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Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays. (English) Zbl 1173.34343

Summary: Consider the following models for Hopfield neural networks (HNNs) with neutral delays:

${x}_{i}^{\text{'}}\left(t\right)=-{c}_{i}\left(t\right){x}_{i}\left(t\right)+\sum _{j=1}^{n}{a}_{ij}\left(t\right){f}_{j}\left({x}_{j}\left(t-{\tau }_{ij}\left(t\right)\right)\right)+\sum _{j=1}^{n}{b}_{ij}\left(t\right){g}_{j}\left({x}_{j}^{\text{'}}\left(t-{\sigma }_{ij}\left(t\right)\right)\right)+{I}_{i}\left(t\right),\phantom{\rule{1.em}{0ex}}i=1,2,\cdots ,n·$

Some sufficient conditions for the existence and uniqueness of almost periodic solutions are established by using the fixed point theorem and differential inequality techniques. The results of this work are new and complement previously known results.

##### MSC:
 34K14 Almost and pseudo-periodic solutions of functional differential equations 92B20 General theory of neural networks (mathematical biology) 34K40 Neutral functional-differential equations 47N20 Applications of operator theory to differential and integral equations
##### References:
 [1] Hopfield, J.: Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national Academy of sciences of the united states of America 79, 2554-2558 (1982) [2] Hopfield, J.: Neurons with graded response have collective computational properties like those of two-stage neurons, Proceedings of the national Academy of sciences of the united states of America 81, 3088-3092 (1984) [3] Liu, B.; Huang, L.: Existence and exponential stability of almost periodic solutions for Hopfield neural networks with delays, Neurocomputing 68, 196-207 (2005) [4] Liu, B.; Huang, L.: Existence and exponential stability of almost periodic solutions for cellular neural networks with time-varying delays, Physics letters A 341, 135-144 (2005) · Zbl 1171.82329 · doi:10.1016/j.physleta.2005.04.052 [5] Huang, X.; Cao, J.: Almost periodic solutions of inhibitory cellular neural networks with time-varying delays, Physics letters A 314, No. 3, 222-231 (2003) · Zbl 1052.82022 · doi:10.1016/S0375-9601(03)00918-6 [6] Hui, Fang; Jibin, Li: On the existence of periodic solutions of a neutral delay model of single-species population growth, Journal of mathematical analysis and applications 259, No. 1, 8-17 (2001) · Zbl 0995.34073 · doi:10.1006/jmaa.2000.7340 [7] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040 [8] Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993) [9] Gui, Zhanji; Ge, Weigao; Yang, Xiao-Song: Periodic oscillation for a Hopfield neural networks with neutral delays, Physics letters A 364, No. 3–4, 267-273 (2007) · Zbl 1203.34109 · doi:10.1016/j.physleta.2006.12.013 [10] Fink, A. M.: Almost periodic differential equations, Lecture notes in mathematics 377, 80-112 (1974) [11] He, C. Y.: Almost periodic differential equation, (1992)