Mathematical Problems in Engineering

VolumeΒ 2009, Article IDΒ 535209, 17 pages

http://dx.doi.org/10.1155/2009/535209

## Singular Positone and Semipositone Boundary Value Problems of Nonlinear Fractional Differential Equations

^{1}School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China^{2}School of Mathematics and Computer, Harbin University, Harbin 150086, Heilongjiang, China

Received 8 January 2009; Accepted 15 April 2009

Academic Editor: VictoriaΒ Vampa

Copyright Β© 2009 Chengjun Yuan 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.

#### Abstract

We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problem ,β , , where , and are continuous, is a real number, and is Riemann-Liouville fractional derivative. Throughout our nonlinearity may be singular in its dependent variable. Two examples are also given to illustrate the main results.

#### 1. Introduction

Fractional calculus has played a significant role in engineering, science, economy, and other fields. Many papers and books on fractional calculus, and fractional differential equations have appeared recently, (see [1β9]). It should be noted that most of papers and books on fractional calculus, are devoted to the solvability of initial fractional differential equations (see [3, 4]). Here, we consider positive solutions of nonlinear fractional differential equation conjugate boundary value problem involving Riemann-Liouville derivative: where , , and are continuous. is a real number, and is the Riemann-Liouville fractional derivative.

It is well known that in mechanics the boundary value problem (1.1) where describes the deflection of an elastic beam rigidly fixed at both ends. The integer order boundary value problem (1.2) has been studied extensively. For details, see for instance, the papers [10β13] and the references therein. In [10, 12], Yao considered and using a Krasnosel'skii fixed-point theorem, derived a -interval such that, for any lying in this interval, the beam equation has existence and multiplicity on positive solution. In this paper, we will consider a more general situation, namely, the boundary value problem (1.1). To the best of our knowledge, there have been few papers which deal with the boundary value problem (1.1) for nonlinear fractional differential equation.

In this paper, in analogy with boundary value problems for differential equations of integer order, we firstly derive the corresponding Green's function named the fractional Green' function. Consequently problem (1.1) is reduced to an equivalent Fredholm integral equation of the second kind. Finally, using Krasnosel'skii's fixed-point theorems, the existence of positive solutions are obtained.

#### 2. Preliminaries

For completeness, in this section, we will demonstrate and study the definitions and some fundamental facts of Riemann-Liouville derivatives of fractional order which can be found in [5].

*Definition 2.1 (see [5, Definition 2.1]). *The integral
where , is called the Riemann-Liouville fractional integral of order .

*Definition 2.2 (see [5, page 36-37]). *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 .

From the definition of Riemann-Liouville derivative, for , we have
giving in particular , where is the smallest integer greater than or equal to .

Lemma 2.3. *Let , then the differential equation
**
has solutions , , , as unique solutions, where is the smallest integer greater than or equal to . **As , from Lemma 2.3, we deduce the following statement.*

Lemma 2.4. *Let , then
**
for some , , is the smallest integer greater than or equal to . **The following Krasnosel'skii's fixed-point theorem will play a major role in our next analysis.*

Theorem 2.5 (see [6]). *Let be a Banach space, and let be a cone in . Assume that are open subsets of with , and let be a completely continuous operator such that, either *(1)*, , , , or*(2)*, , .**Then has a fixed-point in β .*

#### 3. Green's Function and Its Properties

In this section, we derive the corresponding Green's function for boundary-value problem (1.1), and obtain some properties of Green's function.

Lemma 3.1. *Let be a given function, then the boundary-value problem,
**
has a unique solution
**
where
**
Here is called Green's function of boundary-value problem (3.1).*

*Proof. *By means of the Lemma 2.4, we can reduce (3.1) to an equivalent integral equation
From , we have and
Therefore, the unique solution of (3.1) is
The proof is finished.

Lemma 3.2. *The function defined by (3.3) has the following properties: *(1)*;*(2)* and for **where .*

*Proof. *Observing the expression of , it is clear that for . In the following, we consider .

For , we have

and

For , since , we have

Thus , for . Combining , we have
This completes the proof.

We note that is a solution of (1.1) if and only if

For our constructions, we will consider the Banach space equipped with standard norm . We define a cone by Define an integral operator by Notice from (3.13) and Lemma 3.2 that, for , on and then .

On the other hand, we have Thus, . In addition, standard arguments show that is completely continuous.

#### 4. Singular Positone Problems

In this section we present some new result for the singular problem where and nonlinearity may be singular at .

Using Theorem 2.5 we establish the following main result.

Theorem 4.1. *Suppose that the following conditions are satisfied. **
here is Green's function and
**
Then (4.1) has two nonnegative solutions with and for .*

*Proof. *First we will show that there exists a solution to (4.1) with for and Let
We now show
To see this, let . Then and for So for , we have
This together with (4.7) yields
so (4.12) is satisfied.

Next we show

To see this, let so , and let for .

We have

This together with (4.9) yields
Thus so (4.15) is held.

Now Theorem 2.5 implies that has a fixed-point , that is, and for . It follows from (4.12) and (4.15) that so we have

Similarly, if we put

we can show that there exists a solution to (4.1) with for and

This completes the proof of Theorem 4.1.

