Existence And Multiplicity Of Solutions To Boundary Value Problems Of Some Fractional Differential Equations(Systems)

Boundary value problems of nonlinear differential equations are important branch in the qualitative theory of differential equations and have a wide application back-ground.In recent years,with the development of fractional calculus theory,fractional differential equations have been widely used in many fields,such as physical mechan-ics,anomalous diffusion,automatic control,biomedicine and so on,which makes peo-ple pay more attention to the study of fractional boundary value problems and many profound results have been established.Based on the existing literature,by applying the generalized set-valued mapping type Leggett-Williams theorem,improved abstract continuation theorem of k-set contractive operator,improved Avery-Henderson fixed point theorem and the classical critical point theory,topological degree theory and other theoretical methods,this dissertation investigates the existence and multiplicity of so-lutions for several kinds of boundary value problems of fractional differential equations(Systems).As an application,this dissertation also discusses the existence of solutions and Ulam's type stability results for fractional boundary value problems on star graph Some new results are obtained,which generalize and enrich the previous work.Further-more,the improved theorem provides a new method for the study of related problems The whole dissertation is divided into seven chaptersChapter 1 introduces the research background and current situation of the frac-tional boundary value problems.Some basic knowledge needed and the main work done in this thesis are also summarizedChapter 2 studies the existence of positive solutions for quasilinear fractional dif-ferential inclusions at resonance.We extend the set-valued mapping type Leggett-Williams theorems proved by O'Regan and Zima,and obtain the Leggett-Williams theorem of set-valued mapping type for quasilinear operators.By using the generalized set-valued mapping type Leggett-Williams theorem,we give the existence results of positive solutions for a class of fractional differential inclusions with p-Laplacian op-erator at resonance.The results in this chapter enrich the theoretical results in related fields and provide a research method for the existence of positive solutions for reso-nance boundary value problems of differential inclusions with quasilinear operatorsChapter 3 discusses the existence of solutions to the boundary value problems for fractional implicit coupled systems.We improve the abstract continuation theorem of k-set contractive operator,which simplifies the verification process for the application of the theorem to the discussion of resonance boundary value problems.By using the improved abstract continuation theorem of k-set contractive operator,we give the ex-istence conditions of solutions for periodic and anti-periodic boundary value problems of a class of fractional implicit coupled systems with perturbations.In addition,the existence of solutions to periodic boundary value problems for a class of fractional im-plicit coupled systems is proved by using the Mawhin continuation theorem.It is noted that there are few works to deal with boundary value problems of fractional implicit differential equations by using continuation theorem.Our work generalizes,improves and revises the results of the existing literatureChapter 4 investigates the existence and multiplicity of solutions to the integral boundary value problems of Hadamard type fractional differential equations on infinite interval.In order to prove the multiplicity of the solutions results,a new fixed point the-orem is established in this chapter,that is,the improved Avery-Henderson fixed point theorem,which gives the conclusion that there are three fixed points(the original theo-rem result is the existence of two fixed points).By using the new fixed point theorem,some other fixed point theorems and monotone iterative method,the existence and mul-tiplicity of positive solutions for integral and multi-point boundary value problems of Hadamard type fractional differential equation on infinite interval are discussed.More-over,we also study the Hadamard type fractional integral boundary value problems at resonance on infinite interval,obtain the existence results,and prove a criterion of the operator compactness for the case where the nonlinear term of differential equation de-pends on the lower derivative(see lemma 4.7).The improved Avery-Henderson fixed point theorem provides a criterion for the study of multiple solutions of boundary value problems.Compared with the existing literature,the problems studied in this chapter are more general and the conditions given by the theorem are weaker.Chapter 5 consider the existence and multiplicity of solutions for fractional bound-ary value problems with instantaneous and non-instantaneous impulses.By using the variational methods,the existence and multiplicity of solutions to impulsive problems and parametric impulsive problems are established,respectively.It is noted that the previous work is only to study the fractional boundary value problems with one form impulse,so the research in this chapter is more extensive,and the results enrich the related work of the boundary value problems of fractional impulsive differential equa-tions.Chapter 6 studies the existence and uniqueness of solutions and Ulam's type sta-bility results for fractional boundary value problems on a star graph.The problem investigates in this chapter is the application of boundary value problems in star graph.By using Schaefer's fixed point theorem and Banach's contraction mapping theorem,the existence and uniqueness of solutions for boundary value problems of system dif-ferential equations on star graph are established,and the Ulam type stability is proved.Compared with the existing literature,our study problem model is more general,and the existence result is established under the weaker conditions.Moreover,we also dis-cuss the Ulam type stability.It is noted that the research on the Ulam type stability for boundary value problems of differential equations on star graph and of the frac-tional differential systems of higher dimension(n>2)has not been involved at present.Therefore,our work in this chapter generalizes,improves and enriches the previous results.Finally,the main results of this dissertation and the following research work are given in the Chapter 7.