Existence Of Solutions For Boundary Value Problems Of Nonlinear Fractional Differential Equations

Fractional calculus is an extension and expansion of integral calculus. It is the theory of researching the mathematics problems which are characteristics and applications of differential and integral operators of arbitrary order (real order or complex order). The development of fractional calculus almost togethers with that of integeral calculus and has the extensive theoretical significance and practical research values. Fractional differential equations have been of great interest recently that depends on both by the intensive development of the theory of fractional calculus itself and by the applications in various fields. In view of the applications of fractional differential equations in various fields, the researches on the existence of solutions for linear and nonlinear fractional order differential equations boundary value problems has caught attention of the domestic and foreign mathematical workers and will become a hot issue.In this paper, we consider the existence of solutions for boundary value problems of nonlinear fractional differential equations and give some new existence theorems. Some examples are presented to illustrate the main results respectively.In chapter 1, we introduce some background materials from fractional calculus theory and main works of this paper. Also, we give some preliminary definitions and lemmas on fractional differential equations which are needed in this thesis.In chapter 2, we study a class of multiplicity of positive solutions for nonlinear fractional differential equations boundary value problem. In first section, we give some preliminary definitions and lemmas on the problem. In second section, by upper and lower solution method, we establish a sufficient condition for the existence of at least one positive solution of the problem. In third section, using Leggett-Williams fixed point theorem, we give some existence of at least three positive solutions.In chapter 3, we consider the existence of positive solutions for two classes of boundary value problems of singular nonlinear fractional differential equations. In first section, by nonlinear alternative of Leray-Schauder type, Guo-Krasnosel'skii fixed point theorem on a cone and a fixed point theorem in partially ordered sets, some new existence results of positive solutions for a class of singular nonlinear fractional differential equation boundary value problem ( f is singular at t = 0) are given. In second section, some new existence of at least one or two positive solutions for a class of singular nonlinear fractional differential equation boundary value problem ( f is singular at u = 0) are obtained by Guo-Krasnosel'skii fixed point theorem.In chapter 4, we discuss the existence of positive solutions for two classes of Caputo-like and Riemann-Liouville-like fractional nonlinear fractional differential equations boundary value problems with parameters. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on a cone, eigenvalue intervals of nonlinear fractional differential equations boundary value problems are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established.In chapter 5, we investigate the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders. In first section, we give some preliminary definitions and lemmas on the problem. In second section, we establish existence results of at least one or two positive solutions for boundary value problem of non singular system by Guo-Krasnosel'skii fixed point theorem and a fixed point theorem on a cone. In third section, by Schauder fixed point theorem and Banach contraction principle, we give the existence and uniqueness of positive solution for boundary value problem of nonsingular system. In fouth section, by nonlinear alternative of Leray–Schauder type, Guo-Krasnosel'skii fixed point theorem on a cone, we obtain existence results of positive solutions for boundary value problem of singular system.