Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

Journal of Differential Equations

Volumn 135, Issue 2, 1997, Pages 315-357

  Wu, Jianhong ^a   Zou, Xingfu ^a  

^a YORK UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001651766     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1996.3232     Document Type: Article

References (66)

1. Ahmad S., Vatsala A. S. Comparison results of reaction-diffusion equations with delay in abstract cones. Rend. Sem. Mat. Univ. Padova. 65:1981;19-34.
2. Alexander J. C., Auchumuty G. Global bifurcation of phase-locked oscillations. Arch. Rational Mech. Anal. 93:1986;253-270.
3. Aronson D. G., Weinberger H. F. Nonlinear diffusion in population genetics, combustion and nerve propagation. Lecture Notes in Mathematics. 1975;Springer-Verlag, Berlin.
4. Atkinson C., Reuter G. E. H. Deterministic epidemic waves. Math. Proc. Cambridge Philos. Soc. 80:1976;315-330.
5. Belair J. Stability in a model of a delayed neural network. J. Dynam. Differential Equations. 5:1993;607-623.
6. Belair J., Campbell S. A., van den Driessche P. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56:1996;245-255.
7. Bell J. Some threshold results for models of myelinated nerves. Math. Biosci. 54:1981;181-190.
8. Bell J., Cosner C. Threshold behaviour and propagation for nonlinear differential- difference systems motivated by modeling myelinated axons. Quart. Appl. Math. 42:1984;1-14.
9. Britton N. F. Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math. 50:1990;1663-1688.
10. Cahn J. W., Chow S.-N., Van Vleck E. S. Spatially discrete nonlinear diffusion equations. Rocky Mountain J. Math. 25:1995;87-118.
11. Chi H., Bell J., Hassard B. Numerical solution of a nonlinear advance-delay differential equation from nerve conduction theory. J. Math. Biol. 24:1986;583-601.
12. Chua L. O., Yang L. Cellular neural networks: theory. IEEE Trans. Circuits Systems. 35:1988;1257-1272.
13. Chua L. O., Yang L. Cellular neural networks: applications. IEEE Trans. Circuits Systems. 35:1988;1273-1290.
14. Chow S.-N., Mallet-Paret J. The Fuller index and global Hopf bifurcation. J. Differential Equations. 29:1978;66-88.
15. Chow S.-N., Mallet-Paret J. Pattern formation and spatial chaos in lattice dynamical systems, I and II. IEEE Trans. Circuits Systems. 42:1995;746-756.
16. Chow S.-N., Shen W. Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J. Appl. Math. 55:1995;1764-1781.
17. Cohen M. A., Grossberg S. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Systems Man Cybernet. 13:1983;815-826.
18. 1. Ann. Polon. Math. 52:1991;243-280.
19. 1. J. Differential Equations. 98:1992;277-298.
20. Fisher R. A. The wave of advance of advantageous genes. Ann. Eugenics. 7:1937;353-269.
21. Geba K., Marzantowicz W. Global bifurcation of periodic solutions. Topol. Methods Nonlinear Anal. 1:1993;67-94.
22. Gopalsamy K., He X. Stability in asymmetric Hopfield nets with transmission delays. Physica D. 76:1994;344-358.
23. Hastings A. Dynamics of a single species in a spatially varying environments: The stabilizing role of high dispersal rates. J. Math. Biology. 16:1982;49-55.
24. Hankerson D., Zinner B. Wave fronts for a cooperative tridiagonal system of differential equations. J. Dynam. Differential Equations. 5:1993;359-373.
25. Herz A., Salzer B, Kühn R., van Hemmen J. L. Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets. Biol. Cybernet. 60:1989;457-467.
26. Hopfield J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA. 79:1982;2554-2558.
27. Hopfield J. J. Neurons with graded response have collective computational properties like those of two-stage neurons. Proc. Natl. Acad. Sci. USA. 81:1984;3088-3092.
28. J. P. Keener, 1986, Propagation and its failure in the discrete Nagumo equation, Proceedings of Conference on Ordinary and Partial Differential Equations, Dundee.
29. Keener J. P. Propagation and its failure to coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47:1987;556-572.
30. Kerscher W., Nagel R. Asymptotic behavior of one-parameter semigroups of positive operators. Acta Appl. Math. 2:1984;297-309.
31. Krawcewicz W., Wu J. Discrete waves and phase-locked oscillations in the growth of a single-species population over a patchy environment. Open Systems Inform. Dynam. 1:1992;127-147.