Similar to the proof of Theorem 4.1, we have the following result.

Theorem 4.2. *Suppose that (4.2)β(4.8) hold. In addition suppose
**
Then (4.1) has a nonnegative solution with and for .*

*Remark 4.3. *If in (4.19) we have , then (4.1) a nonnegative solution with and for .

It is easy to use Theorem 4.2 and Remark 4.3 to write theorems which guarantee the existence of more than two solutions to (4.1). We state one such result.

Theorem 4.4. *Suppose that (4.2)β(4.6) and (4.8) hold. Assume that and constants with , and
**
In addition suppose for each that
**
and
**
hold. Then (4.1) has nonnegative solutions with for .*

*Example 4.5. *Consider the boundary value problem
where is such that
here
Then (4.23) has two solutions with for

To see this we will apply Theorem 4.1 with (here will be chosen below)

Clearly (4.2)β(4.6) and (4.8) hold, and . Now (4.7) holds with since
Finally notice (4.9) is satisfied for small and large since
as , since . Thus all the conditions of Theorem 4.1 are satisfied so existence is guaranteed.

#### 5. Singular Semipositone Problems

In this section we present a new result for the singular semipositone problem: where and nonlinearity may be singular at .

Before we prove our main result, we first state a result.

Lemma 5.1. *Suppose with on . Then the boundary value problem,
**
has a solution with
**
here
**
In fact, from Lemma 3.1, (5.2) has solution
**
According to Lemma 3.2, we have
**
The above Lemma together with Theorem 2.5 establish our main result.*

Theorem 5.2. *Suppose that the following conditions are satisfied. **
here is any constant (choose and fix it) so that (note exists since in fact we can have ) ) and is Green's function and
**
Then (5.1) has a solution with for .*

*Proof. *To show that (5.1) has a nonnegative solution we will look at the boundary value problem
where
( is as in Lemma 5.1).

We will show, using Theorem 2.5, that there exists a solution to (5.16) with for . If this is true then is a nonnegative solution (positive on ) of (5.1), since

As a result, we will concentrate our study on (5.16). Let as in Section 2, and let
Next let ββbe defined by
In addition, standard argument shows that and is completely continuous.

We now show

To see this, let . Then and for Now for , the Lemma 5.1 implies
so for we have
This together with (5.12) yields
so (5.21) is satisfied.

Next we show

To see this let so and for . Also for we have
As a result
We have
This together with (5.14) yields
Thus so (5.25) is held.

Now Theorem 2.5 implies that has a fixed-point , that is, and for . Thus is a solution of (5.16) with for . Thus (5.1) has a positive solution for .

*Example 5.3. *Consider the boundary value problem
where is such that
here

Then (5.30) has a solution with for

To see this we will apply Theorem 5.2 with (here will be chosen later, in fact here we choose so that works, i.e., we choose so that ),

Clearly (5.7)β(5.11) and (5.13) hold. Now (5.12) holds with since
from (5.31). Finally notice (5.14) is satisfied for large since
as , since . Thus all the conditions of Theorem 5.2 are satisfied so existence is guaranteed.

#### Acknowledgments

This paper is supported by Key Subject of Chinese Ministry of Education (no. 109051) and Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032).

#### References

- O. P. Agrawal, βFormulation of Euler-Lagrange equations for fractional variational problems,β
*Journal of Mathematical Analysis and Applications*, vol. 272, no. 1, pp. 368β379, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Zhang, βExistence of positive solution for some class of nonlinear fractional differential equations,β
*Journal of Mathematical Analysis and Applications*, vol. 278, no. 1, pp. 136β148, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - D. Delbosco and L. Rodino, βExistence and uniqueness for a nonlinear fractional differential equation,β
*Journal of Mathematical Analysis and Applications*, vol. 204, no. 2, pp. 609β625, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH Β· View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach, Yverdon, Switzerland, 1993. View at Zentralblatt MATH Β· View at MathSciNet - M. A. Krasnosel'skiΔ,
*Positive Solutions of Operator Equations*, P. Noordhoff, Groningen, The Netherlands, 1964. View at Zentralblatt MATH Β· View at MathSciNet - R. W. Leggett and L. R. Williams, βMultiple positive fixed points of nonlinear operators on ordered Banach spaces,β
*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673β688, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Z. Bai and H. LΓΌ, βPositive solutions for boundary value problem of nonlinear fractional differential equation,β
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495β505, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Zhang, βThe existence of a positive solution for a nonlinear fractional differential equation,β
*Journal of Mathematical Analysis and Applications*, vol. 252, no. 2, pp. 804β812, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Q. Yao, βPositive solutions for Eigenvalue problems of fourth-order elastic beam equations,β
*Applied Mathematics Letters*, vol. 17, no. 2, pp. 237β243, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - W. Li and M. Zhang, βNon-degeneracy and uniqueness of periodic solutions for some superlinear beam equations,β
*Applied Mathematics Letters*, vol. 22, no. 3, pp. 314β319, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Q. Yao, βExistence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 8, pp. 2683β2694, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Galewski, βOn the nonlinear elastic beam equation,β
*Applied Mathematics and Computation*, vol. 202, no. 1, pp. 427β434, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet