An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation*

In this paper, we focus on the existence of positive solutions for the following eigen- value problem of Hadamard-type singular fractional differential equations with multipoint boundary conditions:

where is coninuous, is the Hadamard fractional derivative of order with, and the constants are nonnegative, the function is continuous.

In recent years, fractional-order nonlinear problems have attracted the attention of many researchers from mathematics and other applied science due to its wide range of applications in applied mathematics, physics, bioscience, engineering, chemistry, etc. A large number of contributions have been made for fractional differential equations in the sense of the Riemann–Liouville fractional derivative or the Caputo fractional derivative, [1, 4–6, 8, 9, 12–20]. However, the Hadamard-type fractional integral and derivative differ from the Riemann–Liouville and the Caputo fractional derivative since the kernels of the Hadamard-type integral and derivative contain logarithmic functions of arbitrary exponent and thus are regarded as a different kind of weakly singular kernels. Thus it is more difficult to explore the existence of solutions for the Hadamard-type fractional differential equations, [2, 10, 11, 21].

In the recent work [21], by analysing the spectral construct of a linear operator and calculating the fixed point index of the corresponding nonlinear operator, Zhang et al. considered the existence of positive solutions for the following Hadamard-type fractional differential equation:

where is a differential operator denoted byare the Hadamard fractional derivatives of order is a continuous function, and the criteria of the existence of positive solutions were established. Recently, based on Leray–Schauder-type continuation, El-Sayed and Gaafar [3] established the existence of positive solutions to a class of singular nonlinear Hadamard-type fractional differential equations with infinite- point boundary conditions or integral boundary conditions.

However when possesses singularities on space variables, especially for the eigenvalue problem, few results are established on Hadamard-type fractional differential equations. Inspired by the above works, the aim of this paper is to establish the existence of positive solutions for the eigenvalue problem of the Hadamard-type fractional differential equation (1) when possesses singularity on space variables.

The rest of this paper is organized as follows. In Section 2, we firstly recall the concepts and properties of Hadamard fractional integral and derivative and then give the logarithmic Green kernel. Our main results are summarized in Section 3.

In this section, we firstly present the definition of Hadamard-type fractional integral and derivatives as given in [7]. Then we give some basic lemmas, which will be used in the rest of the paper.

Suppose is a finite or infinite interval of .

The -order left Hadamard fractional integral is defined by

and the left Hadamard fractional derivative is defined by

The relationship between fractional integration and derivative is introduced as follows.

Lemma 1. (See [7].). Suppose .

1. (i) If for any

2. (ii) The equality holds for every .

3. (iii) Let . The following formula holds:

4. (iv)

Lemma 2. (See [3].) For , the boundary value problem

subject to the multi-point boundary conditions

has a unique solution if and only if x is a solution of the integral equation

where

and

Lemma 3. (See [15]). Let . The Green’s functions has the following properties:

1. (i)

2. (ii) For all , the following inequalities hold:

Definition 1. A continuous function is called a lower solution of (1) if it satisfies

Definition 2. A continuous function is called a upper solution of the eigenvalue problem (1) if it satisfies

We make the following assumptions throughout this paper:

1. (H1) is continuous and is nonincreasing in ;

2. (H2) For all , there exists a constant such that, for any

Remark 1. For , by (H2), we have the following equivalent conclusion: for any

In fact, for and any, one has .

Lemma 4 [Maximal principle]. If satisfies

Proof. By Lemma 2, the conclusion is obvious, and we thus omit the proof here.

Let

then we state our main result as follows.

Theorem 1. Suppose (H1) and (H2) hold, and

Then there are constants such that for any , the eigenvalue problem (1) has at least one positive solution satisfying the asymptotic property

Proof. Firstly, define a function space and a subset

(3) Obviously, is a nonempty since ..

Define an operator

It follows from Lemma 2 that the fixed point of the operator is the solution of the eigenvalue problem (1).

In what follows, we prove that the operator is well defined and . To do this, for any , it follows from the definition of that there exists a positive number such that . Choose , then we have. So by Lemma 3, (H2) and (H3), we gets

Next, take , then it follows from Lemma 3 and (H2)

(5) and (7) indicate that is well defined and .

Now we shall try to construct the upper and lower solutions of the eigenvalue problem (1). As the operator is decreasing on , let

then, similar to 7, for all , one gets

i.e.,

where

On the other hand, notice that is decreasing in , thus, for any , it follows from Lemma 3 and (H3) that

Now take and

Let

By (H2), for any , we have

Thus it follows from Lemma 3 that

Let

then by Lemma 2, for any , we have

It follows from (8) and (9) that and

which implies

Thus, by (10), (11), we have

(8) and (9) imply that satisfy the boundary value conditions of the eigenvalue problem (1). Thus it follows from (11)–(13) that are upper and lower solutions of the eigenvalue problem (1) when

Next, construct a function :

For any , consider the following modified eigenvalue problem:

We define an operator

It follows from the assumption that is continuous. Thus it is clear that a fixed point of the operator is a solution of the modified eigenvalue problem (15).

For all , it follows from Lemma 3, (14) and that

So is bounded. It is easy to see that is continuous from the continuity of

On the other hand, for any bounded, since is uniformly continuous on , we know that is equicontinuous. Thus the Arzela–Ascoli theorem implies thatis completely continuous. It follows from the Schauder fixed point theorem that has at least one fixed point such that

Now we show

To do this, let . Since is the upper solution of theeigenvalue problem (1) and y is a fixed point of , we have

It follows from the definition of , (10) and (11) that

i.e.,

which implies that . It follows from Lemma 4 that , that is, . By the same way, we have , thus we get

By (14), we have . Consequently, is a positive solution of the eigenvalue problem (1).

Finally, we prove the asymptotic properties of solutions. Firstly, from (19) we get

On the other hand, it follows from (20) and Lemma 3 that

Thus we get the asymptotic properties of solutions .

where

Proof. Let

Then (H1) holds, and for all and for any,

which implies that (H2) also holds.

Also, by direct calculation, we have

Hence (H3) holds. Hence, by Theorem 1, there are two constants such that for any , the singular eigenvalue problem (21) has at least one positive solution , and there exists a constant such that

The authors would like to thank the referee for his/her comments that improve the results and the quality of the paper.