Classification Schemes And Existence For Solutions Of Dynamic Equations On Time Scales

The theory of dynamic equations on time scales can not only unify differential and difference equations and understand deeply the essential difference between them,but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time.Hence,the studying of dynamic equations on time scales is worth with theoretical and practical values.At the same time,many difficulties occur when considering dynamic equations on time scales.For example,basic tools from calculus such as Fermat theorem,Rolle theorem and the intermediate value theorem may not necessarily hold and it is difficult to find a universal program for simulation in a model with various timescales,which attract attention of great deal researchers.In this PhD thesis,we first consider classification schemes for positive solutions of the first and second order dynamic systems.In terms of their asymptotic magnitudes, classification schemes for positive solutions of the first-order and second-order nonlinear dynamic systems are given.By Schander's fixed point theorem,Knaster's fixed point theorem and constructing various Banach space,we obtained some sufficient and/or necessary conditions for the existence of solutions with designated asymptotic properties.Such schemes are important since further investigations of qualitative behaviors of solutions can then be reduced to only a number of cases, which make one can forecast and control the direction of development for the dynamic system by adjusting parameter.Next,we study the existence of positive solution for a class singular p-Laplacian dynamic equation with multi-point boundary condition on time scales.As considering the difference equations and differential equations,it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign.Compared with the known results,the nonlinearity is allowed to change sign and is involved with the first-order derivative explicitly.In particular,the boundary condition includes the Dirichlet boundary condition and Robin boundary condition.The singularity may occur at the clearer location.By using Schauder fixed point theorem and upper and lower solution method,we ob- tain some new existence criteria for positive solution to a singular p-Laplacian BVPs with sign-changing nonlinearity involving derivative.We consider how to construct a lower solution and an upper solution in certain conditions and obtained a specific method.As an application,an example is given to illustrate our results.Finally,we discuss the existence of symmetric positive solutions for p-Laplacian two point boundary value problem on time scales.By using Krasnosel'skii's fixed point theorem in a cone,we first give the sufficient condition for single positive symmetric solutions of the problem under six possible cases.Using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem respectively, the existence criteria for at least three positive and arbitrary odd positive symmetric solutions of p-Laplacian dynamic equations are established and the difference between two methods is pointed out.Two simple examples are given to illustrate our results.