## Multiplicity of periodic solutions for Duffing equations under nonuniform non-resonance conditions

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- by Chengwen Wang
- Proc. Amer. Math. Soc.
**126**(1998), 1725-1732 - DOI: https://doi.org/10.1090/S0002-9939-98-04520-1
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## Abstract:

This paper is devoted to the study of multiple $2 \pi$-periodic solutions for Duffing equations \begin{equation*} x'' +cx’ + g(t,x) = s(1+h(t)) \end{equation*} under the condition of nonuniform non-resonance related to the positive asymptotic behavior of $g(t,x)x ^{-1}$ at the first two eigenvalues $0$ and $1$ of the periodic BVP on $[0,2 \pi ]$ for the linear operator $L = - x''$, and the condition on the negative asymptotic behavior of $g(t,x)$ at infinity. The techniques we use are degree theory and the upper and lower solution method.## References

- Manuel A. del Pino, Raúl F. Manásevich, and Alejandro Murúa,
*On the number of $2\pi$ periodic solutions for $u''+g(u)=s(1+h(t))$ using the Poincaré-Birkhoff theorem*, J. Differential Equations**95**(1992), no. 2, 240–258. MR**1165422**, DOI 10.1016/0022-0396(92)90031-H - Manuel del Pino, Raúl Manásevich, and Alberto Montero,
*$T$-periodic solutions for some second order differential equations with singularities*, Proc. Roy. Soc. Edinburgh Sect. A**120**(1992), no. 3-4, 231–243. MR**1159183**, DOI 10.1017/S030821050003211X - Pavel Drábek and Sergio Invernizzi,
*On the periodic BVP for the forced Duffing equation with jumping nonlinearity*, Nonlinear Anal.**10**(1986), no. 7, 643–650. MR**849954**, DOI 10.1016/0362-546X(86)90124-0 - Robert E. Gaines and Jean L. Mawhin,
*Coincidence degree, and nonlinear differential equations*, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR**0637067** - J. Mawhin and J. R. Ward,
*Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations*, Rocky Mountain J. Math.**12**(1982), no. 4, 643–654. MR**683859**, DOI 10.1216/RMJ-1982-12-4-643 - Chengwen Wang,
*Generalized upper and lower solutions method for the forced Duffing equation*, Proc. Amer. Math. Soc.**125**(1997), no. 2, 397–406. MR**1403119**, DOI 10.1090/S0002-9939-97-03947-6 - Chengwen Wang, Multiplicity of periodic solutions for Duffing equation,
*Communications on Applied Nonlinear Analysis*(to appear). - D. Hao and S.Ma, Semi-linear Duffing equations crossing resonance points,
*J. Differential Equations*133(1997),*pp*98-116. - H. Wang and Y. Li,
*Periodic solutions for Duffing equations*, Nonlinear Anal.**24**(1995), no. 7, 961–979. MR**1321737**, DOI 10.1016/0362-546X(94)00114-W - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523**

## Bibliographic Information

**Chengwen Wang**- Affiliation: Department of Mathematics and Computer Science Rutgers University, Newark, New Jersey 07102
- Email: chengwen@andromeda.rutgers.edu
- Received by editor(s): November 20, 1996
- Communicated by: Hal L. Smith
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 1725-1732 - MSC (1991): Primary 34C25, 34B15
- DOI: https://doi.org/10.1090/S0002-9939-98-04520-1
- MathSciNet review: 1458269