Positive Solutions For Boundary Value Problems Of Nonlinear Fractional Differential Equations

Nonlinear functional analysis is a research field of mathematics which has pro-found theories and extensive applications. It takes the nonlinear problems appeared inmathematics and the natural sciences as background to establish some general theoriesand methods to deal with nonlinear problems. Its rich theory and advanced methodhave provided the efective theory tool for solving many kinds of nonlinear diferentialequations, nonlinear integral equations and some other types of equations, and han-dling many nonlinear problems in computational mathematics, cybernetics, optimizedtheory, dynamic system, economical mathematics, etc. At present, the contents ofnonlinear functional analysis mainly have topology degree theory, critical point theory,partial order method, analysis method, monotone mapping theory and so on.The boundary value problems of nonlinear diferential equations for ordinary orderare important subjects in the theory of diferential equations. Owing to the impor-tance in both theory and application, boundary value problems for ordinary difren-tial equations have attracted many researchers, and a large number of results havebeen obtained. In recent years, fractional diferential equations have been widely usedin difusion and transport theory, chaos and turbulence, viscoelastic mechanics, non-newtonian fluid mechanics etc. It has received highly attention of the domestic andforeign mathematics and natural science field, and becomes one of the hottest issues inthe international research field. So the research on boundary value problem of nonlinearfractional diferential equations is very important in both theory and application.The present paper employs the theory and method of nonlinear functional anal-ysis, such as cone theory, fixed point theory, fixed point index theory, Krasnoselskiifixed point theorem, monotone iterative technique and the method of lower and uppersolutions, to investigate the positive solutions to several kinds of (singular) boundaryvalue problems of nonlinear fractional diferential equations (system), including somehigh singularity boundary value problems, nonlocal problems, semipositone problems,etc. In this paper, some new interesting results are obtained.The dissertation is divided into five chapters. In Chapter Ⅰ, we introduce the back-ground of fractional calculus and some basic concepts and theorems. In Chapter Ⅱ,the existence and uniqueness results for positive solutions are derived to three kinds ofnonlocal boundary value problem of singular fractional diferential equations. In§2.1,we establish the existence and uniqueness of positive solutions to a kind of m-pointsingular boundary value problem. In§2.2, we establish the existence of positive solu-tions to a high-order m-point singular boundary value problem. In§2.3, by using the first eigenvalue corresponding to the relevant linear operator, we consider the nonlinearhigh-order singular boundary value problems of fractional order with nonlocal condi-tion which is given by Riemann-Stieltjes integral with a signed measure. In ChapterⅢ, we study two kinds of semipositone boundary value problems of singular fractionaldiferential equation. In Chapter Ⅳ, the fractional derivative is the Caputo fractionalderivative. In§4.1, we obtain the existence of positive solutions for singular fractionaldiferential equations boundary value problems with sign-changing nonlinear term andmay be unbounded from below. In§4.2, we obtain the existence of positive solutionsfor singular fractional diferential system with sign-changing nonlinear term and maybe unbounded from below. In§4.3, we give a necessary and sufcient condition forthe existence of positive solutions to a boundary value problem of fractional diferen-tial equation. In Chapter Ⅴ, we discuss the existence of positive solutions for a kindof nonlocal boundary value problems for fractional diferential equations in BanachSpaces.