Existence of Periodic Solutions for Neutral-Type Neural Networks with Delays on Time Scales

Abstract

In this paper, we employ a fixed point theorem due to Krasnosel’skii to attain the existence of periodic solutions for neutral-type neural networks with delays on a periodic time scale. Some new sufficient conditions are established to show that there exists a unique periodic solution by the contraction mapping principle.

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Huang, Z. and Cai, J. (2013) Existence of Periodic Solutions for Neutral-Type Neural Networks with Delays on Time Scales. Journal of Applied Mathematics and Physics, 1, 1-5. doi: 10.4236/jamp.2013.14001.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] C. J. Cheng, T. L. Liao, J. J. Yan and C. C. Hwang, “Globally Asymptotic Stability of a Class of Neutral-Type Neural Networks with Delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol. 36, No. 5, 2006, pp. 1191-1195. http://dx.doi.org/10.1109/TSMCB.2006.874677
[2] J. H. Park, O. M. Kwon and S. M. Lee, “LMI Optimiza- tion Approach on Stability for Delayed Neural Network of Neutral-Type,” Applied Mathematics and Computation, Vol. 196, No. 1, 2008, pp. 224-236.
[3] H. G. Zhang, Z. W. Liu and G. B. Huang, “Novel Delay-Dependent Robust Stability Analysis for Switched Neutral-Type Neural Networks with Time-Varying Delays via SC Technique,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol. 40, No. 6, 2010, pp. 1480-1491. http://dx.doi.org/10.1109/TSMCB.2010.2040274
[4] R. Samli and S. Arik, “New Results for Global Stability of a Class of Neutral-Type Neural Systems with Time Delays,” Applied Mathematics and Computation, Vol. 210, No. 2, 2009, pp. 564-570. http://dx.doi.org/10.1016/j.amc.2009.01.031
[5] P. Rakkiyappan and P. Balasubramaniam, “New Global Exponential Stability Results for Neutral Type Neural Networks with Distributed Time Delays,” Neurocomput ing, Vol. 71, No. 4-6, 2008, pp. 1039-1045. http://dx.doi.org/10.1016/j.neucom.2007.11.002
[6] Y. N. Raffoul, “Stability in Neutral Nonlinear Differential Equations with Functional Delays Using Fixed Point Theory,” Mathematical and Computer Modelling, Vol. 40, No. 7-8, 2004, pp. 691-700. http://dx.doi.org/10.1016/j.mcm.2004.10.001
[7] W. Kelley and A. Peterson, “Difference Equations: An Introduction with Applications,” Harcourt Academic Press, San Diego, 2001.
[8] S. Mohamad, “Global Exponential Stability in Continuous-Time and Discrete-Time Delayed Bidirectional Neural Networks,” Physica D: Nonlinear Phenomena, Vol. 159, No. 3-4, 2001, pp. 233-251. http://dx.doi.org/10.1016/S0167-2789(01)00344-X
[9] Z. Huang, Y. Xia and X. Wang, “The Existence of k-Almost Periodic Sequence Solutions of Discrete Time Neural Networks,” Nonlinear Dynamics, Vol. 50, No. 1-2, 2007, pp. 13-26. http://dx.doi.org/10.1007/s11071-006-9139-4
[10] M. Bohner and A. Peterson, “Dynamic Equations on Time Scales,” An Introduction with Applications, Birkhauser, Boston, 2001.
[11] M. Bohner and A. Peterson, “Advances in Dynamic Equations on Time Scales,” Birkhauser, Boston, 2003. http://dx.doi.org/10.1007/978-0-8176-8230-9
[12] A. P. Chen and F. L. Chen, “Periodic Solution to BAM Neural Network with Delays on Time Scales,” Neuro-computing, Vol. 73, No. 1-3, 2009, pp. 274-282. http://dx.doi.org/10.1016/j.neucom.2009.08.013
[13] Y. K. Li, X. R. Chen and L. Zhang, “Stability and Existence of Periodic Solutions to Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales,” Neu-rocomputing, Vol. 72, No. 7-8, 2009, pp. 1621-1630. http://dx.doi.org/10.1016/j.neucom.2008.08.010
[14] A. Ardjouni and A. Djoudi, “Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Variable Delay on a Time Scale,” Communication in Nonli-near Science and Numerical Simulation, Vol. 17, No. 7, 2012, pp. 3061-3069. http://dx.doi.org/10.1016/j.cnsns.2011.11.026
[15] E. R. Kaufmann and Y. N. Raffoul, “Periodic Solutions for a Neutral Nonlinear Dynamical Equation on a Time Scale,” Journal of Mathematical Analysis and Applications, Vol. 319, No. 1, 2006, pp. 315-325.
[16] D. R. Smart, “Fixed Points Theorems,” Cambridge University Press, Cambridge, UK, 1980.
[17] K. Gopalsamy, “Leakage delays in BAM,” Journal of Mathematical Analysis and Applications, Vol. 325, No. 2, 2007, pp. 1117-1132. http://dx.doi.org/10.1016/j.jmaa.2006.02.039

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