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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 916260, 6 pages

http://dx.doi.org/10.1155/2014/916260

## Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions

^{1}Department of Applied Mathematics, Xidian University, Xi'an, Shaanxi 710071, China^{2}School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

Received 26 November 2013; Accepted 13 February 2014; Published 24 March 2014

Academic Editor: Hossein Jafari

Copyright © 2014 Guotao Wang 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

By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.

#### 1. Introduction

Fractional calculus has been receiving more and more attention in view of its extensive applications in the mathematical modelling coming from physical and other applied sciences; see books [1–5]. Recently, the existence of solutions (or positive solutions) of nonlinear fractional differential equation has been investigated in many papers (see [6–28] and references cited therein). However, in terms of the eigenvalue problem of fractional differential equation, there are only a few results [29–33].

To the best of author’s knowledge, no paper has considered the eigenvalue problem of the following nonlinear fractional differential equation with integral boundary conditions: where , is the Caputo fractional derivative, and is a continuous function.

Our proof is based upon the properties of the Green function and Guo-Krasnoselskii’s fixed point theorem given in [34]. Our purpose here is to give the eigenvalue interval for nonlinear fractional differential equation with integral boundary conditions. Moreover, according to the range of the eigenvalue , we establish some sufficient conditions for the existence and nonexistence of at least one positive solution of the problem (1).

#### 2. Preliminaries

For the convenience of the readers, we first present some background materials.

*Definition 1. *For a function , the Caputo derivative of fractional order is defined as
where denotes the integer part of the real number .

*Definition 2. *The Riemann-Liouville fractional integral of order for a function is defined as
provided that such integral exists.

Lemma 3. *Let ; then
**
for some , , .*

Lemma 4 (see [34]). *Let be a Banach space, and let be a cone. Assume that , are open subsets of with , , and let be a completely continuous operator such that*(i)*, , and , , or*(ii)*, , and , .**Then has a fixed point in .*

Lemma 5. *Let , , , and . Assume ; then the unique solution of the problem
**
is given by the expression
**
where
*

*Proof. *It is well known that the equation can be reduced to an equivalent integral equation:
for some .

By the conditions and , we can get that and
Hence, we have

Put ; then, from (10), we deduce that
which implies that

Replacing this value in (10), we obtain the following expression for function :
This completes the proof.

Lemma 6. *Let be the Green function, which is given by the expression (7). For , the following property holds:
*

The proof is similar to that of Lemma 2.4 in [7], so we omit it.

Consider the Banach space with general norm Define the cone .

Suppose is a solution of (1). It is clear from Lemma 5 that

Define the operator as follows:

Lemma 7. * is completely continuous.*

*Proof. *Since , it is obvious that . So we have
Therefore, . The other proof is similar to that in [7], so we omit it.

#### 3. Main Result

For convenience, we list the denotation:

Next, we will establish some sufficient conditions for the existence and nonexistence of positive solution for problem (1).

Theorem 8. *Let be a constant. Then for each
**
problem (1) has at least one positive solution.*

*Proof. *First, for any , from (20) we have

On the one hand, by the definition of , there exists such that, for any , we have
Choose . For , we have

On the other hand, by the definition of , there exists such that, for any , we have
Take . For , we have
According to (23), (25), and Lemma 4, has at least one fixed point with , which is a positive solution of (1).

*Remark 9. *If and , then we can get
Theorem 8 implies that, for , problem (1) has at least one positive solution.

Theorem 10. *Let be a constant. Then for each
**
problem (1) has at least one positive solution.*

*Proof. *First, it follows from (27) that, for any ,

By the definition of , there exists such that, for any , we have
Choose . For , we have . Similar to the proof in Theorem 8, it holds from (28) and (29) that

Note . There exists , such that
We consider the problem on two cases. (I) Suppose is bounded. There exists , such that , . Choose . Let . For , we have

(II) Suppose is unbounded. There exists such that

Let . For , we have
Combining (I) and (II), take ; here, . Then for , we have

Hence, (30) and (42) together with Lemma 4 imply that has at least one fixed point with , which is a positive solution of (1).

