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May  2012, 11(3): 945-958. doi: 10.3934/cpaa.2012.11.945

Multiple solutions of second-order ordinary differential equation via Morse theory

Qiong Meng 1, and  X. H. Tang 2,

1. 

School of Mathematics Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
2. 

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083

Received  September 2010 Revised  September 2011 Published  December 2011

In this paper, we consider the the second-order ordinary differential equation with periodic boundary problem $ - \ddot{x}(t)=f(t,x(t))$, subject to $x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0$, where $f:C([0, 2\pi]\times R, R)$. The operator $K=(-\frac{d^2}{dt^2}+I)^{-1}$ plays an important role. By using Morse index, Leray-Schauder degree and Morse index theorem of the type Lazer-Solimini, we obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.
Keywords: Leray-Schauder degree, Lazer Solimini., Periodic solution, Morse index.
Mathematics Subject Classification: 34C25, 37B30, 37J4.
Citation: Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
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show all references

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Academic Press, New York, 1975. Google Scholar

[2]

Birkhäuser, Boston, 1993.  Google Scholar

[3]

Topol. Methods Nonlinear Anal., 3 (1994), 179-187.  Google Scholar

[4]

Topol. Methods Nonlinear Anal., 71 (2009), 66-71. doi: 10.1016/j.na.2008.10.031.  Google Scholar

[5]

Topol. Methods Nonlinear Anal., 157 (1990), 99-116.  Google Scholar

[6]

Advanced Nonlinear Studies, 10 (2010), 819-836.  Google Scholar

[7]

Trans. of the AMS, 311 (1989), 711-726. doi: 10.1090/s0002-9947-1989-0951886-3.  Google Scholar

[8]

(A First Course) in Text and Reading in Mathematics, vol. 28, Hindustan Book Agency, India, 2004. Google Scholar

[9]

J. Math. Mech., 19 (1970), 609-623. Google Scholar

[10]

Topol. Methods Nonlinear Anal., 12 (1988), 761-775.  Google Scholar

[11]

Sichuan University Publishers, 1995. Google Scholar

[12]

Manuscripta Math., 124 (2007), 507-531. doi: 10.1007/s00229-007-0127-x.  Google Scholar

[13]

J. Math. Anal. Appl., 235 (1999), 237-259. doi: 10.1016/jmaa.1999.6396.  Google Scholar

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Springer-Verlag, New York, 1989.  Google Scholar

[17]

CBMS Issues Math. Ed., vol. 65, 1986.  Google Scholar

[18]

Nonlinear Analysis, Theory, Methods and Applications, 21 (1993), 407-424.  Google Scholar

[19]

Electron. J. Differential Equations, 1 (1995), 1-14.  Google Scholar

[20]

Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/s0362-54x100100221-2.  Google Scholar

[21]

Nonlinear Anal., 70 (2009), 1520-1527. doi: 10.1016/j.na.2008.04.012.  Google Scholar

[22]

Proc. Amer. Math. Soc., 126 (1998), 3263-3270. doi: 10.1090/S0002-9939-98-04706-6.  Google Scholar

[23]

Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

[24]

J. Math. Anal. Appl., 279 (2003) 290-307. doi: 10.1016/S0022-247X(03)00012-X.  Google Scholar

[25]

Journal of Differential Equations, 170 (2001), 68-95. doi: 10.1006/jdeq.2000.3812.  Google Scholar

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