# American Institute of Mathematical Sciences

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

 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.
Citation: Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. Google Scholar [6] A. Lasota and Z. Opial, Sur les solutions périodiques des equations differentielles ordinaires,, Ann. Polon. Math., 16 (1964), 69. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [6] A. Lasota and Z. Opial, Sur les solutions périodiques des equations differentielles ordinaires,, Ann. Polon. Math., 16 (1964), 69. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jmaa.1997.5383. Google Scholar
 [1] Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587 [2] Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701 [3] Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635 [4] Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137 [5] Zhongjie Liu, Duanzhi Zhang. Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $\mathbb{R}^\text{2n}$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4187-4206. doi: 10.3934/dcds.2019169 [6] Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure & Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811 [7] Nikos I. Karachalios, Athanasios N Lyberopoulos. On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case. Conference Publications, 2007, 2007 (Special) : 531-540. doi: 10.3934/proc.2007.2007.531 [8] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [9] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 [10] Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294 [11] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [12] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813 [13] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71 [14] Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439 [15] Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007 [16] Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122 [17] Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314 [18] R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435 [19] Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134 [20] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

2018 Impact Factor: 1.143