vanishing and decaying potentials

Ambrosetti, A. and Wang, Z.-Q., Differential and Integral Equations, 2005 - Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay

Li, Qiang, Li, Yongxiang, and Chen, Pengyu, Kodai Mathematical Journal, 2016 - Dynamical Behavior of the Stochastic Delay Mutualism System

Xia, Peiyan, Jiang, Daqing, and Li, Xiaoyue, Abstract and Applied Analysis, 2014 - EXISTENCE, ASYMPTOTICS AND UNIQUENESS OF TRAVELING WAVES FOR NONLOCAL DIFFUSION SYSTEMS WITH DELAYED NONLOCAL RESPONSE

Yu, Zhixian and Yuan, Rong, Taiwanese Journal of Mathematics, 2013