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Anti-periodic solutions for state-dependent impulsive recurrent neural networks with time-varying and continuously distributed delays

  • S.I.: OR in Neuroscience
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Abstract

In this paper, we address a new model of neural networks related to the impulsive phenomena which is called state-dependent impulsive recurrent neural networks with time-varying and continuously distributed delays. We investigate sufficient conditions on the existence and uniqueness of exponentially stable anti-periodic solution for these neural networks by employing method of coincide degree theory and an appropriate Lyapunov function. Moreover, we present an illustrative example to show the effectiveness and feasibility of the obtained theoretical results.

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References

  • Abdurahman, A., & Jiang, H. (2015). The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effect. Neurocomputing, 149, 22–28.

    Article  Google Scholar 

  • Akalın, E., & Akhmet, M. U. (2005). The principles of B-smooth discontinuous flows. Computers and Mathematics with Applications, 49, 981–995.

    Article  Google Scholar 

  • Akhmet, M. U. (2005). Perturbations and Hopf bifurcation of the planar discontinuous dynamical system. Nonlinear Analysis: Theory, Methods & Applications, 60, 163–178.

    Article  Google Scholar 

  • Akhmet, M. (2010). Principles of discontinuous dynamical systems. New York: Springer.

    Book  Google Scholar 

  • Akhmet, M. U., & Perestyuk, N. A. (1990). The comparison method for differential equations with impulse action. Differential Equations, 26(9), 1079–1086.

    Google Scholar 

  • Bainov, D. D., & Stamova, I. M. (1998). Stability of sets for impulsive differential–difference equations with variable impulsive perturbations. Applied Nonlinear Analysis, 5, 69–81.

    Google Scholar 

  • Bainov, D. D., & Stamova, I. M. (1999). Existence, uniqueness and continuability of solutions of impulsive differential–difference equations. Journal of Applied Mathematics and Stochastic Analysis, 12(3), 293–300.

    Article  Google Scholar 

  • Ballinger, G., & Liu, X. (1999). Existence and uniqueness results for impulsive delay differential equations. DCDIS, 5, 579–591.

    Google Scholar 

  • Benchohra, M., Henderson, J., Ntouyas, S. K., & Ouahab, A. (2004). Impulsive functional differential equations with variable times. Computers and Mathematics with Applications, 47, 1659–1665.

    Article  Google Scholar 

  • Benchohra, M., & Ouahab, A. (2003). Impulsive neutral functional differential equations with variable times. Nonlinear Analysis: Theory, Methods & Applications, 55, 679–693.

    Article  Google Scholar 

  • Bouzerdoum, A., & Pinter, R. B. (1993). Shunting inhibitory cellular neural networks: Derivation and stability analysis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40, 215–221.

    Article  Google Scholar 

  • Cabada, A., & Vivero, D. R. (2004). Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations. Advances in Difference Equations, 4, 291–310.

    Google Scholar 

  • Chen, W., & Gong, S. (2014). Global exponential stability of antiperiodic solution for impulsive high-order hopfield neural networks. In Abstract and applied analysis, p. 11. Article ID 138379.

  • Chen, H. L. (1996). Antiperiodic wavelets. Journal of Computational Mathematics, 14, 32–39.

    Google Scholar 

  • Chen, Y., Nieto, J. J., & ORegan, D. (2007). Anti-periodic solutions for fully nonlinear first-order differential equations. Mathematical and Computer Modelling, 46, 1183–1190.

    Article  Google Scholar 

  • Chua, L. O., & Yang, L. (1988). Cellular neural networks: Theory. IEEE Transactions on Circuits and Systems, 35, 1257–1272.

    Article  Google Scholar 

  • Cohen, M. A., & Grossberg, S. (1983). Absolute stability of global pattern formulation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems, Man and Cybernetics, SMC–13, 815–826.

    Article  Google Scholar 

  • Delvos, F. J., & Knoche, L. (1999). Lacunary interpolation by antiperiodic trigonometric polynomials. BIT Numerical Mathematics, 39, 439–450.

    Article  Google Scholar 

  • Du, J. Y., Han, H. L., & Jin, G. X. (2004). On trigonometric and paratrigonometric Hermite interpolation. Journal of Approximation Theory, 131, 74–99.

    Article  Google Scholar 

  • Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations. New York: Springer.

    Book  Google Scholar 

  • Haykin, S. (2001). Neural networks: A comprehensive foundations. Beijing: Tsinghua Press.

    Google Scholar 

  • Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 79, 2554–2558.

