Existence And Multiplicity Of Solutions For Three Classes Of Operator Equations

This thesis investigates the existence and multiplicity of solutions for two classes of unbounded self-adjoint operator equations and a class of bounded self-adjoint operator equations.The thesis consists of six chapters.In Chapter 1,we briefly introduce some research background and main results of the thesis,and give some preparatory knowledge.In Chapters 2-4,we focus on a class of semi-unbounded self-adjoint operator equations which possess only discrete spectrum.More specifically,in Chapter 2,we investigates the existence of subquadratic convex or B-concave operator equations,and obtain four existence results by the index theory and the critical point theory.In Chapter 3,by using the index theory,critical point reduction method and three critical points theorems,we obtain four existence results of three solutions for operator equations.In Chapter 4,under the nonlinear terms without the evenness assumption,we obtain two existence results of infinite many solutions for operator equation by the multiple critical point theorem.Specially,as applications,we use the above existence results to study the second order Hamiltonian systems satisfying generalized periodic boundary value conditions and Sturm-Liouville boundary value conditions,and obtain some new existence and multiplicity results.At the same time,some examples are given to illustrate the effectiveness of our results.In Chapters 5,we focus on a class of unbounded self-adjoint operator equations which possess only discrete spectrum.By using the index theory and the dual least action principle,we investigates the existence of solutions for subquadratic convex operator equations at resonance.Applying the results to subquadratic convex first or second Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions,generalized periodic boundary value conditions and Sturm-Liouville boundary value conditions yield some new theorems concerning existence of solutions or nontrivial solutions.In Chapters 6,we focus on a class of bounded self-adjoint operator equations.By using the Ekeland variational principle and concentration compactness principle,we obtain a nonzero solution of such operator equations.As applications,we discuss the existence of nontrivial homoclinic orbitals for first order Hamiltonian systems and second-order indefinite Hamiltonian systems.