New existence results on periodic solutions of non-autonomous second order Hamiltonian systems

Abstract In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems u ( t ) + ∇ F ( t , u ( t ) ) = 0 ,  a.e.  t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ( 0 ) − u ( T ) = 0 , where F : R × R N → R is T -periodic ( T > 0 ) in its first variable for all x ∈ R N . When potential function F ( t , x ) is either locally in t asymptotically quadratic or locally in t superquadratic, we show that the above mentioned problem possesses at least one T -periodic solutions via the minimax methods in critical point theory, specially, a new saddle point theorem which is introduced in Schechter (1998).

[1]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[2]  Jihui Zhang,et al.  Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential , 2009 .

[3]  C. Azizieh,et al.  A Note on the Moving Hyperplane Method , 2001 .

[4]  R. Ortega.  The pendulum equation: from periodic to almost periodic forcings , 2009, Differential and Integral Equations.

[5]  Martin Schechter.  Periodic solutions of second-order nonautonomous dynamical systems , 2006 .

[6]  Chun-Lei Tang,et al.  Periodic solutions for nonautonomous second order systems with sublinear nonlinearity , 1998 .

[7]  M. Schechter,et al.  Non-autonomous second order Hamiltonian systems , 2014 .

[8]  Giovanna Cerami.  Un criterio di esistenza per i punti critici su varieta'illimitate , 1978 .

[9]  Xianhua Tang,et al.  New existence of periodic solutions for second order non-autonomous Hamiltonian systems , 2010 .

[10]  Some notes on a superlinear second order Hamiltonian system , 2017 .

[11]  Zhiyong Wang,et al.  On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems , 2015, Appl. Math. Lett..

[12]  Paul H. Rabinowitz,et al.  On subharmonic solutions of hamiltonian systems , 1980 .

[13]  Jihui Zhang,et al.  Periodic solutions of a class of second order non-autonomous Hamiltonian systems , 2010 .

[14]  Jean Mawhin,et al.  Multiple Solutions of the Periodic Boundary Value Problem for Some Forced Pendulum-Type Equations , 1984 .

[15]  M. Schechter.  New linking theorems , 1998 .

[16]  R. Agarwal,et al.  Nonconstant periodic solutions for a class of ordinary p-Laplacian systems , 2016 .

[17]  Chun-Lei Tang,et al.  Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems ✩ , 2007 .

[18]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[19]  Chun-Lei Tang,et al.  Periodic solutions for a class of new superquadratic second order Hamiltonian systems , 2014, Appl. Math. Lett..

[20]  Guihua Fei,et al.  On periodic solutions of superquadratic Hamiltonian systems , 2002 .

[21]  Ivar Ekeland,et al.  Selected new aspects of the calculus of variations in the large , 2002 .

[22]  J. Mawhin,et al.  Critical Point Theory and Hamiltonian Systems , 1989 .