Research Article | Open Access

# The Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations

**Academic Editor:**Giovanni Anello

#### Abstract

We consider the existence of positive solutions for the nonlinear fractional differential equations boundary value problem where is a real number, is the Riemann-Liouville fractional derivative of order , and is a given continuous function. Our analysis relies on the fixed point index theory in cones.

#### 1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, or polymer rheology; see [1â€“5]. The interest of the study of fractional-order differential equations lies in the fact that fractional-order models are more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models. Recently, there are some papers dealing with the existence of solutions (or positive solution) of nonlinear initial value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, lower and upper solution method, Adomian decomposition method, ect.); see [6â€“15].

The famous viscous liquid flow problems in the fields of integer-order differential equations can be described by third-order ordinary differential equation boundary value problem where is continuous [16â€“18]. However, there are only a few exisitng contributions, as far as we know, in the field of fractional-order differential equation. In this paper, we discuss the existence of positive solution for the nonlinear fractional differential equations boundary value problem (BVP) where is a real number, is the Riemann-Liouville fractional derivative, and is a continuous function.

For a more general case, specially, where is continuous with , and is the Riemann-Liouville fractional derivative; El-Shahed [20] obtained the existence and nonexistence of positive solutions by employing the well-known Guo-Krasnoselskii fixed point theorem of cone extension or compression. The purpose of this paper is to extend this result. Our argument is based on the fixed point index theory, which is more precise than the fixed point theorem of cone extension or compression. We will employ the theory of fixed point index in cones to present some more extensive conditions on guaranteeing the existence of positive solution of the BVP (2). As far as we know, the method of the fixed point index theory is firstly applied to BVP (2).

This paper is organized as follows. In Section 2, we introduce some basic definitions and properties, preliminary results that will be used to prove our main results. In Section 3, we obtain the existence of the positive solutions for BVP (2) by using the fixed point index theory.

#### 2. Preliminaries

In this section, we introduce some preliminary facts which are used throughout this paper. For details, see [19].

*Definition 1 (see [19]). *The Riemann-Liouville fractional derivative of order of a continuous function is given by
where is Gamma function and , denotes the integer part of number , provided that the right side is pointwise defined on .

*Definition 2 (see [19]). *The Riemann-Liouville fractional integral of order of a function is given by
provided that the right side is pointwise defined on .

Lemma 3. *Let , if ; then the fractional differential equation
**
has unique solutions , , , where is the smallest integer greater than or equal to .*

Lemma 4. *Assume that with a fractional derivative of order that belongs to . Then
**
for some , , where is the smallest integer greater than or equal to .**In the following, we present the Greenâ€™s function of fractional differential equation boundary value problem.*

Lemma 5 5 (see [20]). *Let and . The linear fractional differential equation boundary value problem
**
has a unique solution
**
where
*

Lemma 6 (see [20]). *Let be Greenâ€™s function related to problem (8), which is given by the expression (10). Then, for all , the following properties are fulfilled:*(1)*, ;*(2)*, ;*(3)*, ;*(4)* is a continuous function, .**Let be the Banach space endowed with the norm . We define the operator by
**
where is the Greenâ€™s function defined in (10).**It is clear, form Lemma 5, that the nontrivial fixed points of operator coincide with the positive solutions of BVP (2).**Let . Define a cone by
*

Lemma 7. * is completely continuous.*

*Proof. *From the continuity and the nonnegativeness of functions and on their domains of definition, we have that if , then and for all ; from properties (2) and (3) of Lemma 6, for all ,
Hence, .

Next, we show that is uniformly bounded.

Let be bounded, which is to say, there exists a positive constant such that for all . Define now
Then, for all , it is satisfied that
That is, the set is bounded in .

Finally, we show that is equicontinuous.

For each , we have
As consequence, for all , , we have
Hence the set is equicontinuous.

Now, from the Arzela-Ascoli Theorem, we conclude that is relatively compact. Hence, is a completely continuous operator.

Define an operator by Clearly, also is a completely continuous linear operator and .

Lemma 8. *The operator defined by (18) satisfies
*

*Proof . *Let . For every , by the definition of ,
Hence,
This implies that
The proof is completed.

