On The Existence Of Solutions Of Boundary Value Problems Of Fractional Differential Equations

As an important branch of ordinary differential equations, In recent years, frac-tional differential equation has been continue to in-depth study because of its theo-retical system of continuous improvement and many practical applications (such as: physics, mechanics, chemistry and engineering, etc.). The experts of mathematical world and the natural science world attach importance to the nonlinear functional analysis. Fractional differential equations has become an important modern mathe-matics research direction.Fractional differential equations is a hot topic in recent years, research in this filed is a very important area. In this paper, using the cone theory, fixed point theorem for nonlinear functional methods discuss the existence of solutions for nonlinear fractional differential equation with different boundary value problems, and obtained some new results.The thesis is divided into four sections according to contents.Chapter1Preference, we introduce the main contents, then give the related concepts and important lemma of this paper.Chapter2In this chapter, we discuss the existence and uniqueness of a solution to boundary value problem of nonlinear fractional differential equation where1<α≤2,0<β<1are real numbers. cD0+α,cD0+β are the Caputo’s differen-tiation, a, b are nonnegative constants,f:[0,1]×R×Râ†'R, R=(-∞,+∞), and f(t,0,0)(?)0, h(u) is continuous. In conclusion, we obtain the existence and uniqueness of solutions for this boundary value problem by using Schauder fixed point theorem and the Banach contraction mapping principle.Chapter3In this chapter, we are concerned with the unbounded solutions to the following boundary value problem of fractional order where n一1<α<n,n∈N+,u∞∈[0,∞),D0+α and D0+α-1are the Riemann-Liouville fractional derivatives and D0+α-1u(∞)=limtâ†'+∞D0+α-1u(t).In this chapter,we not only obtain the existence of unbounded solutions by Leray-Schauder nonlinear alternative theorem but also establish iterative schemes for approximating the solutions.Chapter4At the basis of the former one chapters,we deal with the existence of solutions for boundary value problem for fractional order differential equation of the form where1<δ≤2,D0δ+,and D0+δ-1are the standard Riemann-Liouville fractional deriva-tives,u∞∈E,D0+α-1u(∞)=limtâ†'+∞D0+α-1u(t).f:J×E×Eâ†'E.We use Monch fixed point theorem to obtain the existence of unbounded solutions for fractional order differential equation on a Banach space.