## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2013 (2013), Article ID 942309, 13 pages.

### Almost Periodic Solutions for Neutral-Type BAM Neural Networks with Delays on Time Scales

Yongkun Li and Li Yang

**Full-text: Open access**

#### Abstract

Using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain some sufficient conditions for the existence and exponential stability of almost periodic solutions for a class of neutral-type BAM neural networks with delays on time scales. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even if the time scale $\mathbb{T}=\mathbb{R}$ or $\mathbb{Z}$ and complementary to the previously known results.

#### Article information

**Source**

J. Appl. Math., Volume 2013 (2013), Article ID 942309, 13 pages.

**Dates**

First available in Project Euclid: 14 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1394808048

**Digital Object Identifier**

doi:10.1155/2013/942309

**Mathematical Reviews number (MathSciNet)**

MR3074331

**Zentralblatt MATH identifier**

1271.92005

#### Citation

Li, Yongkun; Yang, Li. Almost Periodic Solutions for Neutral-Type BAM Neural Networks with Delays on Time Scales. J. Appl. Math. 2013 (2013), Article ID 942309, 13 pages. doi:10.1155/2013/942309. https://projecteuclid.org/euclid.jam/1394808048

#### References

- B. Kosko, “Bidirectional associative memories,”
*IEEE Transactions on Systems, Man, and Cybernetics*, vol. 18, no. 1, pp. 49–60, 1988. - Y. Li, “Global exponential stability of BAM neural networks with delays and impulses,”
*Chaos, Solitons & Fractals*, vol. 24, no. 1, pp. 279–285, 2005.Mathematical Reviews (MathSciNet): MR2110036

Zentralblatt MATH: 1099.68085

Digital Object Identifier: doi:10.1016/j.chaos.2004.09.027 - H. Zhao, “Global stability of bidirectional associative memory neural networks with distributed delays,”
*Physics Letters A*, vol. 297, no. 3-4, pp. 182–190, 2002.Mathematical Reviews (MathSciNet): MR1912460

Zentralblatt MATH: 0995.92002

Digital Object Identifier: doi:10.1016/S0375-9601(02)00434-6 - Q. Zhou, “Global exponential stability of BAM neural networks with distributed delays and impulses,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 1, pp. 144–153, 2009.Mathematical Reviews (MathSciNet): MR2451697

Zentralblatt MATH: 1154.34391

Digital Object Identifier: doi:10.1016/j.nonrwa.2007.08.019 - Y. Li, “Existence and stability of periodic solution for BAM neural networks with distributed delays,”
*Applied Mathematics and Computation*, vol. 159, no. 3, pp. 847–862, 2004.Mathematical Reviews (MathSciNet): MR2098232

Zentralblatt MATH: 1073.34080

Digital Object Identifier: doi:10.1016/j.amc.2003.11.007 - B. Zheng, Y. Zhang, and C. Zhang, “Global existence of periodic solutions on a simplified BAM neural network model with delays,”
*Chaos, Solitons & Fractals*, vol. 37, no. 5, pp. 1397–1408, 2008.Mathematical Reviews (MathSciNet): MR2412344

Zentralblatt MATH: 1142.34370

Digital Object Identifier: doi:10.1016/j.chaos.2006.10.029 - Y. Xia, J. Cao, and M. Lin, “New results on the existence and uniqueness of almost periodic solution for BAM neural networks with continuously distributed delays,”
*Chaos, Solitons & Fractals*, vol. 31, no. 4, pp. 928–936, 2007. - L. Zhang and L. Si, “Existence and exponential stability of almost periodic solution for BAM neural networks with variable coefficients and delays,”
*Applied Mathematics and Computation*, vol. 194, no. 1, pp. 215–223, 2007.Mathematical Reviews (MathSciNet): MR2385844

Zentralblatt MATH: 1193.34158

Digital Object Identifier: doi:10.1016/j.amc.2007.04.044 - A. Chen, L. Huang, and J. Cao, “Existence and stability of almost periodic solution for BAM neural networks with delays,”
*Applied Mathematics and Computation*, vol. 137, no. 1, pp. 177–193, 2003.Mathematical Reviews (MathSciNet): MR1949131

