The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects

Abstract In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results.

[1]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[2]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[3]  J. Nieto,et al.  Impulsive periodic boundary value problems of first-order differential equations , 2007 .

[4]  Yongkun Li,et al.  Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects , 2010 .

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

[6]  G. Rao,et al.  Three periodic solutions for perturbed second order Hamiltonian systems , 2009 .

[7]  Haibo Chen,et al.  Variational Method to the Impulsive Equation with Neumann Boundary Conditions , 2009 .

[8]  Juan J. Nieto,et al.  Existence and global attractivity of positiveperiodic solution of periodic single-species impulsive Lotka-Volterra systems , 2004, Math. Comput. Model..

[9]  Biagio Ricceri,et al.  A further three critical points theorem , 2009 .

[10]  PERIODIC SOLUTIONS FOR A CLASS OF SECOND-ORDER HAMILTONIAN SYSTEMS , 2005 .

[11]  W. Ge,et al.  Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods , 2010 .

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

[13]  Juan J. Nieto,et al.  New comparison results for impulsive integro-differential equations and applications , 2007 .

[14]  Lansun Chen,et al.  A delayed epidemic model with stage-structure and pulses for pest management strategy , 2008 .

[15]  Yiming Long,et al.  Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials , 1995 .

[16]  M. Benchohra,et al.  Impulsive differential equations and inclusions , 2006 .

[17]  W. Ge,et al.  APPLICATIONS OF VARIATIONAL METHODS TO BOUNDARY-VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS , 2008, Proceedings of the Edinburgh Mathematical Society.

[18]  Juan J. Nieto,et al.  Complexity of a Delayed predator-prey Model with impulsive Harvest and Holling Type II Functional Response , 2008, Adv. Complex Syst..

[19]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[20]  Juan J. Nieto,et al.  Impulsive periodic solutions of first‐order singular differential equations , 2008 .

[21]  Biagio Ricceri,et al.  Existence of three solutions for a class of elliptic eigenvalue problems , 2000 .

[22]  Juan J. Nieto,et al.  Solvability of impulsive neutral evolution differential inclusions with state-dependent delay , 2009, Math. Comput. Model..

[23]  S. Yau.  Mathematics and its applications , 2002 .

[24]  Gabriele Bonanno,et al.  Multiple periodic solutions for Hamiltonian systems with not coercive potential , 2010 .

[25]  B. Ahmad,et al.  Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions , 2008 .

[26]  D. O’Regan,et al.  Variational approach to impulsive differential equations , 2009 .

[27]  Biagio Ricceri,et al.  A three critical points theorem revisited , 2009 .

[28]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[29]  Juan J. Nieto,et al.  Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses , 2009 .