Existence Of Solutions For Several Dynamic Equations On Measure Chains

The theory of differential and difference equations can be unified by dynamicequations on measure chains，by using the theory of measure chains，we canunderstand this two kinds of equations better，and give a better insight of their essentialdifference. Because of the specificity of measure chains--including continuous caseand discrete case，studies for dynamic equations on measure chains have obtained highattention. This thesis consists of five chapters，mainly discuss several dynamicequations on measure chains (including initial value problem，boundary value problem,and terminal value problem). By using the theorems of nonlinear functional analysis，we discuss the existence of solutions for these equations.In Chapter1，we introduce the historical background of measure chains，state themain results of this thesis and list some important definitions and theorems on measurechains that we will use.In Chapter2，we study initial vale problems on time scales，comparing with initialvalue problems of ordinary differential equation, firstly we give the definition of uppersolution and lower solution and obtain the relationship between them under appropriateconditions，then by using lower and upper solution method，we discuss the existence ofsolutions for the initial value problem.In Chapter3，we study boundary value problems on time scales. In section1，westudy the boundary value problem on finite intervals，by using Leray–Schauder fixedpoint theorem，we obtain the sufficient conditions that there exist at least one positivesolution. In section2，we study the boundary value problem on infinite intervals，including the proofs of the theorems and their applications. By using Leray-SchauderNonlinear Alternative and Leggett-Williams fixed point theorem，we establish thecriteria of the existence of at least one positive solution and at least three positivesolutions for the boundary value problem. Finally we illustrate our results by examples.In Chapter4，we study terminal value problems on time scales. There are twosections. In section1，we consider first-order terminal value problems on time scales，comparing with terminal value problems of ordinary differential equations, weintroduce the definition of upper solution and lower solution for it and discuss therelationship between them，then by using lower and upper solution method，we study the existence of solutions for the terminal value problem. In section2，we study twokinds of second-order terminal value problems on time scales，the nonlinear terminclude x Δ(t)and not include x Δ(t)respectively，and we also discuss the existenceof solutions by using lower and upper solution method.In Chapter5，we give a summary and the outlook of this thesis.