Applications Of Critical Point Theory To Boundary Value Problems For Second-order Impulsive Differential Equations

This Ph.D. thesis aims at studying the existence and multiplicity of solutions for sev-eral classes of second-order impulsive differential equations. Different variational struc-tures are made for different impulsive differential equations. By using critical point theory, some sufficient conditions are obtained to guarantee the existence of at least one or multiple solutions for these equations. The thesis is divided into six chapters and the main contents are as follows:Chapter1is preface. The historical background and the research status of impulsive differential equations are introduced. Some preliminaries of critical point theory and the main results of this thesis are also introduced.In Chapter2, a class of second-order nonlinear impulsive differential equations with Dirichlet boundary conditions has been studied. Via critical point theories such as the mountain pass lemma, we discuss the existence of solutions for the impulsive bound-ary value problem. Several important theorems are obtained which give the sufficient conditions to guarantee the existence of at least one solution, two solutions or infinitely many solutions for the problem. Meanwhile, some examples are given to apply the main theorems, which shows that the main results extend the existing results in the related references.In Chapter3, a class of second-order nonlinear impulsive differential equations on the half-line with a parameter has been studied. Firstly, we make a variational structure for the system. By using variational methods and a variant fountain theorem, we obtain some sufficient conditions to guarantee the existence of infinitely many solutions for the problem when f is sub-linear. Finally, some numerical examples are given to verify the main theorems, which illustrates the feasibility of the conclusion. Our results are new.In Chapter4, periodic boundary value problem for a class of second-order nonlinear impulsive differential equations with two parameters has been studied. By using vari-ational methods, the existence of infinitely many solutions for the impulsive boundary value problem has been studied when f is some special super-linear function. Some sufficient conditions are get to guarantee the existence of infinitely many solutions for the impulsive problem when two parameters are normal numbers, which improves and extends the related results in the literature.In Chapter5, a class of second-order nonlinear impulsive damped differential equa- tions has been studied. We apply the variational methods and critical point theory to study the existence of solutions for the equations for the first time, and earlier establish a variational structure for the equations. By using critical point theory, some sufficient conditions are obtained, which guarantees the existence and multiplicity of solutions for the equations. Our results include many existing conclusions which are the special cases of our conclusions.In Chapter6, the existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems with impulsive effects has been studied. By using critical point theory, we study the existence of infinitely many non-zero periodic solutions generated by impulses for the system when f is some special sub-linear function or asymptotic linear function. However, the system has no non-zero periodic solution, if there is no impulsive effect.