Periodic Boundary Value Problems And P-Laplacian Boundary Value Problems For Impulsive Dynamic Equations On Time Scales

In recent years, with the rapid development of modern science and technology,people have built a large number of non-continuous model. This requires a tool to connectcontinuous and discrete cases, so the time scale theory came into being. The theory cannot only unifies the nature of differential equations and difference equations, which avoidsduplication of research, but also reveal the differences in continuous and discrete cases ofsome other sides. In application, dynamic equations has a wide base in nature, for instance,epidemic models, neuronal network model and insectile numbers models. In addition tothese applications in biology, the mathematical tool has been used to prove calculationalmode about stock market. Therefore, the theory of time scales is a relatively new branchand it has a great value of research. Thus, the research of time scales has importantsignificance in theory and application.This paper focuses on the research of existence of solutions and positive solutions forperiodic boundary value problems for impulsive dynamic equations on time scales,existence of positive solutions and quasi-symmetric positive solutions for boundary valueproblems for p-Laplacian impulsive dynamic equations on time scales, the existence anduniqueness of nontrivial solutions for boundary value problems for p-Laplacian impulsivedynamic equations on time scales.Firstly, by the using upper and lower solutions and monotone iterative technique,periodic boundary value problems for first-order impulsive dynamic equations on timescales is researched. And some sufficient conditions of the existence of solutions forperiodic boundary value problems for first-order impulsive dynamic equations on timescales are given. At the same time, some example is also given. Then, existence of positivesolutions for periodic boundary value problems for a class of first-order nonlinearp-Laplacian impulsive dynamic equations on time scales are considered. By employingthe Green's function, some criterions are established. The main result is explained by anapplication example.Secondly, existence of positive solutions for boundary value problems for second- order p-Laplacian impulsive dynamic equations on time scales are discussed. And alsomany sufficient conditions of positive solutions on such equations are obtained. At thesame time, some example is also given. Further, the existence and multiplicity of thequasi-symmetric positive solutions for boundary value problems for second-orderp-Laplacian impulsive dynamic equations on time scales are studied, and the sufficientconditions of the existence and multiplicity of the quasi-symmetric positive solutions forsuch equations have been obtained. Moreover, an application example is provided in thelast part.Finally, the eigenvalue ranges of existence and uniqueness of nontrivial solutions forthree-order mixed derivatives eigenvalue boundary value problems of p-Laplacianimpulsive dynamic equations on time scales are investigated, and the eigenvalue rangesare also gained. Also, an application example is offered.