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Periodic solutions of some differential delay equations created by Hamiltonian systems

Published online by Cambridge University Press:  17 April 2009

Jibin Li
Affiliation:
Kunming University of Science and TechnologyInstitute of Applied Mathematics of Yunnan ProvinceKunming, 650093Peoples Rrepublic of China
Zhengrong Liu
Affiliation:
Department of MathematicsYunnan UniversityInstitute of Applied Mathematics of Yunnan ProvinceKunming, 650091Peoples Republic of China
Xuezhong He
Affiliation:
School of Mathematics and StatisticsThe University of SydneySydney, NSW 2006Australia
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Abstract

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This paper is concerned with finding periodic solutions of differential delay systems

and

where ri (i = 1, 2,…, n − 1) are positive constants. By using the theory of Hamiltonian systems, we obtain some sufficient conditions under which these systems have many periodic solutions with known periods.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Arino, O. and Cherif, A.A., ‘An exact formula for the branch of period-4-solutions of x = − λf(x(t − 1)) which bifurcates at λ = π/2’, Differential Integral Equations 2 (1989), 162169.CrossRefGoogle Scholar
[2]Arino, O. and Cherif, A.A., ‘More on ordinary differential equations which yield periodic solutions of delay differential equations’, J. Math. Anal. Appl. 180 (1993), 361385.CrossRefGoogle Scholar
[3]Chen, Y., ‘The existence of periodic solutions of the equation x′(t) = −f(x(t), x(t − τ))J. Math. Anal. Appl. 163 (1992), 227237.CrossRefGoogle Scholar
[4]Chen, Y., ‘The existence of periodic solutions for a class of neutral differential difference equations’, J. Austral. Math. Soc. Ser. B 33 (1992), 508516.CrossRefGoogle Scholar
[5]Dormayer, P., ‘Exact formulae for periodic solutions of x(t + 1) = α(−x(t) + x 3(t))’, Z. Angew. Math. Phys. 37 (1986), 765775.CrossRefGoogle Scholar
[6]Ge, W., ‘The number of simple periodic solutions to the differential-difference equation x(t) = f(x(t − 1))’, Chinese Ann. Math. Ser. A 14 (1993), 472479.Google Scholar
[7]Ge, W., ‘Periodic solutions of differential delay equations with multiple lags’, Acta. Math. Appl. Sinica 17 (1994), 172181.Google Scholar
[8]Ge, W., ‘Further results on the existence of periodic solutions to DDE 3 with multiple lags’, Acta. Math. Sinica (to appear).Google Scholar
[9]Gopalsamy, K., Li, J. and He, X., ‘On the construction of periodic solutions of Kaplan-Yorke type for some differential delay equations’, Appl. Anal. 59 (1995), 6580.CrossRefGoogle Scholar
[10]Herz, A.V., ‘Solution of x(t) = −g(x(t − 1)) approach the Kaplan-Yorke orbits for odd sigmoid’, J. Differential Equations 118 (1995), 3653.CrossRefGoogle Scholar
[11]Kaplan, J.L. and Yorke, J.A., ‘Ordinary differential equations which yield periodic solutions of differential-delay equations’, J. Math. Anal. Appl. 48 (1974), 317324.CrossRefGoogle Scholar
[12]Li, J. and He, X., ‘Proof and generalisation of Kaplan-Yorke's conjecture on periodic solution of differential delay equations’, Sci. Sinica Ser. A 47 (1999), 19.Google Scholar
[13]Nussbaum, R.D., ‘Periodic solutions of special differential equation: an example in nonlinear functional analysis’, Proc. Royal. Soc. Edinburgh Sect. A 81 (1978), 131151.CrossRefGoogle Scholar
[14]Nussbaum, R.D., ‘Uniqueness and onouniqueness for periodic solutions x′(t) = −g(x(t − 1))’, J. Differential Equations 34 (1979), 2554.CrossRefGoogle Scholar
[15]Wen, L., ‘Existence of periodic solutions of a class of differential-difference equations’, Chinese Ann. Math. Ser. A 10 (1989), 249254.Google Scholar
[16]Wen, L. and Xia, H., ‘Existence of periodic solutions for differential-difference equation with two time laps’, Sci. Sinica Ser. A 31 (1988), 777786.Google Scholar