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Existence of periodic solutions to integro-differential equations of neutral type via limiting equations

Published online by Cambridge University Press:  24 October 2008

Jianhong Wu  and
Huaxing Xia
Jianhong Wu
Affiliation:
Department of Mathematics and Statistics, York University, North York, Ontario, CanadaM3J 1P3
Huaxing Xia
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
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Abstract

In this paper, we present some results on the existence of periodic solutions to Volterra integro-differential equations of neutral type. The main idea is to show the convergence of an equibounded sequence of periodic solutions of certain limiting equations which are of finite delay. This makes it possible to apply the existing Liapunov–Razumikhin technique for neutral equations with finite delay to obtain existence of periodic solutions of Volterra neutral integro-differential equations (of infinite delay). Some comparisons between our results and the existing ideas are also provided.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 403 - 418
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Arino, O. A., Burton, T. A. and Haddock, J. R.. Periodic solutions to functional differential equations. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 253–271.CrossRefGoogle Scholar
[2]Burton, T. A.. Stability and Periodic Solutions of Ordinary and Functional Differential Equations (Academic Press, 1985).Google Scholar
[3]Burton, T. A.. Periodic solutions of integro-differential equations. J. London Math. Soc. (2) 31 (1985), 537–548.CrossRefGoogle Scholar
[4]Hale, J. K.. Theory of Functional Differential Equations (Springer-Verlag, 1977).CrossRefGoogle Scholar
[5]Hale, J. K. and Kato, J.. Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21 (1978), 11–41.Google Scholar
[6]Horn, W. A.. Some fixed point theorems for compact maps and flows in Banach spaces. Trans. Amer. Math. Soc. 149 (1970), 391–404.Google Scholar
[7]Islam, M. N.. Periodic solutions of nonlinear integral equations. Ann. Math. Pura Appl. (4) 150 (1988), 129–140.CrossRefGoogle Scholar
[8]Li, Z., Wang, Z. and Wu, J.. Stability properties of solutions of linear Volterra integro-differential equations. Tôhoku Math. J. 37 (1985), 455–462.Google Scholar
[9]Li, Z., Wang, Z. and Wu, J.. The variation of constants formula and periodicity for linear neutral integrodifferential equations. Funkcial. Ekvac. 29 (1986), 121–130.Google Scholar
[10]Lopes, O.. Forced oscillations in nonlinear neutral differential equations. SIAM J. Appl. Math. 29 (1975), 196–201.CrossRefGoogle Scholar
[11]Murakami, S.. Perturbation theorem for functional differential equations with infinite delay via limiting equations. J. Differential Equations 59 (1985), 314–335.CrossRefGoogle Scholar
[12]Schmacher, K.. Existence and continuous dependence for differential equations with unbounded delay. Arch. Rational Mech. Anal. 67 (1978), 315–335.CrossRefGoogle Scholar
[13]Wang, Z. and Wu, J.. Neutral functional differential equations with infinite delay. Funkcial. Ekvac. 28 (1985), 157–170.Google Scholar
[14]Wu, J.. Periodic solutions of nonconvolutional neutral integrodifferential equations. Acta Math. Sci. 8 (1988), 307–313.CrossRefGoogle Scholar
[15]Wu, J.. Globally stable periodic solutions of linear neutral Volterra integrodifferential equations. J. Math. Anal. Appl. 130 (1988), 474–483.CrossRefGoogle Scholar
[16]Wu, J.. Uniform behaviours of solutions of neutral equations with infinite delay (I). Ann. Differential Equations 4 (1988), 189–229.Google Scholar
[17]Wu, J.. Uniform behaviours of solutions of neutral equations with infinite delay (II). Ann. Differential Equations 6 (1990), 331–346.Google Scholar
[18]Wu, J.. Unified treatment of local theory of NFDEs with infinite delay. Tankang J. Math. 22 (1991), 51–72.CrossRefGoogle Scholar
[19]Yoshizawa, T.. Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions (Springer-Verlag, 1975).CrossRefGoogle Scholar