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Generalized upper and lower solution method for the forced Duffing equation

Author(s): Chengwen Wang
Journal: Proc. Amer. Math. Soc. 125 (1997), 397-406.
MSC (1991): Primary 34B15, 34C25
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Abstract: This paper gives the generalized upper and lower solution method for the forced Duffing equation

\begin{displaymath}x '' + k x ' + f (t,x) = 0 ,\end{displaymath}

and obtains existence theorems for $T$-periodic solutions, where $f$ is a Carathéodory function. Our results generalize or extend some famous results obtained by Mawhin(1985), Habets(1990), Nkashama(1989) and Nieto(1990).

References:

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Gains, R.E., and Mawhin, J., Concidence Degree and Nonlinear Differential Equations. Lecture Notes in Mathematics 56. Springer, Berlin, 1977. MR 58:30551

2.
Habets, P., and Metzen, G., Existence of periodic solutions of Duffing equations, J. of Diff. Eqns. Vol 78(1989) $pp$. 1-32. MR 90c:34040

3.
Habets, P., and Sanchez, L., Periodic solutions of some Liénard equations with singularities. Proceedings of the American Mathematical Society, Vol 109(1990) $pp$. 1035-1044. MR 90k:34049

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Hartman,P., Ordinary Differential Equations. Second Edition. Birkhäuser, Boston, 1982. MR 83e:34002

5.
Mawhin, J., Points fixes, ponits critiques et probléme aux limites. Semin. Math. Sup. no. 92. Press Univ. de Montreál, 1985.
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Nkashama, M.N., A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. J. of Math. Anal. Applic. Vol 140(1989) $pp$. 381-395. MR 90e:34006

7.
Nieto, J.J.,et al A generalization of the monotone iterative technique for nonlinear second order oeriodic boundary value problem. J. of Math. Anal. Applic. Vol 151(1990) $pp$. 181-189. MR 91h:34025

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Additional Information:

Chengwen Wang
Affiliation: Institute of Mathematics, Academia Sinica, Beijing 100080, People's Republic of China - Department of Mathematics & Computer Science, Rutgers University, Newark, New Jersey 07102
Email: chengwen@pegasus.rutgers.edu

DOI: 10.1090/S0002-9939-97-03947-6
PII: S 0002-9939(97)03947-6
Received by editor(s): November 16, 1994
Communicated by: Hal L. Smith
Copyright of article: Copyright 1997, American Mathematical Society

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