# American Institute of Mathematical Sciences

## Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator

 1 School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, Hunan, China 2 School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author: Chuangxia Huang and Jinde Cao

Received  October 2019 Revised  December 2019 Published  May 2020

Fund Project: The work is partially supported by the National Natural Science Foundation of China (Nos. 11971076, 51839002); International Cooperation and Expansion Project of "Double First-class" (No. 2019IC37)

Taking into account the effects of multi-proportional delays and D operator, this paper investigates the stability issue of a general class of neutral-type SICNNs (shunting inhibitory cellular neural networks). With the help of fixed point theorem and some novel differential inequality techniques, we derive a new sufficient conditions to ensure the existence, uniqueness and exponential stability of weighted pseudo almost periodic solutions (WPAPS) of the considered model. The obtained main results are totally new and generalize some published results. At the end of this work, we also give some numerical simulations to support the proposed approach and demonstrate the correctness of the main conclusions.

Citation: Chuangxia Huang, Hedi Yang, Jinde Cao. Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020372
##### References:
 [1] N. S. Al-Islam, S. M. Alsulami and T. Diagana, Existence of weighted pseudo anti-periodic solutions to some non-autonomous differential equations, Applied Mathematics and Computation, 218 (2012), 6536-6548.  doi: 10.1016/j.amc.2011.12.026.  Google Scholar [2] A. Bouzerdoum and R. B. Pinter, Analysis and analog implementation of directionally sensitive shunting inhibitory cellular neural networks, in: Visual Information Processing: From Neurons to Chips, in: SPIE, (1991), 29–38. Google Scholar [3] A. Bouzerdoum and R. B. Pinter, Nonlinear lateral inhibition applied to motion detection in the fly visual system, in: R.B. Pinter, B. Nabet (Eds.), Nonlinear Vision, CRC Press, Boca Raton, FL, (1992), 423–450. Google Scholar [4] A. Bouzerdoum and R. B. Pinter, Shunting inhibitory cellular neural networks: Derivation and stability analysis, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 40 (1993), 215-221.  doi: 10.1109/81.222804.  Google Scholar [5] F. 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Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Analysis: Theory, Methods and Applications, 68 (2008), 2250-2260.  doi: 10.1016/j.na.2007.01.054.  Google Scholar [10] J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., Florida, 1980.  Google Scholar [11] H. Hu and X. Zou, Existence of an extinction wave in the fisher equation with a shifting habitat, Proceedings of the American Mathematical Society, 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar [12] H. Hu, T. Yi and X. Zou, On spatial-temporal dynamics of a Fisher-KPP equation with a shifting environment, Proceedings of the American Mathematical Society, 148 (2020), 213-221.  doi: 10.1090/proc/14659.  Google Scholar [13] H. Hu, X. Yuan, L. Huang and C. 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Zhang, Global convergence on asymptotically almost periodic sicnns with nonlinear decay functions, Neural Processing Letters, 49 (2019), 625-641.  doi: 10.1007/s11063-018-9835-3.  Google Scholar [18] C. Huang, J. Cao, F. Wen and X. Yang, Stability analysis of sir model with distributed delay on complex networks, Plos One, 11 (2016), e0158813. doi: 10.1371/journal.pone.0158813.  Google Scholar [19] C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, Journal of Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.  Google Scholar [20] C. Huang, H. Zhang, J. Cao and H. Hu, Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, International Journal of Bifurcation and Chaos, 29 (2019), 1950091, 23 Pages. doi: 10.1142/S0218127419500913.  Google Scholar [21] C. Huang, X. Long, L. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canadian Mathematical Bulletin, 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.  