Periodic Solutions for Damped Vibration Problems
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Keywords

Homoclinic solutions
critical point
variational methods
mountain pass theorem
genus.

How to Cite

Ziheng Zhang, & Hongjun Li. (2015). Periodic Solutions for Damped Vibration Problems. Journal of Advances in Applied &Amp; Computational Mathematics, 1(2), 54–61. https://doi.org/10.15377/2409-5761.2014.01.02.4

Abstract

In this paper we are concerned with the following damped vibration problem

 

where ,  with  and ,  is -periodic in  such that  is a -periodic, positive definite symmetric matrix and  satisfies the global Ambrosetti-Rabinowitz condition or is subquadratic at infinity. By use of the Mountain Pass Theorem or the genus properties in the critical theory, we establish some new criteria to guarantee the existence and multiplicity of periodic solutions. Recent results in the literature are generalized and significantly improved.
https://doi.org/10.15377/2409-5761.2014.01.02.4
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References

Ambrosetti G, Mancini. Solutions of minimal periodic for a class of convex Hamiltonian systems. Math Ann 1981; 255: 405-421. http://dx.doi.org/10.1007/BF01450713

Antonacci F. Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign. Nonlinear Anal 1997; 29: 1353-1364. http://dx.doi.org/10.1016/S0362-546X(96)00190-3

Ambrosetti, Rabinowitz PH. Dual variational methods in critical point theory and applications. J Funct Anal 1973; 14: 349-381. http://dx.doi.org/10.1016/0022-1236(73)90051-7

Bai L, Dai BX. Existence of nonzero solutions for a class of damped vibration problems with impulsive effects. Applications of Mathematics 2014; 59: 145-165. http://dx.doi.org/10.1007/s10492-014-0046-6

Chen HW, He ZM. Variational approach to some damped Dirichlet problems with impulses. Math Methods Appl Sci 2013; 36 (2013), 2564-2575. http://dx.doi.org/10.1002/mma.2777

Ding YH, Girardi M. Periodic and homoclinic solutions to a class of Hamiltonian systems with the potential changing sign. Dyn Systems Appl 1993; 2: 131-145.

Faraci F, Livrea R. Infinitely many periodic solutions for a second-order nonautonomous system. Nonlinear Anal 2003; 54: 417-429. http://dx.doi.org/10.1016/S0362-546X(03)00099-3

Izydorek M, Janczewska J. Homoclinic solutions for a class of the second order Hamiltonian systems. J Differential Equations 2005; 219: 375-389. http://dx.doi.org/10.1016/j.jde.2005.06.029

Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Springer-Verlag. New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989. http://dx.doi.org/10.1007/978-1-4757-2061-7

Nieto JJ. Variational formulation of a damped Dirichlet impulsive problem. Appl Math Lett 2010; 23: 940-942. http://dx.doi.org/10.1016/j.aml.2010.04.015

Rabinowitz PH. Homoclinic orbits for a class of Hamiltonian systems. Proc Roy Soc Edinburgh Sect A 1990; 114: 33-38. http://dx.doi.org/10.1017/S0308210500024240

Rabinowitz PH. Minimax Methods in Critical Point Theory with Applications to Differential Equations. in: CBMS REgional Conf Ser in Math vol. 65. American Mathematical Society Provodence RI 1986.

Salvatore. Homoclinic orbits for a special class of nonautonomous Hamiltonian systems. Nonlinear Anal 1997; 30(8): 4849-4857. http://dx.doi.org/10.1016/S0362-546X(97)00142-9

Shilgba LK. Existence result for periodic solutions of a class Hamiltonian systems with superquadratic potential. Nonlinear Anal 2005; 63: 565-574. http://dx.doi.org/10.1016/j.na.2005.05.018

Tang L. Periodic solutions of non-autonomous second order systems with ! -quasisubadditive potential. J Math Anal Appl 1995; 189: 671-675. http://dx.doi.org/10.1006/jmaa.1995.1044

Tang L. Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc Amer Math Soc 1998; 126: 3263-3270. http://dx.doi.org/10.1090/S0002-9939-98-04706-6

Tang L. Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlinear Anal 1998; 32: 299-304. http://dx.doi.org/10.1016/S0362-546X(97)00493-8

Tang L, Wu X. Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. J Math Anal Appl 2002; 275: 870-882. http://dx.doi.org/10.1016/S0022-247X(02)00442-0

Wu X. Saddle point characterization multiplicity of periodic solutions of non-autonomous second order systems. Nonlinear Anal 2004; 58: 899-907. http://dx.doi.org/10.1016/j.na.2004.05.020

Wu X, Chen JL. Existence theorems of periodic solutions for a class of damped vibration problems. Appl Math Comput 2009; 207: 230-235. http://dx.doi.org/10.1016/j.amc.2008.10.020

Xiao J, Nieto JJ. Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J Franklin Inst 2011; 348: 369-377. http://dx.doi.org/10.1016/j.jfranklin.2010.12.003

Wu X, Chen SX, Teng KM. On variational methods for a class of damped vibration problems. Nonlinear Anal 2008; 68(6): 1432-1441. http://dx.doi.org/10.1016/j.na.2006.12.043

Xu YT, Guo ZM. Existence of periodic solutions to secondorder Hamiltonian systems with potential indefinite in sign. Nolinear Anal 2002; 51: 1273-1283. http://dx.doi.org/10.1016/S0362-546X(01)00895-1

Zou W, Li S. Infinitely many solutions for Hamiltonian systems. J Differential Equations 2002; 186: 141-164. http://dx.doi.org/10.1016/S0022-0396(02)00005-0

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