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

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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

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

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

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