On the Existence of Solutions of Boundary Value Problems of Fractional Differential Equations 

Author  LiLei 
Tutor  ZhangKeMei 
School  Qufu Normal University 
Course  Applied Mathematics 
Keywords  Fractional differential equation Boundary value problems halfline Fixed point theorem Kuratowski noncompactness measure 
CLC  O175.8 
Type  Master's thesis 
Year  2012 
Downloads  82 
Quotes  0 
As an important branch of ordinary differential equations, In recent years, fractional differential equation has been continue to indepth study because of its theoretical 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 mathematics 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 differentiation, a, b are nonnegative constants,f:[0,1]×R×R→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+α1are the RiemannLiouville fractional derivatives and D0+α1u(∞)=limt→+∞D0+α1u(t).In this chapter,we not only obtain the existence of unbounded solutions by LeraySchauder 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+δ1are the standard RiemannLiouville fractional derivatives,u∞∈E,D0+α1u(∞)=limt→+∞D0+α1u(t).f:J×E×E→E.We use Monch fixed point theorem to obtain the existence of unbounded solutions for fractional order differential equation on a Banach space.