    Article  Google Scholar 

  • Hu, M., & Wang, L. (2011). Anti-periodic solutions of impulsive Cohen–Grossberg SICNNs on time scales. Journal of Mathematics Research, 3, 131–139.

    Article  Google Scholar 

  • Kang, H., & Si, L. (2010). Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays. IEEE, 978, 1–5.

    Google Scholar 

  • Kosko, B. (1988). Bidirectional associative memories. IEEE Transactions on Systems, Man and Cybernetics, 18, 49–60.

    Article  Google Scholar 

  • Kosko, B. (1989). Adaptive bidirectional associative memories. Applied Optics, 26, 4947–4960.

    Article  Google Scholar 

  • Li, Y. K., Chen, X. R., & Zhao, L. (2009). Stability and existence of periodic solutions to delayed Cohen–Grossberg BAM neural networks with impulses on time scales. Neurocomputing, 72, 1621–1630.

    Article  Google Scholar 

  • Li, Y., Liu, C., & Zhu, L. (2005). Global exponential stability of periodic solution for shunting inhibitory CNNs with delays. Physics Letters A, 337, 46–54.

    Article  Google Scholar 

  • Li, Y., & Shu, J. (2011). Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales. Communications in Nonlinear Science and Numerical Simulation, 16, 3326–3336.

    Article  Google Scholar 

  • Liu, B. (2009). Stability of shunting inhibitory cellular neural networks with unbounded time-varying delays. Applied Mathematics Letters, 22, 1–5.

    Article  Google Scholar 

  • Liu, Y. (2012). Anti-periodic solutions of nonlinear first order impulsive functional differential equations. Mathematica Slovaca, 62, 695–720.

    Google Scholar 

  • Liu, X., & Ballinger, G. (2002). Existence and continuability of solutions for differential equations with delay and state-dependent impulses. Nonlinear Analysis: Theory, Methods & Applications, 51, 633–647.

    Article  Google Scholar 

  • Liu, X., & Ballinger, G. (2004). Continuous dependence on initial values for impulsive delay differential equations. Applied Mathematics Letters, 17, 483–490.

    Article  Google Scholar 

  • Liu, X., & Cao, J. (2009). Exponential stability of anti-periodic solutions for neural networks with multiple discrete and distributed delay. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 223, 299–308.

    Google Scholar 

  • Liu, C., Li, C., & Liao, X. (2011). Variable-time impulses in BAM neural networks with delay. Neurocomputing, 74, 3286–3295.

    Article  Google Scholar 

  • Liu, Y., & Liu, X. (2013). New existence results of anti-periodic solutions of nonlinear impulsive functional differential equations. Mathematica Bohemica, 138, 337–360.

    Google Scholar 

  • Liu, J., Song, X., & Zhang, L. (2015). Existence of anti-periodic mild solutions to semilinear nonautonomous evolution equations. Journal of Mathematical Analysis and Applications, 425, 295–306.

    Article  Google Scholar 

  • Liu, X., & Wang, Q. (2006). Stability of nontrivial solution of delay differential equations with state-dependent impulses. Applied Mathematics and Computation, 174, 271–288.

    Article  Google Scholar 

  • Li, Y., & Xing, Z. (2007). Existence and global exponential stability of periodic solution of CNNs with impulses. Chaos, Solitons & Fractals, 33, 1686–1693.

    Article  Google Scholar 

  • Li, Y., Xing, W., & Lu, L. (2006). Existence and global exponential stability of periodic solution of a class of neural networks with impulses. Chaos, Solitons & Fractals, 27, 437–445.

    Article  Google Scholar 

  • Li, Y., Yang, L., & Wu, W. (2015). Anti-periodic solution for impulsive BAM neural networks with time-varying leakage delays on time scales. Neurocomputing, 149, 536–545.

    Article  Google Scholar 

  • Li, Y., Zhang, T., & Bai, S. (2010). Anti-periodic solutions for Cohen–Grossberg shunting inhibitory neural networks with delays. World Academic Science, Engineering and Technolology, 4, 1030–1035.

    Google Scholar 

  • O’regan, D., Cho, Y. J., & Chen, Y. Q. (2006). Topological degree theory and application. Boca Raton, London, New York: Taylor and Francis.

    Google Scholar 

  • Ou, C. (2008). Anti-periodic solutions for high-order Hopfield neural networks. Computers and Mathematics with Applications, 56, 1838–1844.

    Article  Google Scholar 

  • Ou, C. (2009). Almost periodic solutions for shunting inhibitory cellular neural networks. Nonlinear Analysis: Real World Applications, 10, 2652–2658.