Zentralblatt MATH: 1034.34087

Digital Object Identifier: doi:10.1016/S0096-3003(02)00095-4 - Y. Li and X. Fan, “Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients,”
*Applied Mathematical Modelling*, vol. 33, no. 4, pp. 2114–2120, 2009.Mathematical Reviews (MathSciNet): MR2488268

Zentralblatt MATH: 1205.34086

Digital Object Identifier: doi:10.1016/j.apm.2008.05.013 - C. Liu, C. Li, and X. Liao, “Variable-time impulses in BAM neural networks with delays,”
*Neurocomputing*, vol. 74, no. 17, pp. 3286–3295, 2011. - Z. Zhang and K. Liu, “Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales,”
*Neural Networks*, vol. 24, no. 5, pp. 427–439, 2011. - Y. Li and S. Gao, “Global exponential stability for impulsive BAM neural networks with distributed delays on time scales,”
*Neural Processing Letters*, vol. 31, no. 1, pp. 65–91, 2010. - J. H. Park, C. H. Park, O. M. Kwon, and S. M. Lee, “A new stability criterion for bidirectional associative memory neural networks of neutral-type,”
*Applied Mathematics and Computation*, vol. 199, no. 2, pp. 716–722, 2008.Mathematical Reviews (MathSciNet): MR2420599

Zentralblatt MATH: 1149.34345

Digital Object Identifier: doi:10.1016/j.amc.2007.10.032 - R. Rakkiyappan and P. Balasubramaniam, “New global exponential stability results for neutral type neural networks with distributed time delays,”
*Neurocomputing*, vol. 71, no. 4–6, pp. 1039–1045, 2008. - R. Rakkiyappan and P. Balasubramaniam, “LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays,”
*Applied Mathematics and Computation*, vol. 204, no. 1, pp. 317–324, 2008.Mathematical Reviews (MathSciNet): MR2458370

Zentralblatt MATH: 1168.34356

Digital Object Identifier: doi:10.1016/j.amc.2008.06.049 - C. Bai, “Global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays,”
*Applied Mathematics and Computation*, vol. 203, no. 1, pp. 72–79, 2008.Mathematical Reviews (MathSciNet): MR2451540

Zentralblatt MATH: 1173.34344

Digital Object Identifier: doi:10.1016/j.amc.2008.04.002 - B. Xiao, “Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays,”
*Applied Mathematics Letters*, vol. 22, no. 4, pp. 528–533, 2009.Mathematical Reviews (MathSciNet): MR2502249

Zentralblatt MATH: 1173.34343

Digital Object Identifier: doi:10.1016/j.aml.2008.06.025 - H. Xiang and J. Cao, “Almost periodic solution of Cohen-Grossberg neural networks with bounded and unbounded delays,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2407–2419, 2009.Mathematical Reviews (MathSciNet): MR2508453

Zentralblatt MATH: 1163.92309

Digital Object Identifier: doi:10.1016/j.nonrwa.2008.04.021 - K. Wang and Y. Zhu, “Stability of almost periodic solution for a generalized neutral-type neural networks with delays,”
*Neurocomputing*, vol. 73, no. 16–18, pp. 3300–3307, 2010. - J. Liu and G. Zong, “New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type,”
*Neurocomputing*, vol. 72, no. 10–12, pp. 2549–2555, 2009. - 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, pp. 564–570, 2009.Mathematical Reviews (MathSciNet): MR2509934

Zentralblatt MATH: 1170.34352

Digital Object Identifier: doi:10.1016/j.amc.2009.01.031 - R. Samidurai, S. M. Anthoni, and K. Balachandran, “Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays,”
*Nonlinear Analysis: Hybrid Systems*, vol. 4, no. 1, pp. 103–112, 2010.Mathematical Reviews (MathSciNet): MR2570187

Zentralblatt MATH: 1179.93143

Digital Object Identifier: doi:10.1016/j.nahs.2009.08.004 - R. Rakkiyappan, P. Balasubramaniam, and J. Cao, “Global exponential stability results for neutral-type impulsive neural networks,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 122–130, 2010.Mathematical Reviews (MathSciNet): MR2570531

