미분방정식의 해의 안정성에 관한 연구

  • 고윤희 (제주대학교 사범대학 수학교육과)
  • Published : 2005.07.01


본 논문에서는 상미분방정식, 지연미분방정식, 중립형 지연미분방정식, 그리고 차분방정식 의 안정성에 관한 연구방법과 최근의 연구동향들을 간략하게 소개하였다.


  1. A. Ardito, and P. Ricciardi, Lyapunov Functions for a Generalized Gauss-Type Model, J. Math. Biology 33 (1995), 816-828
  2. P. Auger, R. B. Parra, S. Morand, and E. Sanchez, A predator-prey model with predators using hawk and dove tactics, Mathe. Biosci, 177 (2002), 185-200
  3. K. Balachandran, D. G. Park, and Y. C. Kwun, Comparision Theorems for Controllability of Nonlinear Volterra Integrodifferential Systems, J. Math. Anal. Appl. 268 (2002), 457-465
  4. R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York and London, 1963
  5. R. G. Bowers and A. White, The adaptive dynamics of Lotka- Voltera systems with trade-offs, Math. Biosci. 175 (2002), 67-8l
  6. T. A. Burton, Stability Theory for Volterra Equations, J. Differential Equations 32 (1979), 101-118
  7. T. A. Burton, Volterra Integral and Differential Equations, Academic Press, 1983
  8. T. A. Burton and T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445-454
  9. T. A. Burton and G. Makay, Asymptotic stability for functional differential equations, Acta Math. Hungar 65 (1994), no. 3, 243-25l
  10. T. A. Burton and B. Zhang, Periodic Solutions of Abstract Differential Equations with Infinite Delay, J. Differential Equations 90 (1991),357-396
  11. D. S. Callaway and A. S. Perelson, HIV-l Infection and Low Steady State Viral Loads, Bull. Math. BioI. 64 (2002), 29-64
  12. R. S. Cantrell, C. Cosner, and W. F. Fagan, Habitat edges and predator-prey interactions: effects on critical patch size, Math. Biosci. 175 (2002), 31-55
  13. Y. Cao, H. I. Freeman, and T. C. Gard, A mapping method for global asymptotic stability of population interaction models with time delays, Nonlinear Anal. 34 (1998), 361-389
  14. T. Caraballo, P. Marin-Rubio, and J. Valero, Autonomous and non-autonomous atiractors for differential equations with delays, J. Differential Equations 208 (2005), 9-4l
  15. W. Chen and Z. Guan, Uniform asymptotic stability for perturbed neutral delay differential equations, J. Math. Anal. Appl. 291 (2004), 578-595
  16. W. Chen and X. Lu, Asymptotic stbility in pertubed delay difference systems, J. Math. Anal. Appl. 229 (2004), 261-272
  17. S. K. Choi, N. J. Koo, and Y. H. Goo, Asymptotic property of nonlinear Volterra difference systems, Nonlinear Anal. 51 (2002), 321-337
  18. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D.C. Health and Company, Boston, 1965
  19. E. N. Dancer and Z. Zhang, Dynamics of Lotka- Volterra Competition Systems with Large Interaction, J. Differential Equations 182 (2002), 470-489
  20. H. A. EI-Morshedy and S. R. Grace, Comparison theorems for second order nonlinear difference equations, J. Math. Anal. Appl. 306 (2005), 106-121
  21. A. Fernandes, C. Gutierrez, and R.Rabanal, Global asymptotic stability for differentiable vector fields, J. Differential Equations 206 (2004), 470-482
  22. Y. Fan, L. Wang, and W. Li, Global behavior of a higher order nonlinear difference equation, J. Math. Anal. Appl. 299 (2004), 113-126
  23. H. I. Freeman and X. Yuantong, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993), 513-527
  24. H. I. Freedman and J. W. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985), 69-86
  25. B. S. Goh, Global Stability in Many-Species Systems, The Amer. Natural. 111 (1977), 135-143
  26. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publisher, 1992
  27. M. Gyllenberg, P. Yan, and J. Jiang, The qualitative behavior of a second-order system with zero diagonal coefficient, J. Math. Anal. Appl. 291 (2004),322-340
  28. J. Haddock and J. Terijeki, On the location of positive limit sets for functional differential equations with infinite delay, J. Differential Equations 86 (1990), 132
  29. W. Hahn, Stability of Motion, Springer-Verlag, 1967
  30. J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969
  31. J. K. Hale, Dynamical Systems and Stability, J. Math. Anal. Appl. 26 (1969), 39-59
  32. J. K. Hale, Theory of Functional Differential Equations, Springer Verlag, 1976
  33. J. K. Hale and J. Kato, Phase Space for Retarded Equations with Infinite Delay, Funkaialaj Ekvacioj, 21 (1978), 11-41
  34. L. Hatvani, Annulus arguments in the stability theory for functional differential equations, Differ. Integral Equ. Appl. 10 (1997), no. 5, 975-1002
  35. S. B. Hsu, On Global Stability of a Predator-Prey System, Math. Biocsi. 39 (1978),1-10
  36. W. Hurewicz, Lectures on Ordinary Differential Equations, John Wiley & Sons, INC, New York, 1958
  37. J. Kalas and L. Barakova, Stability and asymptotic behaviour of a twodimensional differential system with delay, J. Math. Anal. Appl. 269 (2002), 278-300
  38. F. Kappel and W. Schappacher, Some Considerations to the Fundamental Theory of Infinte Delay Equations, J. Differential Equations 37 (1980), 141-183
