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May  2013, 33(5): 2155-2168. doi: 10.3934/dcds.2013.33.2155

Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations

Pedro J. Torres 1, , Zhibo Cheng 2, and  Jingli Ren 3,

1. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada
2. 

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454000, China
3. 

Dept. of Math., Zhengzhou University, Zhengzhou 450001

Received  December 2011 Revised  August 2012 Published  December 2012

We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.
Keywords: superlinear, uniqueness, $2n$-order differential equation., Non-degeneracy, semilinear.
Mathematics Subject Classification: 34C2.
Citation: Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155
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show all references

References:
[1]

Nonlinear Anal. TMA, 70 (2009), 516-523. Google Scholar

[2]

J. Math. Anal. Appl., 241 (2000), 1-9. doi: 10.1006/jmaa.1999.6471.  Google Scholar

[3]

Nonlinear Anal. TMA, 32 (1998), 787-793. doi: 10.1016/S0362-546X(97)00517-8.  Google Scholar

[4]

Proc. Royal Soc. Edinburgh Sect. A, 112 (1989), 145-153. doi: 10.1017/S0308210500028213.  Google Scholar

[5]

Sci. Math. Jpn., 65 (2007), 333-359.  Google Scholar

[6]

Ann. Polon. Math., 16 (1964), 69-94  Google Scholar

[7]

Appl. Math. Lett., 22 (2009), 314-319. doi: 10.1016/j.aml.2008.03.027.  Google Scholar

[8]

Adv. Nonlinear Stud., 6 (2006), 563-590.  Google Scholar

[9]

Proc. Royal Soc. Edinburgh Sect. A, 135 (2005), 119-132. doi: 10.1017/S0308210500003796.  Google Scholar

[10]

J. Math. Anal. Appl., 343 (2008), 904-918. doi: 10.1016/j.jmaa.2008.01.096.  Google Scholar

[11]

Comput. Math. Appl., 57 (2009), 324-331. doi: 10.1016/j.camwa.2008.10.071.  Google Scholar

[12]

Nonlinear Anal., 71 (2009), 6182-6193. doi: 10.1016/j.na.2009.06.011.  Google Scholar

[13]

J. Math. Anal. Appl., 326 (2007), 1161-1173. doi: 10.1016/j.jmaa.2006.03.078.  Google Scholar

[14]

Proc. Amer. Math. Soc., 81 (1981), 415-420. doi: 10.2307/2043477.  Google Scholar

[15]

J. Math. Anal. Appl., 209 (1997), 291-298. doi: 10.1006/jmaa.1997.5383.  Google Scholar

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