American Institute of Mathematical Sciences

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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
References:
[1]

Z. B. Cheng and J. L. Ren, Periodic solutions for a fourth-order Rayleigh type $p$-Laplacian delay equation, Nonlinear Anal. TMA, 70 (2009), 516-523. Google Scholar

[2]

F. Z. Cong, Q. D. Huang and S. Y. Shi, Existence and uniqueness of periodic solution for $(2n+1)^{th}$-order differential equation, J. Math. Anal. Appl., 241 (2000), 1-9. doi: 10.1006/jmaa.1999.6471.  Google Scholar

[3]

F. Z. Cong, Periodic solutions for $2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. TMA, 32 (1998), 787-793. doi: 10.1016/S0362-546X(97)00517-8.  Google Scholar

[4]

A. Fonda and J. Mawhin, Quadratic forms, weighted eigenfunctions and boundary value problems for non-linear second order ordinary differential equations, Proc. Royal Soc. Edinburgh Sect. A, 112 (1989), 145-153. doi: 10.1017/S0308210500028213.  Google Scholar

[5]

Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality, Sci. Math. Jpn., 65 (2007), 333-359.  Google Scholar

[6]

A. Lasota and Z. Opial, Sur les solutions périodiques des equations differentielles ordinaires, Ann. Polon. Math., 16 (1964), 69-94  Google Scholar

[7]

W. Li and M. R. Zhang, Non-degeneracy and uniqueness of periodic solutions for some superlinear beam equations, Appl. Math. Lett., 22 (2009), 314-319. doi: 10.1016/j.aml.2008.03.027.  Google Scholar

[8]

G. Meng, P. Yan, X. Y. Lin and M. R. Zhang, Non-degeneracy and periodic solutions of semilinear differential equations with deviation, Adv. Nonlinear Stud., 6 (2006), 563-590.  Google Scholar

[9]

R. Ortega and M. Zhang, Some optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Royal Soc. Edinburgh Sect. A, 135 (2005), 119-132. doi: 10.1017/S0308210500003796.  Google Scholar

[10]

L. J. Pan, Periodic solutions for higher order differential equations with deviating argument, J. Math. Anal. Appl., 343 (2008), 904-918. doi: 10.1016/j.jmaa.2008.01.096.  Google Scholar

[11]

J. L. Ren and Z. B. Cheng, On high-order delay differential equation, Comput. Math. Appl., 57 (2009), 324-331. doi: 10.1016/j.camwa.2008.10.071.  Google Scholar

[12]

J. L. Ren and Z. B. Cheng, Periodic solutions for generalized high-order neutral differential equation in the critical case, Nonlinear Anal., 71 (2009), 6182-6193. doi: 10.1016/j.na.2009.06.011.  Google Scholar

[13]

K. Wang and S. P. Lu, On the existence of periodic solutions for a kind of high-order neutral functional differential equation, J. Math. Anal. Appl., 326 (2007), 1161-1173. doi: 10.1016/j.jmaa.2006.03.078.  Google Scholar

[14]

J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81 (1981), 415-420. doi: 10.2307/2043477.  Google Scholar

[15]

M. R. Zhang, An abstract result on asympotitically positively homogeneous differential equations, J. Math. Anal. Appl., 209 (1997), 291-298. doi: 10.1006/jmaa.1997.5383.  Google Scholar

show all references

References:
[1]

Z. B. Cheng and J. L. Ren, Periodic solutions for a fourth-order Rayleigh type $p$-Laplacian delay equation, Nonlinear Anal. TMA, 70 (2009), 516-523. Google Scholar

[2]

F. Z. Cong, Q. D. Huang and S. Y. Shi, Existence and uniqueness of periodic solution for $(2n+1)^{th}$-order differential equation, J. Math. Anal. Appl., 241 (2000), 1-9. doi: 10.1006/jmaa.1999.6471.  Google Scholar

[3]

F. Z. Cong, Periodic solutions for $2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. TMA, 32 (1998), 787-793. doi: 10.1016/S0362-546X(97)00517-8.  Google Scholar

[4]

A. Fonda and J. Mawhin, Quadratic forms, weighted eigenfunctions and boundary value problems for non-linear second order ordinary differential equations, Proc. Royal Soc. Edinburgh Sect. A, 112 (1989), 145-153. doi: 10.1017/S0308210500028213.  Google Scholar

[5]

Y. Kametaka, H. Yamagishi, K. Watanabe, A. Nagai and K. Takemura, Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality, Sci. Math. Jpn., 65 (2007), 333-359.  Google Scholar

[6]

A. Lasota and Z. Opial, Sur les solutions périodiques des equations differentielles ordinaires, Ann. Polon. Math., 16 (1964), 69-94  Google Scholar

[7]

W. Li and M. R. Zhang, Non-degeneracy and uniqueness of periodic solutions for some superlinear beam equations, Appl. Math. Lett., 22 (2009), 314-319. doi: 10.1016/j.aml.2008.03.027.  Google Scholar

[8]

G. Meng, P. Yan, X. Y. Lin and M. R. Zhang, Non-degeneracy and periodic solutions of semilinear differential equations with deviation, Adv. Nonlinear Stud., 6 (2006), 563-590.  Google Scholar

[9]

R. Ortega and M. Zhang, Some optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Royal Soc. Edinburgh Sect. A, 135 (2005), 119-132. doi: 10.1017/S0308210500003796.  Google Scholar

[10]

L. J. Pan, Periodic solutions for higher order differential equations with deviating argument, J. Math. Anal. Appl., 343 (2008), 904-918. doi: 10.1016/j.jmaa.2008.01.096.  Google Scholar

[11]

J. L. Ren and Z. B. Cheng, On high-order delay differential equation, Comput. Math. Appl., 57 (2009), 324-331. doi: 10.1016/j.camwa.2008.10.071.  Google Scholar

[12]

J. L. Ren and Z. B. Cheng, Periodic solutions for generalized high-order neutral differential equation in the critical case, Nonlinear Anal., 71 (2009), 6182-6193. doi: 10.1016/j.na.2009.06.011.  Google Scholar

[13]

K. Wang and S. P. Lu, On the existence of periodic solutions for a kind of high-order neutral functional differential equation, J. Math. Anal. Appl., 326 (2007), 1161-1173. doi: 10.1016/j.jmaa.2006.03.078.  Google Scholar

[14]

J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81 (1981), 415-420. doi: 10.2307/2043477.  Google Scholar

[15]

M. R. Zhang, An abstract result on asympotitically positively homogeneous differential equations, J. Math. Anal. Appl., 209 (1997), 291-298. doi: 10.1006/jmaa.1997.5383.  Google Scholar

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