Abstract and Applied Analysis

Volume 2012, Article ID 512127, 16 pages

doi:10.1155/2012/512127

Research Article

Positive Solutions of Eigenvalue

Problems for a Class of Fractional Differential

Equations with Derivatives

Xinguang Zhang,1 Lishan Liu,2

Benchawan Wiwatanapataphee,3 and Yonghong Wu4

1

School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China 3

Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand 4

Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia 2

Correspondence should be addressed to Xinguang Zhang, zxg123242@sohu.com and Benchawan Wiwatanapataphee, scbww@mahidol.ac.th

Received 2 January 2012; Accepted 15 March 2012

Academic Editor: Shaoyong Lai

Copyright q 2012 Xinguang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional diﬀerential equations is discussed. Some suﬃcient conditions for the existence of positive solutions are established.

1. Introduction

In this paper, we discuss the existence of positive solutions for the following eigenvalue problem of a class fractional diﬀerential equation with derivatives −Dt α x t

λf t, x t , Dt β x t ,

β

γ

t ∈ 0, 1 ,

1.1

p −2

Dt x 0

0,

γ

Dt x 1

aj Dt x ξj ,

j1

where λ is a parameter, 1 < α ≤ 2, α − β > 1, 0 < β ≤ γ < 1, 0 < ξ1 < ξ2 < · · · < ξp−2 < 1, aj ∈ 0, ∞ with c

p −2

j1

α−γ −1

aj ξj

< 1, and Dt is the standard Riemann-Liouville derivative.

2

Abstract and Applied Analysis

f : 0, 1 × 0, ∞ × 0, ∞ → 0, ∞ is continuous, and f t, u, v may be singular at u 0, v 0, and t 0, 1.

As fractional order derivatives and integrals have been widely used in mathematics, analytical chemistry, neuron modeling, and biological sciences 1–6 , fractional diﬀerential equations have attracted great research interest in recent years 7–17 . Recently, ur Rehman and Khan 8 investigated the fractional order multipoint boundary value problem: Dt α y t

y0

f t, y t , Dt β y t ,

0,

Dt β y 1 −

t ∈ 0, 1 ,

1.2

m−2

ζi Dt β y ξi

y0 ,

i1

α−β−1

where 1 < α ≤ 2, 0 < β < 1, 0 < ξi < 1, ζi ∈ 0, ∞ with m−2 ζi ξi < 1. The Schauder ﬁxed

i1

point theorem and the contraction mapping principle are used to establish the existence and uniqueness of nontrivial solutions for the BVP 1.2 provided that the nonlinear function f : 0, 1 × R × R is continuous and satisﬁes certain growth conditions. But up to now, multipoint boundary value problems for fractional diﬀerential equations like the BVP 1.1 have seldom been considered when f t, u, v has singularity at t 0 and or 1 and also at u 0, v 0. We will discuss the problem in this paper.

The rest of the paper is organized as follows. In Section 2, we give some deﬁnitions and several lemmas. Suitable upper and lower solutions of the modiﬁed problems for the BVP 1.1 and some suﬃcient conditions for the existence of positive solutions are established in Section 3.

2. Preliminaries and Lemmas

For the convenience of the reader, we present here some deﬁnitions about fractional calculus. Deﬁnition 2.1 See 1, 6 . Let α > 0 with α ∈ R. Suppose that x : a, ∞ → R. Then the αth Riemann-Liouville fractional integral is deﬁned by

I αx t

t

1

Γα

t−s

α−1

2.1

x s ds

a

whenever the right-hand side is deﬁned. Similarly, for α ∈ R with α > 0, we deﬁne the αth Riemann-Liouville fractional derivative by

Dα x t

1

Γ n−α

d

dt

n

t

t−s

n−α−1

x s ds,...