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Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application
Advances in Difference Equations volume 2013, Article number: 159 (2013)
In this paper, we consider the properties of Green’s function for the singular nonlinear fractional differential equation boundary value problem
where is a real number and is the standard Riemann-Liouville differentiation. As an application of the properties of Green’s function, we give the existence of multiple positive solutions for the above mentioned singular boundary value problems. Our tools are Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed-point theorem on cones.
MSC:34A08, 34B18, 45B05.
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 selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. For details, see [1–6]. Now the boundary value problem for fractional differential equations attracts lots of attention. Especially, Jiang and Yuan  considered the nonlinear fractional differential equation Dirichlet-type boundary value problem and established the existence of positive solutions for the corresponding BVP. Xu et al.  dealt with the following equation:
Motivated by the above mentioned works, we consider the properties of Green’s function for
where is a real number and is the standard Riemann-Liouville differentiation. As an application of Green’s function, we will give the existence of multiple positive solutions for singular boundary value problems (1.1), (1.2). As far as we know, no contributions concerning BVP (1.1), (1.2) exist.
The outline of this paper is as follows. In Section 2, we derive the corresponding Green’s function and some of its properties. In Section 3, by using Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed-point theorem in a cone, we offer criteria for the existence of positive solutions for singular BVP (1.1), (1.2).
2 Background materials and Green’s function
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literature such as .
Definition 2.1 
The Riemann-Liouville fractional integral of order of a function is given by
provided the right-hand side is pointwise defined on .
Definition 2.2 
The Riemann-Liouville fractional derivative of order of a continuous function is given by
where , denotes the integer part of the number α, provided that the right-hand side is pointwise defined on .
From the definition of the Riemann-Liouville derivative, we can obtain the statement.
Lemma 2.1 
Let . If we assume that , then the fractional differential equation
has , , , as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2 
Assume that with a fractional derivative of order that belongs to . Then
for some , , N is the smallest integer greater than or equal to α.
In the following, we present Green’s function of the fractional differential equation boundary value problem (1.1), (1.2).
Lemma 2.3 Given and , the unique solution of
Here is called Green’s function of BVP (2.1), (2.2).
Proof We may apply Lemma 2.2 to reduce (2.1) to an equivalent integral equation
for some . Consequently, the general solution of (2.1) is
By (2.2), we get that , and
Therefore, the unique solution of (2.1), (2.2) is
This completes the proof. □
From the expression of , we can obtain the following properties.
Lemma 2.4 Green’s function defined by (2.3) has the following properties:
Proof of (2.4)
where and are used. Thus, (2.4) is verified. □
Proof of (2.5) By direct calculation, we get
On the one hand, for , from (2.4), we can get
On the other hand, for , we have
In summary, the property (2.5) holds. □
Proof of (2.6) For , we have
So, we have
For , we first prove that
For fixed , let
Obviously, and . Equation (2.4) together with
implies that , . Hence (2.7) holds.
By (2.5) we know that is increasing in t on . Hence, for ,
This completes the proof of property (2.6). □
As an application of properties of Green’s function, we will establish the existence of positive solutions for BVP (1.1), (1.2) in Section 3, in which Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed-point theorem on cones are our main tools. For the convenience of the reader, we recall the two famous theorems here.
Lemma 2.5 Assume that Ω is a relative subset of a convex set K in a normed space X. Let be a compact map with . Then
A has a fixed point in , or
there are and such that .
Lemma 2.6 Let E be a Banach space, and let be a cone in E. Assume that , are open subsets of E with , and let be a completely continuous operator such that either
, ; , ; or
, ; , .
Then T has a fixed point in .
3 Positive solution of a singular problem
In this section, we establish the existence of positive solutions for singular BVP (1.1), (1.2). Throughout this section, we always assume that is continuous. Given , we write if for and it is positive in a subset of positive measure.
Let be endowed with the maximum norm, .
Theorem 3.1 Suppose that the following hypotheses hold:
(H1) for each constant , there exists a continuous function such that for all ;
(H2) there exist continuous, nonnegative functions and such that
where is nonincreasing and is nondecreasing in ;
(H3) there exists a constant such that for all ;
(H5) there exists a positive number r such that
Then BVP (1.1), (1.2) has at least one positive solution u with .
Proof Choose such that . For fixed , consider the family of integral equations
where , and , . We claim that any solution of (3.1) for any must satisfy . Otherwise, assume that is a solution of (3.1) for some such that . Note that by (2.6),
Hence, for all , by (2.6) again, we have
Thus, it follows from the choice of , (3.2), (2.6), (H2) and (H3) that for all ,
which contradicts (H5) and the claim is proved.
It is easy to check from (2.5), (2.6) and (H4) that the operator
is completely continuous, where . We omit the details here.
Now Lemma 2.5 guarantees that the integral equation has a solution, denoted by , in .
From (H1), we know that there exists a function such that for all ,
By using (2.5), (3.5) and a similar calculation as in (3.3), we have that for all ,
The Arzela-Ascoli theorem guarantees that has a subsequence converging uniformly on to a function . By the Lebesgue dominated convergence theorem, we have that
where (3.5) is used. Therefore, is a positive solution of BVP (1.1), (1.2). □
Theorem 3.2 Suppose that (H2), (H3), (H4) and (H5) are satisfied. Furthermore, assume that (H6) there exists a positive number such that the following inequality holds:
Then BVP (1.1), (1.2) has another solution with .
Proof To show the existence of , we will use Lemma 2.6. Define
It is obvious that K is a cone on . Let
Next, the operator is defined by
It is easy to check that the operator T maps into K. In fact, for any , we have from (2.6) that for ,
This implies that , that is, . In addition, by a similar argument as in Theorem 3.1, it is not difficult to prove that the operator is completely continuous. Now we prove that
For any , from (2.6), (H2), (H3), (3.6) and (H5), we have that for ,
Therefore, (3.7) holds. Next, we will prove that
For any , from (2.6), (H2), (H3), (3.6) and (H6), we have that
This implies that (3.8) holds.
It follows from Lemma 2.6 that the operator T has a fixed point . Clearly, this fixed point is a positive solution of BVP (1.1), (1.2) satisfying . □
Example 3.1 Consider the boundary value problem
where , and is a given parameter.
Corollary 3.1 Assume that , .
If , then (3.9) has at least one nonnegative solution for each .
If , then (3.9) has at least one nonnegative solution for each , where is some positive constant.
If , then (3.9) has at least two nonnegative solutions for each .
Proof We will apply Theorems 3.1 and 3.2 to obtain our desired results. Note that (H1) holds with . Let
Then (H2) and (H3) are satisfied. Since , (H4) is also satisfied. Now for (H5) to be satisfied, we need
for some , where
Therefore, (3.9) has at least one nonnegative solution for
Note that if , . If , set
The function possesses a maximum at
then . We have the desired results (i) and (ii). If , condition (H6) becomes
for some , where
Since , the right-hand side of (3.10) tends to 0 as . Thus, for any given , it is always possible to find an such that (3.10) is satisfied. Therefore, (3.9) has another nonnegative solution . This implies that (iii) holds. □
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The author Yuguo Lin would like to express his gratitude to Professor Daqing Jiang for his careful direction. This work was partially supported by NSFC of China (No. 11201008).
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.
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Jin, M., Lin, Y. Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application. Adv Differ Equ 2013, 159 (2013). https://doi.org/10.1186/1687-1847-2013-159
- fractional differential equation
- positive solution
- fractional Green’s function
- fixed-point theorem