Solutions For Some Differential Solutions And Operator Equations

This thesis investigates three kinds of problems:periodic perturbations of linear differential equations at resonance,instantaneous and non-instantaneous impulsive differential equations and selfadjoint operator equations.It consists of seven chapters.In Chapter 1,we introduce some research background and main results briefly and list on some working spaces and their signs.In Chapters 2 and 3,linear differential equations with periodic perturbation are focused on.More precise,In Chapter 2,periodic perturbation of linear second Order Hamiltonian systems at resonance is investigated.And an existence result of solutions is obtained from a version of saddle point theorem.Meanwhile,we present two ex-amples in order to illustrate our results.Moreover,two extensions are obtained.In Chapter 3,periodic perturbations of linear 2p-order differential equation at resonance is considered.A multiplicity result of solutions is obtained from Lyapunov-Schmidt reduction and a careful analysis of the oscillatory behaviour of the bifurcation equa-tion induced by original equation.In Chapters 4 and 5,impulsive equations are discussed.In Chapter 4,asymp-totically linear 2p-order Hamiltonian systems with impulsive effects are investigated.We present a new definition of weak solution for the concerned impulsive systems.Then,we verify the existence of a variational construction for the system.Next,two multiplicity results of solutions are obtained from an index theory and some critical point theorems.In Chapter 5,superlinear second-order system with non-instantaneous impulses is considered.Similarly,we present a new definition of weak solution for the concerned systems with non-instantaneous impulsive effects.Then,a multiplicity result of solutions is obtained via one critical point theorem.In Chapters 6 and 7,selfadjiont operator equations are concerned on.Specifical-ly,linear operator possesses only a discrete spectrum with bounded from below.In Chapter 6,a superquadratic case is investigated.A multiplicity result is obtained via the same critical point theorem stated in Chapter 5.In Chapter 7,some general cases are considered.Some existence results of solutions not belonging to the kernel spaces of selfadjiont operator are obtained by linking methods of Schechter and a critical point theorem.