On Positive Solutions For Multi-point Boundary Vaule Problems Of Fractional Differential Equation

Recently, more and more authors begin to pay attention to and study fractional calculus. Compared with integral calculus, fractional calculus has been much more widely used in natural science and engineering, such as Dynamical System, Control Theory, Stochastic Equation, Polymer Solution Chain, Flaccid Vibration, etc.By using topological degree, cone theory and monotone iterative technique, this paper investigates the existence and multiplicity of positive solutions for boundary value problems of nonlinear fractional differential equation. Generally speaking, a is defined asα∈(n,n+1], n≥2. In other words, we expandαin the Dirichlet-Type Equation in this paper to an arbitrary order.In this paper, we investigate the existence and multiplicity of positive solutions for boundary value problems of nonlinear fractional differential equation: where , Dα0+is the standard Riemann-Liouville differentiation, and the function f is con-tinuous on [In Chapter 1 we describe the orign of fractional calculus, with the development of fractional calculus in in natural science and engineering, we introduce the efforts and result that people have made for fractional calculus.In Chapter 2 we introduce some useful lemmas in nonlinear functional analysis, for they play an important role in the following proof.In Chapter 3 we present necessary definitions and properties of fractional cal-culus, which establishes a theoretical foundation for the following proof by using nonlinear functional analysis.While in Chapter 4, we introduce first two related fractional differential equa-tions. On the other hand, by some mathematical transformations, we present Green function of the problem in this paper. Additionally, we deduce properties of the Green function, such as nonnegativity, monotonicity, maximum value, etc.In Chapter 5, which is the principal chapter in this paper, we investigate non solution, one and only one solution, at least one solution, at least two solutions, at least three solutions of the problem in this paper comprehensively, by means of nonlinear functional analysis theory, such as contraction mapping theorem, Schauder fixed point theorem, topological degree, cone theory, Amann theorem, etc.