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)
Traveling waves for nonlinear cellular neural networks with distributed delays. (English) Zbl 1228.34122
The authors study traveling waves of a nonlinear cellular neural network model, which is described by a lattice equation with finite or infinite distributed delays. They prove the existence of monostable traveling waves by employing Schauder’s fixed point theorem coupled with the upper and lower solutions method. They also establish the nonexistence of monostable traveling waves and the exponential asymptotic behavior of the obtained monotone wave profiles as the wave coordinate goes to infinity. Their work improves and covers some previous results.

MSC:
34K31Lattice functional-differential equations
34K10Boundary value problems for functional-differential equations
35C07Traveling wave solutions of PDE
47N20Applications of operator theory to differential and integral equations
92B20General theory of neural networks (mathematical biology)
WorldCat.org
Full Text: DOI
References:
[1] Chua, L. O.: CNN: A paradigm for complexity, World sci. Ser. nonlinear sci. Ser. A 31 (1998) · Zbl 0916.68132
[2] Chua, L. O.; Yang, L.: Cellular neural networks: theory, IEEE trans. Circuits syst. 35, 1257-1272 (1988) · Zbl 0663.94022 · doi:10.1109/31.7600
[3] Chua, L. O.; Yang, L.: Cellular neural networks: applications, IEEE trans. Circuits syst. 35, 1273-1290 (1988)
[4] Diekmann, O.; Kaper, H. G.: On the bounded solutions of a nonlinear convolution equation, Nonlinear anal. 2, 721-737 (1978) · Zbl 0433.92028 · doi:10.1016/0362-546X(78)90015-9
[5] Faria, T.; Huang, W.; Wu, J.: Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. lond. Ser. A math. Phys. eng. Sci. 462, 229-261 (2006) · Zbl 1149.35368 · doi:10.1098/rspa.2005.1554
[6] Hsu, C. H.; Li, C. H.; Yang, S. Y.: Diversity of traveling wave solutions in delayed cellular neural networks, Internat. J. Bifur. chaos appl. Sci. engrg. 18, 3515-3550 (2008) · Zbl 1165.34393 · doi:10.1142/S0218127408022561
[7] Hsu, C. H.; Lin, S. S.: Existence and multiplicity of traveling waves in a lattice dynamical systems, J. differential equations 164, 431-450 (2000) · Zbl 0954.34029 · doi:10.1006/jdeq.2000.3770
[8] Hsu, C. H.; Lin, S. S.; Shen, W.: Traveling waves in cellular neural networks, Internat. J. Bifur. chaos appl. Sci. engrg. 9, 1307-1319 (1999) · Zbl 0964.34033 · doi:10.1142/S0218127499000912
[9] Hsu, C. H.; Yang, S. Y.: Structure of a class of traveling waves in delayed cellular neural networks, Discrete contin. Dyn. syst. 13, 339-359 (2005) · Zbl 1085.34049 · doi:10.3934/dcds.2005.13.339
[10] Hsu, C. H.; Yang, S. Y.: Traveling wave solutions in cellular neural networks with multiple time delays, Discrete contin. Dyn. syst. Suppl., 410-419 (2005) · Zbl 1155.34340
[11] Hsu, C. H.; Yang, S. Y.: On camel-like traveling wave solutions in cellular neural networks, J. differential equations 196, 481-514 (2004) · Zbl 1057.34068 · doi:10.1016/S0022-0396(03)00135-9
[12] Hsu, S. B.; Zhao, X.: Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. anal. 40, 776-789 (2008) · Zbl 1160.37031 · doi:10.1137/070703016
[13] Li, B.; Lewis, M.; Weinberger, H.: Existence of traveling waves for integral recursions with nonmonotone growth functions, J. math. Biol. 58, 323-338 (2009) · Zbl 1162.92030 · doi:10.1007/s00285-008-0175-1
[14] Liu, X. X.; Weng, P. X.; Xu, Z. T.: Existence of traveling wave solutions in nonlinear delayed cellular neural networks, Nonlinear anal. Real world appl. 10, 277-286 (2009) · Zbl 1154.34364 · doi:10.1016/j.nonrwa.2007.09.010
[15] Ma, S.: Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. differential equations 171, 294-314 (2001) · Zbl 0988.34053 · doi:10.1006/jdeq.2000.3846
[16] Ou, C.; Wu, J.: Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. differential equations 235, 219-261 (2007) · Zbl 1117.35037 · doi:10.1016/j.jde.2006.12.010
[17] Schaaf, K.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. amer. Math. soc. 302, 587-615 (1987) · Zbl 0637.35082 · doi:10.2307/2000859
[18] Thieme, H.; Zhao, X.: Asymptotic speed of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. differential equations 195, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X
[19] Volpert, A.; Volpert, V.; Volpert, V.: Travelling wave solutions of parabolic systems, Transl. math. Monogr. 140 (1994) · Zbl 1001.35060 · http://www.ams.org/online_bks/mmono140/
[20] Wang, Z.; Li, W.; Ruan, S.: Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. differential equations 222, 185-232 (2006) · Zbl 1100.35050 · doi:10.1016/j.jde.2005.08.010
[21] Weng, P. X.; Wu, J.: Deformation of traveling waves in delayed cellular neural networks, Internat. J. Bifur. chaos appl. Sci. engrg. 13, 797-813 (2003) · Zbl 1062.35163 · doi:10.1142/S0218127403006947
[22] Wu, J.; Zou, X.: Traveling wave front solutions in reaction-diffusion systems with delay, J. dynam. Differential equations 13, 651-687 (2001) · Zbl 0996.34053 · doi:10.1023/A:1016690424892
[23] Wu, J.: Theory and applications of partial functional differential equations, (1996) · Zbl 0870.35116
[24] Yu, Z. -X.; Yuan, R.: Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications, Discrete contin. Dyn. syst. Ser. B 13, 709-728 (2010) · Zbl 1206.35244 · doi:10.3934/dcdsb.2010.13.709
[25] Z.-X. Yu, R. Yuan, Traveling waves of delayed reaction diffusion systems with applications, Nonlinear Anal. Real World Appl., doi:10.1016/j.nonrwa.2011.02.005, in press.
[26] Zhao, X.: Dynamical systems in population biology, (2003) · Zbl 1023.37047