zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays. (English) Zbl 1218.34096

For the system

x ˙ i (t)=-a i (t)x i (t)+f i (x 1 (t),,x n (t),x 1 (t-τ i1 ),,x n (t-τ in )),tt k ,
Δx i (t k )=x i (t k + )-x i (t k - )=J k (x i (t k ))

existence and exponential stability of a periodic solution are established.

The obtained results are compared with some known ones.

MSC:
34K45Functional-differential equations with impulses
34K20Stability theory of functional-differential equations
34K13Periodic solutions of functional differential equations
47N20Applications of operator theory to differential and integral equations
References:
[1]Zhao, H. Y.: Global stability of bidirectional associative memory neural networks with distributed delays, Phys. lett. A 297, 182-190 (2002) · Zbl 0995.92002 · doi:10.1016/S0375-9601(02)00434-6
[2]Liao, X. F.; Wong, K. W.; Yang, S. Z.: Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays, Phys. lett. A 316, 55-64 (2003) · Zbl 1038.92001 · doi:10.1016/S0375-9601(03)01113-7
[3]Li, Y. K.; Liu, P.: Existence and stability of positive periodic solution for BAM neural networks with delays, Math. comput. Modelling 40, 757-770 (2004) · Zbl 1197.34125 · doi:10.1016/j.mcm.2004.10.007
[4]Zhao, H.: Global stability of associative memory neural networks with distributed delays, Phys. lett. A 13, 182-190 (2003) · Zbl 0995.92002 · doi:10.1016/S0375-9601(02)00434-6
[5]Sun, C.; Feng, C.: Exponential periodicity and stability of delayed neural networks, Math. comput. Simulation 66, 469-478 (2004) · Zbl 1057.34097 · doi:10.1016/j.matcom.2004.03.001
[6]Wu, R. C.: Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays, Nonlinear anal. RWA 11, No. 1, 562-573 (2010) · Zbl 1186.34121 · doi:10.1016/j.nonrwa.2009.02.003
[7]Xia, Y. H.; Wong, P. J. Y.: Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos solitons fractals 39, 440-453 (2009) · Zbl 1197.34146 · doi:10.1016/j.chaos.2007.04.005
[8]Li, Y. K.: Global exponential stability of BAM neural networks with delays and impulses, Chaos solitons fractals 24, 279-285 (2005) · Zbl 1099.68085 · doi:10.1016/j.chaos.2004.09.027
[9]Gui, Z.; Yang, X.; Ge, W.: Existence and global exponential stability of periodic solutions of recurrent cellular neural networks with impulses and delays, Math. comput. Simulation 79, 14-29 (2008) · Zbl 1144.92002 · doi:10.1016/j.matcom.2007.09.001
[10]Wang, H.; Liao, X. F.; Li, C. D.: Existence and exponential stability of periodic solution of BAM neural networks with impulses and time-varying delay, Chaos solitons fractals 33, 1028-1039 (2007) · Zbl 1148.34049 · doi:10.1016/j.chaos.2006.01.112
[11]Wu, H. Q.; Shan, C. H.: Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. math. Model. 33, 2564-2574 (2009) · Zbl 1205.34017 · doi:10.1016/j.apm.2008.07.022
[12]Samidurai, R.; Sakthivel, R.; Anthoni, S. M.: Global asymptotic stability of BAM neural networks with mixed delays and impulses, Appl. math. Comput. 212, 113-119 (2009) · Zbl 1173.34346 · doi:10.1016/j.amc.2009.02.002
[13]Zhou, Q. H.: Global exponential stability of BAM neural networks with distributed delays and impulses, Nonlinear anal. RWA 10, 144-153 (2009) · Zbl 1154.34391 · doi:10.1016/j.nonrwa.2007.08.019
[14]Mohamad, S.; Gopalsamy, K.; Acka, H.: Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear anal. RWA 9, 872-888 (2008) · Zbl 1154.34042 · doi:10.1016/j.nonrwa.2007.01.011
