- Open Access
Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application
© Jin and Lin; licensee Springer 2013
- Received: 22 May 2013
- Accepted: 22 May 2013
- Published: 6 June 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 equation
- positive solution
- fractional Green’s function
- fixed-point theorem
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).
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 
provided the right-hand side is pointwise defined on .
Definition 2.2 
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 
has , , , as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2 
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).
Here is called Green’s function of BVP (2.1), (2.2).
This completes the proof. □
From the expression of , we can obtain the following properties.
where and are used. Thus, (2.4) is verified. □
In summary, the property (2.5) holds. □
implies that , . Hence (2.7) holds.
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.
A has a fixed point in , or
there are and such that .
, ; , ; or
, ; , .
Then T has a fixed point in .
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 ;
where is nonincreasing and is nondecreasing in ;
(H3) there exists a constant such that for all ;
Then BVP (1.1), (1.2) has at least one positive solution u with .
which contradicts (H5) and the claim is proved.
is completely continuous, where . We omit the details here.
Now Lemma 2.5 guarantees that the integral equation has a solution, denoted by , in .
where (3.5) is used. Therefore, is a positive solution of BVP (1.1), (1.2). □
Then BVP (1.1), (1.2) has another solution with .
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 . □
where , and is a given parameter.
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 .
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. □
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).
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