Hereafter, we use to denote the spectral radius of the operator .

Lemma 9. *Suppose that is defined by (18); then the spectral radius .*

*Proof. *Set . Then, by (18) and the positivity of , we have and
Inductively, we obtain that
Consequently,
So,
By this and Gelfandâ€™s formula of spectral radius we have
The proof of Lemma 9 is completed.

Now, since the operator is a completely continuous linear operator, by the well-known Krein-Rutman theorem ([21], Theorem 19.3), the operator has the maximum positive real eigenvalue ; then there exists a eigenfunction such that .

Set ; then . Thus, is the minimum positive real eigenvalue of the linear equation (8).

To prove the existence of at least one positive solution of BVP (2), we will find the nonzero fixed point of (defined in (11)) by using the fixed point index theory in cones.

We recall some concepts and conclusions on the fixed point index in cones in [21, 22], which will be used in the argument later. Let be a Banach space and let be a closed convex cone in . Assume that is a bounded open subset of with boundary and . Let be a completely continuous mapping. If for every , then the fixed point index is well defined. One important fact is that if , then has a fixed point in .

For , let , and , which is the relative boundary of in . The following two lemmas are needed in our argument.

Lemma 10 (see [22]). *Let be a completely continuous mapping. If there exists an such that
**
then the fixed point index .*

Lemma 11 (see [22]). *Let be a completely continuous mapping. If
**
then the fixed point index .*

Lemma 12 (see [22]). *Let be a completely continuous mapping and it satisfies that for every . If , , then the fixed point index .*

#### 3. Main Results

In this section we show the existence of positive solutions of BVP (2) by using the fixed point index theory in cones.

Theorem 13. *Assume is continuous and satisfies the following conditions.*(F1)*There exist and , such that
*(F2)*There exist and , such that
**Then, the BVP (2) has at least one positive solution.*

*Proof. *Let be the positive eigenfunction of corresponding to ; thus , where is defined by (18).

Choose , where is the constant in assumption (F1). For every , from assumption (F1), we have
Namely,

Suppose that has no fixed point on (otherwise, the proof is completed). Now we show that
If it is not true, there exist and (if , the proof is completed) such that
Then,
That is,
Let
It is easy to see that and . Taking into account the positivity of the Greenâ€™s function and definition of the operator , it is easy to know that is a nondecreasing linear operator, so
Therefore by (33)
which contradicts the definition of . Hence (34) holds and we have from Lemma 10 that

On the other hand, we choose . Now we show that if is large enough, then
From (F2), ; then there exist , such that .

Let , . Then is a bounded linear operator and .

Let
In the following, we prove that is bounded.

For any , we have
Thus,

Since , therefore ; it is easy to get that the inverse operator exists and
It follows from that . Hence we have
and is bounded. Let ; then, by Lemma 11, we have
Now by the additivity of fixed point index, (41) and (48), we have
Therefore has a fixed point in , which is a positive solution of BVP (2).

The proof of Theorem 13 is completed.

For convenience, we set .

Theorem 14. *Assume that is continuous and satisfies the following conditions.*(F3)*There exist and , such that
*(F4)*There exist and , such that
**Then, BVP (2) has at least one positive solution.*

*Proof. *Let , , where is defined in (18). Then, is a bounded linear operator; we have
Choose , where is the constant in assumption (F3). For every , from assumption (F3), we have
Namely, , .

Suppose that has no fixed point on (otherwise, the proof is completed). Now we show that
If it is not true, there exist and such that
Then, , . Therefore, we have
Thus, . Using Gelfandâ€™s theorem, we have
which contradicts (52). Hence (54) holds; from Lemma 11, we have

On the other hand, from (F4), we have
We choose . For every , by the definitions of and , we have and
Hence, form (59), it follows that
Now, by (11) and (61), we have
This means that
Hence, by Lemma 12, we have that
Now by the additivity of fixed point index, (58) and (64), we have
Therefore has a fixed point in , which is a positive solution of BVP (2).

The proof of Theorem 14 is completed.

*Example 15. *Consider the following fractional differential equations boundary value problem:
where , are constants. We assume that is large enough such that . Then it is easy to verify that
satisfies conditions (F1) and (F2) of Theorem 13. Hence, by Theorem 13, (66) has at least one positive solution.

*Example 16. *Consider the fractional differential equations boundary value problem
where , are constants. Let . We can easily see that
satisfies conditions (F3) and (F4) of Theorem 14. By Theorem 14, (68) has at least one positive solution.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research is supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).

#### References

- S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integral And Derivatives (Theory and Applications)*, Gordon and Breach, Yverdon, Switzerland, 1993. View at: MathSciNet - K. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, Wiley, New York, NY, USA, 1993. View at: MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at: MathSciNet - Y. Wang, L. Liu, and Y. Wu, â€śPositive solutions for a nonlocal fractional differential equation,â€ť
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 11, pp. 3599â€“3605, 2011. View at: Publisher Site | Google Scholar | MathSciNet - C. S. Goodrich, â€śExistence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,â€ť
*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 191â€“202, 2011. View at: Publisher Site | Google Scholar | MathSciNet - J. Mao, Z. Zhao, and N. Xu, â€śOn existence and uniqueness of positive solutions for integral boundary value problems,â€ť
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 16, pp. 1â€“8, 2010. View at: Google Scholar | MathSciNet - D. Jiang and C. Yuan, â€śThe positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,â€ť
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 2, pp. 710â€“719, 2010. View at: Publisher Site | Google Scholar | MathSciNet - S. Zhang and X. Su, â€śThe existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order,â€ť
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1269â€“1274, 2011. View at: Publisher Site | Google Scholar | MathSciNet - B. Ahmad and J. J. Nieto, â€śExistence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,â€ť
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838â€“1843, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z. Bai and H. Lü, â€śPositive solutions for boundary value problem of nonlinear fractional differential equation,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495â€“505, 2005. View at: Publisher Site | Google Scholar | MathSciNet - S. Zhang, â€śPositive solutions for boundary-value problems of nonlinear fractional differential equations,â€ť
*Electronic Journal of Differential Equations*, vol. 2006, no. 36, 12 pages, 2006. View at: Google Scholar | MathSciNet - Z. Wei, Q. Li, and J. Che, â€śInitial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 367, no. 1, pp. 260â€“272, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Lin, X. Liu, and H. Fang, â€śMethod of upper and lower solutions for fractional differential equations,â€ť
*Electronic Journal of Differential Equations*, vol. 100, pp. 1â€“13, 2012. View at: Google Scholar | MathSciNet - C. Bai, â€śImpulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,â€ť
*Journal of Mathematical Analysis and Applications*, vol. 384, no. 2, pp. 211â€“231, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. K. Ntouyas, G. Wang, and L. Zhang, â€śPositive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments,â€ť
*Opuscula Mathematica*, vol. 31, no. 3, pp. 433â€“442, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. C. Troy, â€śSolutions of third-order differential equations relevant to draining and coating flows,â€ť
*SIAM Journal on Mathematical Analysis*, vol. 24, no. 1, pp. 155â€“171, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Yao and Y. Feng, â€śThe existence of solution for a third-order two-point boundary value problem,â€ť
*Applied Mathematics Letters*, vol. 15, no. 2, pp. 227â€“232, 2002. View at: Publisher Site | Google Scholar | MathSciNet - F. Bernis and L. A. Peletier, â€śTwo problems from draining flows involving third-order ordinary differential equations,â€ť
*SIAM Journal on Mathematical Analysis*, vol. 27, no. 2, pp. 515â€“527, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier Science, Amsterdam, The Netherlands, 2006. View at: MathSciNet - M. El-Shahed, â€śPositive solutions for boundary value problem of nonlinear fractional differential equation,â€ť
*Abstract and Applied Analysis*, vol. 2007, Article ID 10368, 8 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin , Germany, 1985. View at: Publisher Site | MathSciNet - D. Guo and V. Lakshmikantham,
*Nonlinear Problems in Abstract Cones*, Academic Press, New York, NY, USA, 1988. View at: MathSciNet

#### Copyright

Copyright © 2014 Yanli Chen and Yongxiang Li. 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.