Abstract
In this paper, we investigate the continuous dependence of solutions of the functional differential equation with infinite delayx′(t)=f(t,x t ) on initial functions. Endowing the phase space ag-norm as well as a supremum norm, we show that if the equation satisfies a mild fading memory dondition, then the continuity off in respect to the topology induced by the supremum norm can yield the continuity of solutions of the equation in respect to the topology induced by theg-norm which is stronger than the ahead one.
Similar content being viewed by others
References
Arino, O., Burton, T. A., and Haddock, J., Periodic solution of functional differential equations, Roy-Soc. Edinburgh Proceedings 101A (1985), 253–271.
Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, Florida, 1985.
Burton, T. A., Phase spaces and boundedness in Volterra equations,J. Integral Equations,10(1975), 61–72.
Burton, T. A. and Dwiggins, D. P., Uniqueness without continuous dependence, Springer Lecture Notes 1192(1986), 115–121 (J. Vosmansky and M. Zlámal, eds.).
Burton, T. A., Dwiggins, D. P., and Feng, Yoube, Periodic solutions of functional differential equations with infinite delay,J. London Math. Soc.,40(1989), 81–88.
Burton, T. A. and Hutson, V., Repellers in systems with infinite delay,J. Math. Anal. Appl.,137(1989), 240–263.
Hale, J. K. and Kato, J., Phase space for retarded equations with infinite delay,Funkcialaj Ekvacioj,21(1978), 11–41.
Kaminogo, T., Continuous dependence of solutions for integro differential equations with infinite delay,J. Math. Anal. Appl.,129(1988), 307–314.
Kappel, F. and Schappacher, W., Some considerations to the fundamental theory of infinite delay equations,J. Differential Equations,37(1980), 141–183.
Sawano, K., Exponential asymptotic stability for functional differential equations with infinite retardation,Tohoku Math. J.,31(1979), 363–382.
Sawano, K., Some considerations on the fundamental theorems for functional differential equations with infinite delay,Funkcialaj Ekvacioj,25(1982), 97–104.
Seifert, G., On Caratheodory conditions for functional differential equations with infinite delay,Rocky Mountain J.,12(1982), 615–619.
Author information
Authors and Affiliations
Additional information
This research was supported in part by an NSF grant with number NSF-DMS-8521408.
On leave from South China Normal University, Guangzhou, PRC. This research was supported in part by the National Science Foundation of PRC.
Rights and permissions
About this article
Cite this article
Burton, T.A., Feng, Y. Continuity in functional differential equations with infinite delay. Acta Mathematicae Applicatae Sinica 7, 229–244 (1991). https://doi.org/10.1007/BF02005972
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02005972