Boundary Value Problems Of Nonlinear Dynamic Equations On Time Scales

This thesis is divided into fifth chapters,which are concerned with some dynamic boundary value problems on time scales.We focus on the application of spectral theory of linear dynamic problems to the existence of solutions for nonlinear dynamic boundary value problems.In the first chapter,we introduce the background material and the structure of this thesis.We also present the main problems with which we are concerned and state the main results.In this chapter,we give a new concept - V difference equation.The second chapter is to focus on the existence of positive solutions for the following nonlinear weighted eigenvalue problem on time scales T by using Rabi-nowitz global bifurcation theory Letλ₁ be the first eigenvalue of the linearization of problem(0.0.5).We answer the question that whatλcan be that which could essure problem(0.0.5)has at least one positive solution.The values ofλrely on the value ofλ₁.The results in this chapter generalize many previous theorems on the existence of positive solutions.The third chapter discusses the existence of nodal solutions and the global structure of the solution set for the following nonlinear eigenvalue problem by using global bifurcation theory on time scales T Letλ_k be the kth eigenvalue of the linearization of problem(0.0.6).We answer the question that whatλcan be that which could essure problem(0.0.6)has at least two nodal solutions with specified number of generalized zeros.The values ofλrely on the value ofλ_k.The results in this chapter could cover many previous theorems on the existence of positive solutions,and generalize some previous existence results of solutions for the similar problems.In order to overcome the difficulty steming from the fact that the operator is not self-adjoint,we introduce a new concept-V difference equation.Let V:= {a₁,…,a_m} be the finite subset of R.Say the difference equation on V the V defference equation.The fourth chapter is concerned with the following nonlinear boundary value problem of V difference system where e:V⁰→Rⁿ,and f∈C¹(Rⁿ)is a potential function satisfying some nonresonance conditions.By using a mini max theorem,we obtain the existence and uniqueness of solutions for problem(0.0.7).Here the conditions which essure the existence of solutions for problem(0.0.7)is sharp.By using connectivity properties of solution sets of parameterized families of compact vector fields,the fifth chapter considers the following nonlinear resonance problem of V difference equation where f∈C(R),r,h:V⁰→R,r(t)＞0,t∈V⁰.Letλ_k be an eigenvalue of the corresponding linear problem.We study the existence of one solution and two solutions of problem(0.0.8)under sign conditions and sublinear growth conditions.The previous work are all studied for the differential equations and difference equations,and lots of existence results need that the nonlinear function is bounded.So the conditions we give here greatly improve the previous ones.