32. Krisztin T., Wu J. Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts. Acta Math. Univ. Comenian. 63:1993;207-220.
33. Krisztin T., Wu J. Asymptotic periodicity, monotonicity and oscillation of solutions of scalar neutral functional differential equations. J. Math. Anal. Appl. 199:1996;502-525.
34. Kunish K., Schappacher W. Order preserving evolution operators of functional differential equations. Boll. Unione Mate. Italina. 16B:1979;480-500.
35. Ladas G., Lakshmikantham V. Differential Equations in Abstract Spaces. 1972;Academic Press, New York.
36. V. Lakshmikantham, S. Leela, 1981, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Oxford.
37. LaSalle J. Regional Conferences Series in Applied Mathematics. 1976.
38. Leela S., Moauro V. Existence of solutions of delay differential equations on closed subsets of a Banach space. Nonlinear Anal. 2:1978;47-58.
39. Levin S. A. Dispersion and population interaction. Amer. Natur. 108:1974;207-228.
40. Levin S. A. Spatial patterning and the structure of ecological communities. Some Mathematical Questions in Biology. 1976;Amer. Math. Soc. Providence.
41. Levin S. A. Population models and community structure in heterogeneous environments. Mathematical Ecology. 1986;Springer-Verlag, New York.
42. Madras N., Wu J., Zou X. Local-nonlocal interaction and spatial-temporal patterns in single species population over a patchy environment. Canad. Appl. Math. Quart. 4:1996;109-134.
43. Marcus C. M., Westervelt R. M. Stability of analog neural networks with delay. Phys. Rev. A. 39:1989;347-359.
44. Mckean H. K. Nagumo's equation. Adv. Math. 4:1970;209-223.
45. Murray J. D. Mathematical Biology. 1989;Springer-Verlag, New York.
46. Nussbaum R. D. Circulant matrices and differential-delay equations. J. Differential Equations. 60:1985;201-217.
47. Nussbaum R. D., Mallet-Paret J. Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Ann. Math. Pura Appl. 145:1986;33-128.
48. Nussbaum R. D. J. Differential Equations. 46:1982;379-408.
49. Pineda F. J. Generalization of back-propagation to recurrent neural networks. Phys. Rev. Lett. 59:1987;2229-2232.
50. L. S. Pontryagin, R. V. Gamkreledze, 1962, The Mathematical Theory of Optimal Processes, Interscience, New York.
51. Rustichini A. Functional differential equations of mixed type: The linear autonomous case. J. Dynam. Differential Equations. 1:1989;121-143.
52. Rustichini A. Hopf bifurcation for functional differential equations of mixed type. J. Dynam. Differential Equations. 1:1989;145-177.
53. Schaaf K. W. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Amer. Math. Soc. 302:1987;587-605.
54. Schumacher K. Traveling-front solutions for integro-differential equations. Lecture Notes in Biomath. 1980;Springer-Verlag, Berlin/New York.
55. Schulman L. S. Correlating arrows of time. Phys. Rev. D. 7:1973;2868-2874.
56. Schulman L. S. Some differential difference equations containing both advance and retardation. J. Math. Phys. 15:1974;195-198.
57. Smith H. L. Monotone semiflows generated by functional differential equations. J. Differential Equations. 66:1987;420-442.
58. Smith H. L. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Amer. Math. Soc. 1995.
59. Smith H. L., Thieme H. R. Monotone semiflows in scalar non-quasimonotone functional differential equations. J. Math. Anal. Appl. 150:1990;289-306.
60. Smith H. L., Thieme H. R. Strongly order preserving semiflows generated by functional differential equations. J. Differential Equations. 93:1991;332-363.
61. Wheeler J. A., Feynman R. P. Classical electrodynamics in terms of direct interparticle action. Rev. Mod. Phys. 21:1949;425-433.
62. Wu J. Bifurcating waves in coupled cells described by delay-differential equations. Pattern Formation: Symmetry Methods and Applications (Waterloo, Ontario, 1993). 1996;Fields Inst. Commun. 5, Amer. Math. Soc. Providence.
63. Wu J., Zou X. Patterns of sustained oscillation in neural networks with delayed interactions. J. Appl. Math. Comput. 73:1995;55-75.
64. Zinner B. Stability of traveling wave fronts for the discrete Nagumo equation. SIAM J. Math. Anal. 22:1991;1016-1020.
65. Zinner B. Existence of traveling wave front solutions for the discrete Nagumo equation. J. Differential Equations. 96:1992;1-27.
66. Zinner B., Harris G., Hudson W. Traveling wavefronts for the discrete Fisher's equation. J. Differential Equations. 105:1993;46-62.