Google Scholar [22] C. Huang, X. Long and J. Cao, Stability of anti-periodic recurrent neural networks with multi-proportional delays, Mathematical Methods in the Applied Sciences, 43 (2020), 6093-6102.  doi: 10.1002/mma.6350.  Google Scholar [23] C. Huang, X. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Mathematics and Computers in Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.  Google Scholar [24] C. Huang, Y. Qiao, L. Huang and R. Agarwal, Dynamical behaviors of a food-chain model with stage structure and time delays, Advances in Difference Equations, 2018 (2018), Paper No. 186, 26 pp. doi: 10.1186/s13662-018-1589-8.  Google Scholar [25] C. Huang, S. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52.   Google Scholar [26] C. Huang, R. Su, J. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators, Mathematics and Computers in Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.  Google Scholar [27] S. Kumari, R. Chugh, J. Cao and C. Huang, On the construction, properties and Hausdorff dimension of random Cantor one $p^th$ set, AIMS Mathematics, 5 (2020), 3138-3155.   Google Scholar [28] X. Li, X. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar [29] X. Longn and S. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Applied Mathematics Letters, 100 (2020), 106027, 6 pp. doi: 10.1016/j.aml.2019.106027.  Google Scholar [30] C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, Journal of Inequalities and Applications, 2020 (2020), Paper No. 13. doi: 10.1186/s13660-019-2275-4.  Google Scholar [31] G. Rajchakit, A. Pratap, R. Raja, J. Cao, J. Alzabut and C. Huang, Hybrid control scheme for projective lag synchronization of riemann–liouville sense fractional order memristive bam neural networks with mixed delays,, Mathematics, 7 (2019), 759. doi: 10.3390/math7080759.  Google Scholar [32] C. Song, S. Fei, J. Cao and C. Huang, Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Mathematics, 7 (2019), 599. doi: 10.3390/math7070599.  Google Scholar [33] Y. Tan, C. Huang, B. Sun and Tao. Wang, Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, Journal of Mathematical Analysis and Applications, 458 (2018), 1115-1130.  doi: 10.1016/j.jmaa.2017.09.045.  Google Scholar [34] Y. Tang, Pseudo almost periodic shunting inhibitory cellular neural networks with multi-proportional delays, Neural Processing Letters, 48 (2018), 167-177.  doi: 10.1007/s11063-017-9708-1.  Google Scholar [35] J. Wang, C. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Analysis: Hybrid Systems, 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004.  Google Scholar [36] W. Wang, C. Huang, C. Huang, J. Cao, J. Lu and L. Wang, Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses, Applied Mathematics and Computation, 370 (2020), 124926, 17 pp. doi: 10.1016/j.amc.2019.124926.  Google Scholar [37] S. Xiao, Global Exponential Convergence of HCNNs with Neutral Type Proportional Delays and D Operator, Neural Processing Letters, 49 (2019), 347-356.  doi: 10.1007/s11063-018-9817-5.  Google Scholar [38] Y. Xu, Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays, Neural Processing Letters, 46 (2017), 507-519.  doi: 10.1007/s11063-017-9595-5.  Google Scholar [39] D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar [40] X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.  Google Scholar [41] G. Yang and W. Wan, Weighted Pseudo Almost Periodic Solutions for Cellular Neural Networks with Multi-proportional Delays, Neural Processing Letters, 49 (2019), 1125-1138.  doi: 10.1007/s11063-018-9851-3.  Google Scholar [42] A. Zhang, Almost periodic solutions for SICNNs with neutral type proportional delays and D operators, Neural Processing Letters, 47 (2018), 57-70.  doi: 10.1007/s11063-017-9631-5.  Google Scholar [43] J. Zhang and C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Advances in Difference Equations, 2020 (2020), Paper No. 120. doi: 10.1186/s13662-020-02566-4.  Google Scholar [44] Y. Zhou, X. Wan, C. Huang and X. Yang, Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control, Applied Mathematics and Computation, 376 (2020), 125157. doi: 10.1016/j.amc.2020.125157.  Google Scholar

