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Stability properties in almost periodic systems of functional differential equations

  • Taro Yoshizawa
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 799)

Keywords

Periodic Solution Asymptotic Stability Fixed Point Theorem Periodic System Exponential Dichotomy 
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References

  1. [1]-
    AMERIO, L., Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitati, Ann. Mat. Pura Appl., 39(1955), 97–119.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]-
    BOCHNER, S., A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48(1962), 2039–2043.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]-
    BROWDER, F.E., On a generalization of the Schauder fixed point theorem, Duke Math. J., 26(1959), 291–303.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]-
    CHOW, S.N., Remarks on one-dimensional delay-differential equations, J. Math. Anal. Appl., 41(1973), 426–429.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]-
    CHOW, S.N. and HALE, J.K., Strong limit-compact maps, Funkcial. Ekvac., 17(1974), 31–38.MathSciNetzbMATHGoogle Scholar
  6. [6]-
    CHOW, S.N. and YORKE, J.A., Lyapunov theory and perturbation of stable and asymptotically stable systems, J. Differential Eqs., 15(1974), 308–321.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]-
    COPPEL, W.A., Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl., 76(1967), 27–49.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]-
    COPPEL, W.A., Dichotomies in Stability Theory, Lecture Notes in Mathematics 629, Springer-Verlag, 1978.Google Scholar
  9. [9]-
    FAVARD, J., Leçons sur les Fonctions Presque-périodiques, Gauthier-Villars, Paris, 1933.zbMATHGoogle Scholar
  10. [10]-
    FINK, A.M., Almost Periodic Differential Equations, Lecture Notes in Mathematics 377, Springer-Verlag, 1974.Google Scholar
  11. [11]-
    FINK, A.M. and FREDERICKSON, P.O., Ultimate boundedness does not imply almost periodicity, J. Differential Eqs., 9(1971), 280–284.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]-
    FRÉCHET, M., Les fonctions asymptotiquement presque-périodiques, Rev. Scientifique, 79(1941), 341–354.zbMATHGoogle Scholar
  13. [13]-
    HALANAY, A., Differential Equations; Stability, Oscillation, Time Lags, Academic Press, New York, 1966.zbMATHGoogle Scholar
  14. [14]-
    HALANAY, A. and YORKE, J.A., Some new results and problems in the theory of differential-delay equation, SIAM Review, 13(1971), 55–80.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]-
    HALE, J.K., Periodic and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 15(1964), 289–304.ADSMathSciNetzbMATHGoogle Scholar
  16. [16]-
    HALE, J.K., Theory of Functional Differential Equations, Applied Math. Sciences, Vol. 3, 2nd ed., Springer-Verlag, 1977.Google Scholar
  17. [17]-
    HALE, J.K. and KATO, J., Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21(1978), 11–41.MathSciNetzbMATHGoogle Scholar
  18. [18]-
    HALE, J.K. and LOPES, O.F., Fixed point theorems and dissipative processes, J. Differential Eqs., 13(1973), 391–402.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]-
    HINO, Y., Stability and existence of almost periodic solutions of some functional differential equations, Tohoku Math. J., 28(1976), 389–409.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]-
    HINO, Y., Favard's separation theorem in functional differential equations with infinite retardations, Tohoku Math. J., 30(1978), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]-
    JONES, G.S., Stability and asymptotic fixed point theory, Proc. Nat. Acad. Sci., U.S.A., 59(1965), 1262–1264.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]-
    KATO, J., Uniformly asymptotic stability and total stability, Tohoku Math. J., 22(1970), 254–269.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]-
    KATO, J., Favard's separation theorem in functional differential equations, Funkcial. Ekvac., 18(1975), 85–92.MathSciNetzbMATHGoogle Scholar
  24. [24]-
    KATO, J. and NAKAJIMA, F., On Sacker-Sell's theorem for a linear skew product flow, Tohoku Math. J., 28(1976), 79–88.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]-
    KATO, J. and SIBUYA, Y., Catastrophic deformation of a flow and non-existence of almost periodic solutions, J. Faculty of Sci., Univ. of Tokyo, Ser. IA, 24(1977), 267–280.MathSciNetzbMATHGoogle Scholar
  26. [26]-
    KATO, J. and YOSHIZAWA, T., A relationship between uniformly asymptotic stability and total stability, Funkcial. Ekvac., 12(1969), 233–238.MathSciNetzbMATHGoogle Scholar
  27. [27]-
    KATO, J. and YOSHIZAWA, T., Stability under the perturbation by a class of functions, Dynamical Systems, An International Symposium, Vol. 2, Academic Press, 1976, 217–222.Google Scholar
  28. [28]-
    LEVINSON, N., Transformation theory of nonlinear differential equations of second order, Ann. of Math., 45(1944), 723–737. Correction, Ann. of Math., 49(1948), 738.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]-
    MASSERA, J.L., The existence of periodic solutions of systems of differential equations, Duke Math. J., 17(1950), 457–475.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]-
    MILLER, R.K., Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions, J. Differential Eqs., 1(1965), 337–345.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]-
    NAKAJIMA, F., Separation conditions and stability properties in almost periodic systems, Tohoku Math. J., 26(1974), 305–314.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]-
    OIAL, Z., Sur une équation différentielle presque-périodique sans solution presque-périodique, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 9(1961), 673–676.zbMATHGoogle Scholar
  33. [33]-
    REUTER, G.E.H., On certain non-linear differential equations with almost periodic solutions, J. London Math. Soc., 26(1951), 215–221.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]-
    SACKER, R.J. and SELL, G.R., Existence of dichotomies and invariant splittings for linear differential systems 1, J. Differential Eqs., 15(1974), 429–458.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]-
    SACKER, R.J. and SELL, G.R., Lifting properties in skew-product flow with applications to differential equations, Memoirs Amer. Math. Soc., 190(1977).Google Scholar
  36. [36]-
    SAWANO, K., Exponentially asymptotic stability for functional differential equations with infinite retardations, Tohoku Math. J., 31(1979).Google Scholar
  37. [37]-
    SEIFERT, G., Almost periodic solutions for almost periodic systems of ordinary differential equations, J. Differential Eqs., 2(1966), 305–319.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]-
    SELL, G.R., Nonautonomous differential equations and topological dynamics I, II, Trans. Amer. Math. Soc., 127(1967), 241–262, 263–283.MathSciNetzbMATHGoogle Scholar
  39. [39]-
    YOSHIZAWA, T., Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 17(1964), 148–170.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]-
    YOSHIZAWA, T., Stability Theory by Liapunov's Second Method, The Mathematical Society of Japan, Tokyo, 1966.zbMATHGoogle Scholar
  41. [41]-
    YOSHIZAWA, T., Asymptotically almost periodic solutions of an almost periodic systems, Funkcial. Ekvac., 12(1969), 23–40.MathSciNetzbMATHGoogle Scholar
  42. [42]-
    YOSHIZAWA, T., Favard's condition in linear almost periodic systems, International Conference on Differential Equations, Academic Press, 1975, 787–799.Google Scholar
  43. [43]-
    YOSHIZAWA, T., Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Math. Sciences, Vol. 14, Springer-Verlag, 1975.Google Scholar

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© Springer-Verlag 1980

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  • Taro Yoshizawa

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