[15]Bai, C. Z.: Global exponential stability and existence of periodic solution of Cohen–Grossberg type neural networks with delays and impulses, Nonlinear anal. RWA 9, 747-761 (2008) · Zbl 1151.34062 · doi:10.1016/j.nonrwa.2006.12.007
[16]Yang, X. S.: Existence and global exponential stability of periodic solution for Cohen–Grossberg shunting inhibitory cellular neural networks with delays and impulses, Neurocomputing 72, 2219-2226 (2009)
[17]Gu, H. B.; Jiang, H. J.; Teng, Z. D.: Existence and globally exponential stability of periodic solution of BAM neural networks with impulses and recent-history distributed delays, Neurocomputing 71, 813-822 (2008)
[18]Li, Y. T.; Wang, J. Y.: An analysis on the global exponential stability and the existence of periodic solutions for non-autonomous hybrid BAM neural networks with distributed delays and impulses, Comput. math. Appl. 56, 2256-2267 (2008) · Zbl 1165.34410 · doi:10.1016/j.camwa.2008.03.048
[19]Tan, M. J.; Tan, Y.: Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays, Appl. math. Model. 33, 373-385 (2009) · Zbl 1167.34373 · doi:10.1016/j.apm.2007.11.010
[20]Huang, L.; Huang, C.; Liu, B.: Dynamics of a class of celluar neural networks with time-varying delays, Phys. lett. A 345, 330-344 (2005)
[21]Li, Y.; Zhu, L.; Liu, P.: Existence and stability of periodic solutions of delayed cellular neural networks, Nonlinear anal. RWA 7, 225-234 (2006) · Zbl 1086.92002 · doi:10.1016/j.nonrwa.2005.02.004
[22]Liu, B.; Huang, L.: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, Phys. lett. A 349, 274-283 (2006) · Zbl 1195.34061 · doi:10.1016/j.physleta.2005.09.064
[23]Liu, Z.; Liao, L.: Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. math. Anal. appl. 290, 247-262 (2004) · Zbl 1055.34135 · doi:10.1016/j.jmaa.2003.09.052
[24]Li, Y. K.; Fan, X. L.: Existence and globally exponential stability of almost periodic solution for Cohen–Grossberg BAM neural networks with variable coefficients, Appl. math. Model. 33, 2114-2120 (2009) · Zbl 1205.34086 · doi:10.1016/j.apm.2008.05.013
[25]Cao, J. D.; Song, Q.: Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19, 1601-1617 (2006) · Zbl 1118.37038 · doi:10.1088/0951-7715/19/7/008
[26]Zhou, Q.; Wan, L.; Sun, J.: Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with time-varying delays, Chaos solitons fractals 32, 129-142 (2007)
[27]Zhang, Y.; Pheng, H.; Vadakkepat, P.: Absolute periodicity and absolute stability of delayed neural networks, IEEE trans. Circuits syst. 49, 256-261 (2002)
[28]Xia, Y.; Han, M. A.: New conditions on the existence and stability of periodic solution in Lotka–Volterra population system, SIAM J. Appl. math. 69, 1580-1597 (2009) · Zbl 1181.92084 · doi:10.1137/070702485
[29]Huang, Z.; Xia, Y.: Exponential periodic attractor of impulsive BAM networks with finite distributed delays, Chaos solitons fractals 39, 373-384 (2009) · Zbl 1197.34124 · doi:10.1016/j.chaos.2007.04.014
[30]Horn, R. A.; Johnson, C. R.: Topics in matrix analysis, (1991)
[31]Fang, B.; Zhou, J.; Li, Y.: Matrix theory, (2004)
[32]Lu, S.; Ge, W.: Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. math. Comput. 146, 195-209 (2003) · Zbl 1037.34065 · doi:10.1016/S0096-3003(02)00536-2
[33]Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equation, (1977)
[34]Rudin, W.: Real and complex analysis, (1987) · Zbl 0925.00005