Periodic Solution for Neutral Type Neural Networks

ABSTRACT

The principle aim of this paper is to explore the existence of periodic solution of neural networks model with neutral delay. Sufficient and realistic conditions are obtained by means of an abstract continuous theorem of k-set contractive operator and some analysis technique.

Cite this paper

W. Zhang, Y. Yan, Z. Gui and K. Wang, "Periodic Solution for Neutral Type Neural Networks,"*Open Journal of Applied Sciences*, Vol. 3 No. 1, 2013, pp. 49-52. doi: 10.4236/ojapps.2013.31B1010.

W. Zhang, Y. Yan, Z. Gui and K. Wang, "Periodic Solution for Neutral Type Neural Networks,"

References

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[2] J. Zhang and Z. J. Gui, “Periodic Solutions of Nonautonomous Cellular Neural Networks with Impulses and Delays,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 3, 2009，pp. 1891-1903. doi：10.1016/j.nonrwa.2008.02.029

[3] K. Gopalsamy and I. Leung, “Delay Induced Periodicity in a Neural Netlet of Excitation and Inhibition,” Physica D: Nonlinear Phenomena, Vol. 89, No. 3-4, 1996，pp. 395-426. doi:10.1016/0167-2789(95)00203-0

[4] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation and Multistability in a System of Two Coupled Neurons with Multiple Delays,” SIAM Journal on Applied Mathematics, Vol. 61, No. 2, 2000，pp. 673-700. doi:0.1137/S0036139998344015

[5] J. Wu, T. Faria and Y. S. Huang, “Synchronization and Stable Phase-locking in a Network of Neurons with Memory,” Mathematical and Computer Modeling, Vol. 30, No. 1-2, 1999，pp. 117-138. doi:0.1016/S0895-7177(99)00120-X

[6] Y. Li and L. Lu, “Global Exponential Stability and Existence of Periodic Solution of Hopfield- type Neural Networks with Impulses,” Physics Letters A, Vol. 333, No. 1-2, 2004，pp. 62-71. doi:0.1016/j.physleta.2004.09.083

[7] Z. J. Gui, S. J. Lu and W. G. Ge, “Existence of Periodic Solution of Two Competition Dynamics System with Neutral Delay and Several Deviating Arguments,” Chinese Journal of Engrg. Mathematics, Vol. 22, 2005，pp. 703-711.

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[9] Z. Liu and Y. Mao, “Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equations,” Journal of Mathematical Analysis and Appications, Vol. 216, No. 2, 1997，pp. 481–490. doi:.1006/jmaa.1997.5669

[10] Z. Liu and L. Liao, “Existence and Global Exponential Stability of Periodic Solution of Cellular Neural Networks with Timevarying Delays,” Journal of Mathematical Analysis and Applications, Vol. 290, No. 1, 2004，pp. 247–262. doi:10.1016/j.jmaa.2003.09.052

[1] X. Y. Liu and J. D. Cao, “On Periodic Solution of Neural Networks via Differential Inclusions,” Neural Networks, Vol. 22, No. 4, 2009，pp. 329-334. doi：10.1016/j.neunet.2008.11.003

[2] J. Zhang and Z. J. Gui, “Periodic Solutions of Nonautonomous Cellular Neural Networks with Impulses and Delays,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 3, 2009，pp. 1891-1903. doi：10.1016/j.nonrwa.2008.02.029

[3] K. Gopalsamy and I. Leung, “Delay Induced Periodicity in a Neural Netlet of Excitation and Inhibition,” Physica D: Nonlinear Phenomena, Vol. 89, No. 3-4, 1996，pp. 395-426. doi:10.1016/0167-2789(95)00203-0

[4] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation and Multistability in a System of Two Coupled Neurons with Multiple Delays,” SIAM Journal on Applied Mathematics, Vol. 61, No. 2, 2000，pp. 673-700. doi:0.1137/S0036139998344015

[5] J. Wu, T. Faria and Y. S. Huang, “Synchronization and Stable Phase-locking in a Network of Neurons with Memory,” Mathematical and Computer Modeling, Vol. 30, No. 1-2, 1999，pp. 117-138. doi:0.1016/S0895-7177(99)00120-X

[6] Y. Li and L. Lu, “Global Exponential Stability and Existence of Periodic Solution of Hopfield- type Neural Networks with Impulses,” Physics Letters A, Vol. 333, No. 1-2, 2004，pp. 62-71. doi:0.1016/j.physleta.2004.09.083

[7] Z. J. Gui, S. J. Lu and W. G. Ge, “Existence of Periodic Solution of Two Competition Dynamics System with Neutral Delay and Several Deviating Arguments,” Chinese Journal of Engrg. Mathematics, Vol. 22, 2005，pp. 703-711.

[8] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.

[9] Z. Liu and Y. Mao, “Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equations,” Journal of Mathematical Analysis and Appications, Vol. 216, No. 2, 1997，pp. 481–490. doi:.1006/jmaa.1997.5669

[10] Z. Liu and L. Liao, “Existence and Global Exponential Stability of Periodic Solution of Cellular Neural Networks with Timevarying Delays,” Journal of Mathematical Analysis and Applications, Vol. 290, No. 1, 2004，pp. 247–262. doi:10.1016/j.jmaa.2003.09.052