  39. Y. Ko, An Aymptotic Stability and a Uniform Asymptotic Stability for Functional Differential Equations, Proc. Amer. Math. Soc. 119 (1993), 535-545
  40. Y. Ko,The uniform asymptotic stability for functional differential equations with finite delay, Acta Sci. Math. (Szeged) 66 (2000), 565-577
  41. Y. Ko, The Instability for delay differential equations, Nonlinear Anal. 47(2001), 4049-4057
  42. N. N. Krasovskii, Stability of Motion, Stanford, 1963
  43. T. Krisztin and O. Arino, The Two-Dimensional Attractor of a Differential Equation with State-Dependent Delay, J. Dynam. Differential Equations 13 (2001), 453-507
  44. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993
  45. Y. Kuang, Global Stability for Infinite Delay Loika- Volterra Type Systems, J. Differential Equations 103 (1993), no. 2, 221-246
  46. V. Lakshimikantham and S. Leela, Differential and Integral Inequalities, vol 1, Academic Press, New York and London, 1969
  47. V. Lakshmikantham and D. Trigiante, Theory of Difference Equations, Academic Press INC, Boston, San Dieago, New York and Tokyo, 1988
  48. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Academic Press, 1961
  49. S. Lefschetz, Stability of Nonlinear Control Systems, Academic Press, London and New York, 1965
  50. A. M. Liapunov, Stability of Motion, Academic Press, 1966
  51. T. Lindstrom, Global stability of a model for competing predators, Nonlinear Anal. 39 (2000), 793-805
  52. A. Marin-Sanguine and N. V. Torres, Modelling, Steady State Analysis and Optimization of the Catalytic Efficiency of the Triosephosphate Isomerase, Bull. Math. BioI. 64 (2002), 301-326
  53. R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, 1982
  54. J. M. Nichols and J. D. Nichols, Attractor reconstruction for nonlinear systems: a methodological note, Math. Biosci. 171 (2001), 21-32
  55. P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-l infection, Math. Biosci. 179 (2002), 73-94
  56. S. Novo, R. Obaya, and A. M. Sanz, Attractor minimal sets for cooperative and strongly convex delay differential systems, J. Differential Equations 208 (2005), 86-123
  57. J. Ortega, V. Planas-Bielsa, and T. S. Ratiu, Asymptotic and Lyapunov stability of constrained and Poisson equilibria, J. Differential Equations 214 (2005), 92-127
  58. N. Ortiz, Necessary conditions for the neutral problem of Bolza with continuously varying time delay, J. Math. Anal. Appl. 305 (2005), 513-527
  59. C. C. Philos and I. K. Purnaras, Asymptotic Behavior of Solution of Second Order Nonlinear Ordinary Differential Equations, Nonlinear Anal. 24 (1995), no. 1, 81-90
  60. R. Rabah, G. M. Sklyar, and A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2005), 391-428
  61. S. Ruan, The Effect of Delays on Stability and Persistence in Plankton Models, Nonlinear Anal. 24 (1995), no. 5, 575-585
  62. Y. Saito, T. Hara, and W. Ma, Harmless delays for permanence and impersistence of a Lotka- Volterra discrete Predator-prey system, Nonlinear Anal. 50 (2002), 703-715
  63. H. Sedaghat, Periodicity and convergence, J. Math. Anal. Appl. 291 (2004), 31-39
  64. A. B. Silva and M. A. Teixera, Global aymptotic stability on Euclidean spaces, Nonlinear Anal. 50 (2002), 91-114
  65. S. Tang and L. Chen, Global Qualitative Analysis for a Ratio-Dependent Predator-Prey Model with Delay, J. Math. Anal. Appl. 266 (2002), 401-419
  66. H. Tian, The exponential asymptotic stability of singurarly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl. 270 (2002), 143-149
  67. A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci. 179 (2002), 21-55
  68. E. Venturino, The Effects of Diseases on Competing Species, Math. Biosci. 174 (2001), 111-131
  69. V. Volterra, Theory of Functionals and of Integral and Integra-differential Equations, Dover, New York, 1959
  70. J. Walker, Dynamical Systems and Evolution Equation, Plenum, New York, 1980
  71. ei-Hui Wang and M. Kot, Speeds of invasion in a model with strang or weak Allee effects, Math. Biosci. 171 (2001), 83-97
  72. Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci. 171 (2001), 59-82
  73. B. Xu, Further Results on the Stability of Linear Systems with Multiple Delays, J. Math. Anal. Appl. 267 (2002), 20-28
  74. R. Xu, M. J. Chaplain, and F. A. Davidson, permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure, J. Math. Anal. Appl. 303 (2005), 602-621
  75. R. Xu, F. A. Davidson, and M. A. J. Chaplain, Persistence and stability for a two-species ratio-dependent predator-prey system with distributed time delay, J. Math. Anal. Appl, 269 (2002), 256-277
  76. T. Yoshizawa, Stability Theory By Liapunov's Second Method, The Mathematical Society of Japan, 1966
  77. J. Zhang, L. Chen, and X. D. Chen, Persistence and global stability for twospecies nonautonomous competition Lotka- Volterra patch-system with time delay, Nonlinear Anal. 37 (1999), 1019-1028
  78. X. Zhang, L. Chen, and A. U. Neuman, The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci, 168 (2000), 201-210