Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1–5]. For the basic theory and recent development of the subject, we refer a text by Lakshmikantham [6]. For more details and examples, see [7–23] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.

In [

23], Zhang used a fixed-point theorem for the mixed monotone operator to show the existence of positive solutions to the following singular fractional differential equation.

subject to the boundary conditions

where
is the standard Rimann-Liouville fractional derivative of order
, the nonlinearity
may be singular at
, and function
may be singular at
. The author derived the corresponding Green's function named by fractional Green's function and obtained some properties as follows.

Proposition 1.1.

Green's function
satisfies the following conditions:

(i)
for all
;

(ii)there exists a positive function
such that

where

and

here
.

It is well known that the cone theoretic techniques play a very important role in applying Green's function in the study of solutions to boundary value problems. In [

23], the author cannot acquire a positive constant taking instead of the role of positive function

with

in (1.3). At the same time, we notice that many authors obtained the similar properties to that of (1.3), for example, see Bai [

12], Bai and L

[

13], Jiang and Yuan [

14], Li et al, [

15], Kaufmann and Mboumi [

19], and references therein. Naturally, one wishes to find whether there exists a positive constant

such that

for the fractional order cases. In Section 2, we will deduce some new properties of Green's function.

Motivated by the above mentioned work, we study the following higher-order singular boundary value problem of fractional differential equation.

where
is the standard Rimann-Liouville fractional derivative of order
and
may be singular at
or/and at
,
is nonnegative, and
.

For the case of
, the boundary value problems () reduces to the problem studied by Eloe and Ahmad in [24]. In [24], the authors used the Krasnosel'skii and Guo [25] fixed-point theorem to show the existence of at least one positive solution if
is either superlinear or sublinear to problem (). For the case of
, the boundary value problems () is related to a m-point boundary value problems of integer-order differential equation. Under this case, a great deal of research has been devoted to the existence of solutions for problem (), for example, see Pang et al. [26], Yang and Wei [27], Feng and Ge [28], and references therein. All of these results are based upon the fixed-point index theory, the fixed-point theorems and the fixed-point theory in cone for strict set contraction operator.

The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, we present the expression and properties of Green's function associated with boundary value problem (). In Section 3, we discuss some characteristics of the integral operator associated with the problem () and state a fixed-point theorem in cones. In Section 4, we discuss the existence of at least one positive solution of boundary value problem (). In Section 5, we will prove the existence of two or
positive solutions, where
is an arbitrary natural number. In Section 6, we study the nonexistence of positive solution of boundary value problem (). In Section 7, one example is also included to illustrate the main results. Finally, conclusions in Section 8 close the paper.

The fractional differential equations related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to fractional differential equations. The readers who are unfamiliar with this area can consult, for example, [1–6] for details.

Definition 1.2 (see [4]).

where
, is called Riemann-Liouville fractional integral of order
.

Definition 1.3 (see [4]).

For a function

given in the interval

, the expression

where
denotes the integer part of number
, is called the Riemann-Liouville fractional derivative of order
.

Lemma 1.4 (see [13]).

Assume that

with a fractional derivative of order

that belongs to

. Then

for some
, where
is the smallest integer greater than or equal to
.