    Article  Google Scholar 

  • Pan, L., & Cao, J. (2011). Anti-periodic solution delayed cellular neural networks with impulsive effects. Nonlinear Analysis: Real World Applications, 12, 3014–3027.

    Google Scholar 

  • Peng, G., & Huang, L. (2009). Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays. Nonlinear Analysis: Real World Applications, 10, 2434–2440.

    Article  Google Scholar 

  • Şaylı, M., & Yılmaz, E. (2014). Global robust asymptotic stability of variable-time impulsive BAM neural networks. Neural Networks, 60, 67–73.

    Article  Google Scholar 

  • Şaylı, M., & Yılmaz, E. (2015). Periodic solution for state-dependent impulsive shunting inhibitory CNNs with time-varying delays. Neural Networks, 68, 1–11.

    Article  Google Scholar 

  • Şaylı, M., & Yılmaz, E. (2016). State-dependent impulsive Cohen–Grossberg neural networks with time-varying delays. Neurocomputing, 171, 1375–1386.

    Article  Google Scholar 

  • Shao, J. (2008). Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Physics Letters A, 372, 5011–5016.

    Article  Google Scholar 

  • Shi, P., & Dong, L. (2010). Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses. Applied Mathematics and Computation, 216, 623–630.

    Article  Google Scholar 

  • Wang, Y., & Han, M. (2014). Antiperiodic solutions to impulsive Cohen–Grossberg neural networks with delays on time scales. In Abstract and applied analysis, vol. 2014, p. 12. Article ID 308768.

  • Wang, Q., Fang, Y., Li, H., Su, L., & Dai, B. (2014). Anti-periodic solutions for high-order Hopfield neural networks with impulse. Neurocomputing, 138, 339–346.

    Article  Google Scholar 

  • Wang, L., & Fu, X. (2007). A new comparison principle for impulsive differential systems with variable impulsive perturbations and stability theory. Computers and Mathematics with Applications, 54, 730–736.

    Article  Google Scholar 

  • Wei, X., & Qui, Z. (2013). Anti-periodic solutions BAM neural networks with time delays. Applied Mathematics and Computation, 221, 221–229.

    Article  Google Scholar 

  • Xu, C., & Zhang, Q. (2014). Existence and exponential stability of anti-periodic solutions for a high-order delayed Cohen–Grossberg neural networks with impulsive effects. Neural Processing Letters, 40, 227–243.

    Article  Google Scholar 

  • Xu, C., & Zhang, Q. (2015). Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay. Neurocomputing, 153, 108–116.

    Article  Google Scholar 

  • Yang, T. (2001). Impulsive control theory. Berlin: Springer.

    Google Scholar 

  • Yang, Z., & Xu, D. (2006). Existence and exponential stability of periodic solution for impulsive delay differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications, 64, 130–145.

    Article  Google Scholar 

  • Yılmaz, E. (2014). Almost periodic solutions of impulsive neural networks at non-prescribed moments of time. Neurocomputing, 141, 148–152.

    Article  Google Scholar 

  • Zhang, L., Fan, R., Liu, A., & Yang, W. (2014). Existence of anti-periodic solution for delayed cellular neural networks with impulsive effects. Applied Mechanics and Materials, 477–478, 1499–1503.

    Google Scholar 

  • Zhang, J., & Gui, Z. (2005). Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays. Nonlinear Analysis: Theory, Methods & Applications, 60, 163–178.

    Article  Google Scholar 

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Acknowledgments

The authors would like to express the sincere appreciation to the referees for their careful comments and valuable suggestions which greatly improved the presentation of the paper. The first author was supported by a Grant from the Middle East Technical University (METU) BAP1 Faculty/Institute Project: BAP-01-01-2015-005. Moreover, the authors were supported by TUBITAK (The Scientific and Technological Research Council of Turkey).

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Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Mustafa Şaylı

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    Mustafa Şaylı

  3. Department of Mathematics, Polatlı Faculty of Science and Arts, Gazi University, 06900, Ankara, Turkey

    Enes Yılmaz

  4. Princeton Neuroscience Institute and Program in Applied and Computational Mathematics, Princeton University, Washington Road, Princeton, NJ, 08544, USA

    Enes Yılmaz

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  1. Mustafa Şaylı
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Correspondence to Enes Yılmaz.

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Şaylı, M., Yılmaz, E. Anti-periodic solutions for state-dependent impulsive recurrent neural networks with time-varying and continuously distributed delays. Ann Oper Res 258, 159–185 (2017). https://doi.org/10.1007/s10479-016-2192-6

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  • DOI: https://doi.org/10.1007/s10479-016-2192-6

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