Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems

In this paper, a fractional multi-point boundary value problem is considered. By using the fixed point index theory and Krein-Rutman theorem, some results on existence are obtained.


Introduction
Fractional differential equations have been of great interest recently.This is due to the intensive development of the theory of fractional calculus itself as well as its applications.Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self similar and porous structures, electrical networks, visco-elasticity, chemical physics, and many other branches of science.For details, see [1]- [7].
It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions.Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional differential equations boundary value problems, see [8]- [14].
Now we list some conditions for convenience. (H1) 0

Preliminary
For the convenience of readers, we provide some background material in this section.{\bf Definition 2.1 [7] The Riemann-Liouville fractional of order α for function y is defined as ( ) (1 ) , ( ) The proof is complete.Lemma 2.6 If (H1) hold, then there exist a constant M such that 2 0 ( , ) (1 (1 ) ( ) ( ) .The proof is completed.
Define an operator : and a linear operator : T X X → as follows: Then the fixed point \ {0} u K ∈ of A is the positive solutions of (1)

Main Results
In order to obtain our main results, we firstly present and prove some lemmas.By the same method, we can get that : T X X → is completely continuous also.Lemma3.2If (H1)-(H2) hold, then ( ) 0 r T > ( r is the spectral radius of T ) Proof: Take ( ) (1 , ( ) lim 0 The proof is completed.By Lemma 2.3, we can get there exists 0 the lebesgue measure of E and the same as follows).

Example
Let's consider the following boundary value problem  6) has at least one solution.This problem can be not solved by the theorem in [11].

Lemma 3 . 1
If (H1)-(H3) hold, then : to the Lebesgue Dominated Convergence Theorem and Lemma 2.6, we have : A K K → is uniformly bounded and equicontinuous.It follows from Ascoli-Arzela theorem that : has at least one positive solution.Proof: It follows from 0 f µ ∞ ≤ < and Lemma 3.4 there exists 0 r > such that ( , , ) 1 r i A K K = .By 0 f µ < < ∞ and Lemma 3.3, we can get there exists 0 r ρ < < such that either there exists u K ∈ ∂ with or ( , , ) 0 u Au i A K K ρ = = .In the second case, A has a fixed point u K ∈ with u r ρ ≤ < by the properties of index.The proof is completed.