Existence Of Solutions And Positive Solutions For Boundary Value Problems Of Differential Equations

Boundary value problems for differential equation is an important branch of the theory of differential equation. It has many different applications in the field of sciences and engineering. In recent years, the research of multi-point boundary value problems for differential equation has been gained more and more attention. Multi-point boundary value problems originated in the 1980 s, since then, many scholars have discussed more general nonlinear multi-point boundary value problems, and a lot of fruitful results which included the results about fractional differential equation have been obtained.Resonance is a common phenomenon of nature. The mathematical model for describing resonance is boundary value problems for differential equation at resonance.Though, many scholars have studied the existence of solutions for multi-point boundary value problem and achieved fruitful results. But for multi-point boundary value problems at resonance, the results are rare, especially on one considered the positive solutions for multi-point boundary value problems of fractional order at resonance.We investigated the existence of solutions and positive solutions for some classes of boundary value problems for differential equation at resonance or non-resonance. Our results are based on the Leggett-Williams fixed point theorem, Mawhin continuation theorem, critical point theory and other theories. In view of the above discussion, this thesis organized as six chapters.In Chapter 1, we stated background for differential equation, current status of results, some basic definitions and the main work of this thesis.In Chapter 2, we studied the existence of positive solutions of boundary value problem for differential equation of fractional order at resonance by using LeggettWilliams fixed point theorem. About boundary value problems for differential equation of fractional order at resonance, there are many results, such as Dirichlet boundary value problems and anti-periodic boundary value problems. But the study did not appear which concern with positive solutions for multi-point boundary value problems of fractional differential equation at resonance. Thus, we further studied from three aspects of this problem based on the previous researches, namely, positive solutions of fractional order multi-point boundary value problems at resonance, positive solutions of two-point boundary value problems for fractional differential equation of higher order at resonance, positive solutions of multi-point boundary value problems for fractional differential equation at resonance on the half-line. Results which we have obtained extended and improved the some known works.In Chapter 3, by using Mawhin continuation theorem, the existence of solutions to boundary value problems for fractional differential equations at resonance has been studied. Previous studies on this problems based on the case of the kernel space is twodimensional and often needed to construct special projector operators. Our results did not need to construct projector operators of this type, so to a certain sense our results is an improvement and promotion for the previous works.In Chapter 4, by using the extension theorem of Mawhin continuation theorem proposed by Ge, we studied the existence of solutions of boundary value problem for fractional p-Laplacian equation at resonance. We studied the existence of solutions for a class of boundary value problem of p-Laplacian equation when the kernel space is two-dimensional and the existence of positive solutions for boundary value problem of fractional p-Laplacian equation at resonance. The first result specially improved the conditions of existence of the projector operator. No one investigated the second result which we obtained at present.In Chapter 5, by using critical theory, we investigated the existence of solutions of second order impulsive differential system and the existence of solutions of boundary value problem for fractional p-Laplacian equation with variational structure. Though,some scholars have studied impulsive differential equation by using the mountain theorem, link theorem, no one studied the boundary value problem for second order impulsive differential system. Our results rich the previous ones and it is an extension or improvement to the known results. We obtained multiplicity of solutions of boundary value problem for fractional p-Laplacian equation with variational structure in contrast with pervious results.In Chapter 6, we stated the conclusion of this thesis and the work in the future.