Existence Of Solutions For Second Order Impulsive Differential Equations By Variational Method

In this thesis, the existence and multiplicity of solutions for a class of second-order impulsive damped differential equations are considered by critical point theory and variational approach.The paper contains four chapters. The ?rst chapter mainly introduced the background and development of variational method to impulsive differential equation, and some basic de?nitions and theorems. In the second chapter, existence and multiplicity of solutions for a class of second nonlinear impulsive damped differential equations with perturbed term has been discussed by using variational approach and critical point theory. Applying the Mountain pass theorem and symmetric Mountain theorem,we provide certain new criteria to guarantee the perturbed problem has at least one nontrivial solution, supposing that the nonlinearity is superquadratic at in?nity, subquadratic at the origin, and the impulsive functions have sublinear growth. Moreover,if the nonlinearity and the impulsive functions are odd, then the perturbed problem has in?nitely many solutions. In chapter three, we discus the existence at least one solution for the nonlinear second-order damped impulsive Hamiltonian system with periodic boundary conditions. By using a new approach via critical point theory and variational approach, the existence results of solutions are obtain. In chapter four,we study the multiplicity of solutions for a class of second-order impulsive Hamiltonian system. By applying the variational approach, combining Fountain theorem, we offer some new criteria to ensure the impulsive Hamiltonian systems have in?nitely many solutions under the assumptions that the nonlinearity satis?es superquadratics,asmptotically quadratic and subquadratics, separately.