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##### References:
 [1] N. S. Al-Islam, S. M. Alsulami and T. Diagana, Existence of weighted pseudo anti-periodic solutions to some non-autonomous differential equations, Applied Mathematics and Computation, 218 (2012), 6536-6548.  doi: 10.1016/j.amc.2011.12.026.  Google Scholar [2] A. Bouzerdoum and R. B. Pinter, Analysis and analog implementation of directionally sensitive shunting inhibitory cellular neural networks, in: Visual Information Processing: From Neurons to Chips, in: SPIE, (1991), 29–38. Google Scholar [3] A. Bouzerdoum and R. B. Pinter, Nonlinear lateral inhibition applied to motion detection in the fly visual system, in: R.B. Pinter, B. Nabet (Eds.), Nonlinear Vision, CRC Press, Boca Raton, FL, (1992), 423–450. Google Scholar [4] A. Bouzerdoum and R. B. Pinter, Shunting inhibitory cellular neural networks: Derivation and stability analysis, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 40 (1993), 215-221.  doi: 10.1109/81.222804.  Google Scholar [5] F. Ch$\acute{e}$rif, Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays, Journal of Applied Mathematics and Computing, 39 (2012), 235-251.   Google Scholar [6] D. Chen, W. Zhang, J. Cao and C. Huang, Fixed time synchronization of delayed quaternion-valued memristor-based neural networks, Advances in Difference Equations, 2020 (2020), Paper No. 92, 16 pp. doi: 10.1186/s13662-020-02560-w.  Google Scholar [7] Z. Chen and A. Zhang, Weighted pseudo almost periodic shunting inhibitory cellular neural networks with multi-proportional delays, Neural Processing Letters, 50 (2019), 1831-1843.  doi: 10.1007/s11063-018-9961-y.  Google Scholar [8] T. Diagana, Weighted pseudo almost periodic functions and applications, Comptes Rendus Mathematique, 343 (2006), 643-646.  doi: 10.1016/j.crma.2006.10.008.  Google Scholar [9] T. Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Analysis: Theory, Methods and Applications, 68 (2008), 2250-2260.  doi: 10.1016/j.na.2007.01.054.  Google Scholar [10] J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., Florida, 1980.  Google Scholar [11] H. Hu and X. Zou, Existence of an extinction wave in the fisher equation with a shifting habitat, Proceedings of the American Mathematical Society, 145 (2017), 4763-4771.  doi: 10.1090/proc/13687.  Google Scholar [12] H. Hu, T. Yi and X. Zou, On spatial-temporal dynamics of a Fisher-KPP equation with a shifting environment, Proceedings of the American Mathematical Society, 148 (2020), 213-221.  doi: 10.1090/proc/14659.  Google Scholar [13] H. Hu, X. Yuan, L. Huang and C. Huang, Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks, Mathematical Biosciences and Engineering, 16 (2019), 5729-5749.  doi: 10.3934/mbe.2019286.  Google Scholar [14] C. Huang and B. Liu, New studies on dynamic analysis of inertial neural networks involving non-reduced order method, Neurocomputing, 325 (2019), 283-287.  doi: 10.1016/j.neucom.2018.09.065.  Google Scholar [15] C. Huang, L. Yang and J. Cao, Asymptotic behavior for a class of population dynamics, AIMS Mathematics, 5 (2020), 3378-3390.  doi: 10.3934/math.2020218.  Google Scholar [16] C. Huang, H. Zhang and L. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Communications on Pure and Applied Analysis, 18 (2019), 3337-3349.  doi: 10.3934/cpaa.2019150.  Google Scholar [17] C. Huang, B. Liu, X. Tian, L. Yang and X. Zhang, Global convergence on asymptotically almost periodic sicnns with nonlinear decay functions, Neural Processing Letters, 49 (2019), 625-641.  doi: 10.1007/s11063-018-9835-3.  Google Scholar [18] C. Huang, J. Cao, F. Wen and X. Yang, Stability analysis of sir model with distributed delay on complex networks, Plos One, 11 (2016), e0158813. doi: 10.1371/journal.pone.0158813.  Google Scholar [19] C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, Journal of Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.  Google Scholar [20] C. Huang, H. Zhang, J. Cao and H. Hu, Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, International Journal of Bifurcation and Chaos, 29 (2019), 1950091, 23 Pages. doi: 10.1142/S0218127419500913.  Google Scholar [21] C. Huang, X. Long, L. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canadian Mathematical Bulletin, 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.  Google Scholar [22] C. Huang, X. Long and J. Cao, Stability of anti-periodic recurrent neural networks with multi-proportional delays, Mathematical Methods in the Applied Sciences, 43 (2020), 6093-6102.  