Zentralblatt MATH: 1186.34101

Digital Object Identifier: doi:10.1016/j.nonrwa.2008.10.050 - Y. Li, L. Zhao, and X. Chen, “Existence of periodic solutions for neutral type cellular neural networks with delays,”
*Applied Mathematical Modelling*, vol. 36, no. 3, pp. 1173–1183, 2012.Mathematical Reviews (MathSciNet): MR2872888

Zentralblatt MATH: 1243.34102

Digital Object Identifier: doi:10.1016/j.apm.2011.07.090 - P. Balasubramaniam and V. Vembarasan, “Asymptotic stability of BAM neural networks of neutral-type with impulsive effects and time delay in the leakage term,”
*International Journal of Computer Mathematics*, vol. 88, no. 15, pp. 3271–3291, 2011.Mathematical Reviews (MathSciNet): MR2834519

Zentralblatt MATH: 1247.34122

Digital Object Identifier: doi:10.1080/00207160.2011.591388 - B. Aulbach and S. Hilger, “Linear dynamic processes with inhomogeneous time scale,” in
*Nonlinear Dynamics and Quantum Dynamical Systems*, vol. 59 of*Mathematical Research*, pp. 9–20, Akademie, Berlin, Germany, 1990. - L. Erbe and S. Hilger, “Sturmian theory on measure chains,”
*Differential Equations and Dynamical Systems*, vol. 1, no. 3, pp. 223–244, 1993. - V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan,
*Dynamic Systems on Measure Chains*, Kluwer Academic, Dordrecht, The Netherlands, 1996.Mathematical Reviews (MathSciNet): MR1419803 - R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,”
*Results in Mathematics*, vol. 35, no. 1-2, pp. 3–22, 1999.Mathematical Reviews (MathSciNet): MR1678096

Zentralblatt MATH: 0927.39003

Digital Object Identifier: doi:10.1007/BF03322019 - S. Hilger, “Analysis on measure chains–-a unified approach to continuous and discrete calculus,”
*Results in Mathematics*, vol. 18, no. 1-2, pp. 18–56, 1990.Mathematical Reviews (MathSciNet): MR1066641

Zentralblatt MATH: 0722.39001

Digital Object Identifier: doi:10.1007/BF03323153 - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales: An Introduction with Applications*, Birkhäuser, Boston, Mass, USA, 2001.Mathematical Reviews (MathSciNet): MR1843232 - Y. Li and C. Wang, “Almost periodic functions on time scales and applications,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 727068, 20 pages, 2011. - Y. Li and C. Wang, “Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 341520, 22 pages, 2011. - J. Zhang, M. Fan, and H. Zhu, “Existence and roughness of exponential dichotomies of linear dynamic equations on time scales,”
*Computers & Mathematics with Applications*, vol. 59, no. 8, pp. 2658–2675, 2010. - V. Lakshmikantham and A. S. Vatsala, “Hybrid systems on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 227–235, 2002.Mathematical Reviews (MathSciNet): MR1908840

Zentralblatt MATH: 1032.34050

Digital Object Identifier: doi:10.1016/S0377-0427(01)00448-4 - Y. Li and K. Zhao, “Robust stability of delayed reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales,”
*Neurocomputing*, vol. 74, no. 10, pp. 1632–1637, 2011. - Y. K. Li, K. H. Zhao, and Y. Ye, “Stability of reaction-diffusion recurrent neural networks with distributed delays and neumann boundary conditions on time scales,”
*Neural Processing Letters*, vol. 36, pp. 217–234, 2012. - J. Shen and J. Cao, “Consensus of multi-agent systems on time scales,”
*IMA Journal of Mathematical Control and Information*, vol. 29, no. 4, pp. 507–517, 2012.Mathematical Reviews (MathSciNet): MR3002709

Zentralblatt MATH: 1256.93068

Digital Object Identifier: doi:10.1093/imamci/dns006 - Y. K. Li, “Periodic solutions of non-autonomous cellular neural networks with impulses and delays on time scales,”
*IMA Journal of Mathematical Control and Information*, 2013.

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