Unified boundedness, periodicity, and stability in ordinary and functional differential equations. (English) Zbl 0626.34038

The uniform boundedness and uniform ultimate boundedness in connection with stability, uniform asymptotic stability, equi-asymptotic stability and periodicity of solutions to a system of ordinary differential equations \(x'=F(t,x)\), with continuous right-hand side are studied. The common differential equations, equations with both bounded and unbounded delay as well as the systems with unbounded initial function are considered. Many examples illustrate the main theorems.
Reviewer: D.Babrowski


34C25 Periodic solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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[1] Arino, O.; Burton, T. A.; Haddock, J., Periodic solutions of functional differential equations, Roy. Soc. Edinburgh, Proc. A, 101, 253-271 (1985) · Zbl 0582.34077
[2] Burton, T. A., Periodicity and limiting equations in Volterra systems, Boll. Un. Mat. Ital., IV, 31-39 (1985) · Zbl 0581.45007
[3] Burton, T. A., Periodic solutions of linear Volterra equations, Funkcial. Ekvac., 27, 229-253 (1984) · Zbl 0552.45011
[4] Burton, T. A., Periodic solutions of nonlinear Volterra equations, Funkcial. Ekvac., 27, 301-317 (1985) · Zbl 0569.45012
[5] Burton, T. A., Periodic solutions of integrodifferential equations, J. London Math. Soc., 31, 527-548 (1985) · Zbl 0577.45008
[6] Burton, T. A., Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc., 68, 195-199 (1978) · Zbl 0378.34058
[7] Burton, T. A., Volterra Integral and Differential Equations (1983), New York: Academic Press, New York · Zbl 0515.45001
[8] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (1985), Orlando, Florida: Academic Press, Orlando, Florida · Zbl 0635.34001
[9] Burton, T. A.; Grimmer, R. C., Oscillation, continuation, and uniqueness of solutions of retarded differential equations, Trans. Amer. Math. Soc., 179, 193-209 (1973) · Zbl 0286.34113
[10] Burton, T. A.; Huang, Q.; Mahfoud, W. E., Liapunov functionals of convolution type, J. Math. Anal. Appl., 106, 249-272 (1985) · Zbl 0587.34042
[11] Cartwright, M. L., Forced oscillations in nonlinear systems, Contra. to the theory of Nonlinear Oscillations, 1, 149-241 (1950) · Zbl 0039.09901
[12] Cronin, J., Fixed Points and Topological Degree in Nonlinear Analysis (1964), Providence, Rhode Island: Amer. Math. Soc., Providence, Rhode Island · Zbl 0117.34803
[13] Driver, R. D., Existence and stability of solutions of a delay-differential system, Archiv. Rat. Mech. Anal., 10, 401-426 (1962) · Zbl 0105.30401
[14] Furumochi, T., Periodic solutions of periodic functional differential equations, Funkcial. Ekvac., 24, 247-258 (1981) · Zbl 0474.34059
[15] Furumochi, T., Periodic solutions of functional differential equations with large delays, Funkcial. Ekvac., 25, 33-42 (1982) · Zbl 0487.34079
[16] Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcial. Ekvac, 21, 11-41 (1978) · Zbl 0383.34055
[17] Hale, J. K.; Lopes, O., Fixed point theorems and dissipative processes, J. Differential Equations, 13, 391-402 (1973) · Zbl 0256.34069
[18] Kaminogo, T., Kneser’s property and boundary value problems for some retarded functional differential equations, Tohoku Math. J., 30, 471-486 (1978) · Zbl 0398.34062
[19] Kappel, F.; Schappacher, W., Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations, 37, 141-183 (1980) · Zbl 0466.34036
[20] Kato, J., Liapunov’s second method in functional differential equations, Tohoku Math. J., 32, 487-497 (1980) · Zbl 0471.34055
[21] Langenhop, C. E., Periodic and almost periodic solutions of Volterra integral differential equations with infinite memory, J. Differential Equations, 58, 391-403 (1985) · Zbl 0564.45005
[22] Sawano, K., Exponential asymptotic stability for functional differential equations with infinite retardations, Tohoku Math. J., 31, 363-382 (1979) · Zbl 0449.34053
[23] Z.Wang,Periodic solutions of linear neutral integro-differential equations, to appear.
[24] L. Z.Wen,On the uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc., (1982), pp. 533-538. · Zbl 0495.34039
[25] J.Wu - Z.Li - Z.Wang,Remarks on periodic solutions of linear Volterra equations, to appear. · Zbl 0576.34035
[26] Yoshizawa, T., Stability Theory by Liapunov’s Second Method (1964), Tokyo: Math. Soc. Japan, Tokyo
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