doi: 10.1002/mma.6350.  Google Scholar [23] C. Huang, X. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Mathematics and Computers in Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.  Google Scholar [24] C. Huang, Y. Qiao, L. Huang and R. Agarwal, Dynamical behaviors of a food-chain model with stage structure and time delays, Advances in Difference Equations, 2018 (2018), Paper No. 186, 26 pp. doi: 10.1186/s13662-018-1589-8.  Google Scholar [25] C. Huang, S. Wen and L. Huang, Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays, Neurocomputing, 357 (2019), 47-52.   Google Scholar [26] C. Huang, R. Su, J. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators, Mathematics and Computers in Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.  Google Scholar [27] S. Kumari, R. Chugh, J. Cao and C. Huang, On the construction, properties and Hausdorff dimension of random Cantor one $p^th$ set, AIMS Mathematics, 5 (2020), 3138-3155.   Google Scholar [28] X. Li, X. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.  Google Scholar [29] X. Longn and S. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Applied Mathematics Letters, 100 (2020), 106027, 6 pp. doi: 10.1016/j.aml.2019.106027.  Google Scholar [30] C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, Journal of Inequalities and Applications, 2020 (2020), Paper No. 13. doi: 10.1186/s13660-019-2275-4.  Google Scholar [31] G. Rajchakit, A. Pratap, R. Raja, J. Cao, J. Alzabut and C. Huang, Hybrid control scheme for projective lag synchronization of riemann–liouville sense fractional order memristive bam neural networks with mixed delays,, Mathematics, 7 (2019), 759. doi: 10.3390/math7080759.  Google Scholar [32] C. Song, S. Fei, J. Cao and C. Huang, Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Mathematics, 7 (2019), 599. doi: 10.3390/math7070599.  Google Scholar [33] Y. Tan, C. Huang, B. Sun and Tao. Wang, Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, Journal of Mathematical Analysis and Applications, 458 (2018), 1115-1130.  doi: 10.1016/j.jmaa.2017.09.045.  Google Scholar [34] Y. Tang, Pseudo almost periodic shunting inhibitory cellular neural networks with multi-proportional delays, Neural Processing Letters, 48 (2018), 167-177.  doi: 10.1007/s11063-017-9708-1.  Google Scholar [35] J. Wang, C. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Analysis: Hybrid Systems, 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004.  Google Scholar [36] W. Wang, C. Huang, C. Huang, J. Cao, J. Lu and L. Wang, Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses, Applied Mathematics and Computation, 370 (2020), 124926, 17 pp. doi: 10.1016/j.amc.2019.124926.  Google Scholar [37] S. Xiao, Global Exponential Convergence of HCNNs with Neutral Type Proportional Delays and D Operator, Neural Processing Letters, 49 (2019), 347-356.  doi: 10.1007/s11063-018-9817-5.  Google Scholar [38] Y. Xu, Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays, Neural Processing Letters, 46 (2017), 507-519.  doi: 10.1007/s11063-017-9595-5.  Google Scholar [39] D. Yang, X. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Analysis: Hybrid Systems, 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.  Google Scholar [40] X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences and Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069.  Google Scholar [41] G. Yang and W. Wan, Weighted Pseudo Almost Periodic Solutions for Cellular Neural Networks with Multi-proportional Delays, Neural Processing Letters, 49 (2019), 1125-1138.  doi: 10.1007/s11063-018-9851-3.  Google Scholar [42] A. Zhang, Almost periodic solutions for SICNNs with neutral type proportional delays and D operators, Neural Processing Letters, 47 (2018), 57-70.  doi: 10.1007/s11063-017-9631-5.  Google Scholar [43] J. Zhang and C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Advances in Difference Equations, 2020 (2020), Paper No. 120. doi: 10.1186/s13662-020-02566-4.  Google Scholar [44] Y. Zhou, X. Wan, C. Huang and X. Yang, Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control, Applied Mathematics and Computation, 376 (2020), 125157. doi: 10.1016/j.amc.2020.125157.  Google Scholar
Numerical solutions of model (3.1), take the initial value (0.5, -0.7, 1, -0.5), (0.6, -0.3, 0.4, -0.6), (-0.5, 0.7, -0.2, 0.4)
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