Variational Approach To Existence Of Solutions For Differential Equations Boundary Value Problems

In this dissertation, the existence and multiplicity of solutions for boundary value problems of differential equations are considered by variational methods and critical point theorems.The dissertation is divided into four chapters. In the first chapter, we present the recent devel-opment of boundary value problems of differential equations, and some definitions and some critical point theorems are briefly introduced, which will be used in the proof of our main results.In Chapter2, some second-order nonlinear impulsive differential equations has been studied. Firstly, we obtain the existence and multiplicity of classical solutions when nonlinearity satisfying the classical Ambrosetti-Rabinowitzn condition. Secondly, we obtain the existence and multiplicity of classical solutions when the functional satisfying Cerami condition. In Chapter3, two classes of second-order nonlinear singular differential equations has been studied. Firstly, we obtain the existence and multiplicity of weak solutions of singular differential equations. Secondly, we obtain the existence and multiplicity of weak solutions of impulsive singular differential equations. In Chapter4, a class of second-order Hamiltonian systems with impulsive damped vibration differential equations has been studied.In details, five classes of differential equations has been discussed. Firstly, a class of second-order nonlinear impulsive differential equations has been studied under Ambrosetti-Rabinowitz condition. The main theorems used are the principle of least action Mountain pass lemma and Symmetric moun-tain pass theorem. We discuss the existence and multiplicity of solutions for the impulsive boundary value problem. Secondly, a class of second-order nonlinear impulsive differential equations has been studied under Cerami condition. The main theorems used are Mountain pass lemma under Cerami condition and Symmetric mountain pass theorem under Cerami condition and Fountain theorem under Cerami condition. We obtain the existence and multiplicity of solutions for the impulsive boundary value problem. Thirdly, two classes of second-order nonlinear singular differential equations has been studied by modifying the nonlinearity to obtain the variational structure. The main tools used are the principle of least action and Mountain pass lemma. Fourthly, we studied the existence of a class of second-order nonlinear singular impulsive differential equations. By modifying the nonlinear func-tion and using variational methods to obtain some sufficient conditions to guarantee the existence of solutions when the impulses are bounded. Finally, a class of second-order Hamiltonian systems with impulsive damped vibration differential equations has been studied. By using variational methods, the existence at least one solution for second-order Hamiltonian systems has been studied by using the principle of least action and Saddle point theorem.