Theorem 11. *Assume and . Problem (1) has no positive solution provided
**
where is a constant defined in (38).*

*Proof. *Since and , together with the definitions of and , there exist positive constants , , , and satisfying such that
Take

It follows that for any . Suppose that is a positive solution of (1). That is,
In sequence,
which is a contradiction. Hence, (1) has no positive solution.

Theorem 12. *Assume and . Problem (1) has no positive solution provided
**
where is a constant defined in (43).*

*Proof. *Since and , together with the definitions of and , there exist positive constants , , , and satisfying such that
Take
It follows that for any . Suppose that is a positive solution of (1). That is,
In sequence,
which is a contradiction. Hence, (1) has no positive solution.

*Example 13. *Consider the fractional differential equation
In this example, take
Obviously, we have

Since and , through a computation, we can get

Choose ; we have
Theorem 8 implies that, for , , the problem (46) has at least one positive solution.

*Remark 14. *In particular, if we take in Example 13, then and . Remark 9 implies that problem (46) has at least one positive solution for .

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the NNSF of China (no. 61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 2012021002-3).

#### References

- I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet - V. Lakshmikantham, S. Leela, and J. V. Devi,
*Theory of Fractional Dynamic Systems*, Cambridge Scientific Publishers, Cambridge, UK, 2009. - J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds.,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, Dordrecht, The Netherlands, 2007. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “On ${L}^{p}$-solutions for a class of sequential fractional differential equations,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 2074–2081, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 389, no. 1, pp. 403–411, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Wang, “Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments,”
*Journal of Computational and Applied Mathematics*, vol. 236, no. 9, pp. 2425–2430, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 25, no. 6, pp. 1019–1024, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Wang, D. Baleanu, and L. Zhang, “Monotone iterative method for a class of nonlinear fractional differential equations,”
*Fractional Calculus and Applied Analysis*, vol. 15, no. 2, pp. 244–252, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Wang, A. Cabada, and L. Zhang, “Integral boundary value problem for nonlinear differential equations 3 of fractional order on an unbounded domain,”
*Journal of Integral Equations and Applications*. In press. - G. Wang, S. Liu, and L. Zhang, “Neutral fractional integro-differential equation with nonlinear term depending on lower order derivative,”
*Journal of Computational and Applied Mathematics*, vol. 260, pp. 167–172, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - S. Liu, G. Wang, and L. Zhang, “Existence results for a coupled system of nonlinear neutral fractional differential equations,”
*Applied Mathematics Letters*, vol. 26, pp. 1120–1124, 2013. View at Publisher · View at Google Scholar - L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, M. Al-Yami, and W. Shammakh, “Nonlocal integrodifferential boundary value problem for nonlinear fractional differential equations on an unbounded domain,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 813903, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - T. Jankowski, “Boundary problems for fractional differential equations,”
*Applied Mathematics Letters*, vol. 28, pp. 14–19, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3046–3052, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. O'Regan and S. Staněk, “Fractional boundary value problems with singularities in space variables,”
*Nonlinear Dynamics*, vol. 71, no. 4, pp. 641–652, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Liu, B. Ahmad, and R. P. Agarwal, “Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half-line,”
*Advances in Difference Equations*, vol. 2013, article 46, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - L. Zhang, B. Ahmad, G. Wang, and R. P. Agarwal, “Nonlinear fractional integro-differential equations on unbounded domains in a Banach space,”
*Journal of Computational and Applied Mathematics*, vol. 249, pp. 51–56, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Benchohra, A. Cabada, and D. Seba, “An existence result for nonlinear fractional differential equations on Banach spaces,”
*Boundary Value Problems*, vol. 2009, Article ID 628916, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,”
*Electronic Journal of Differential Equations*, vol. 2006, pp. 1–12, 2006. View at Zentralblatt MATH · View at MathSciNet - B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 2, pp. 599–606, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Feng, X. Zhang, and W. Ge, “New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions,”
*Boundary Value Problems*, vol. 2011, Article ID 720702, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. A. H. Salem, “Fractional order boundary value problem with integral boundary conditions involving Pettis integral,”
*Acta Mathematica Scientia B*, vol. 31, no. 2, pp. 661–672, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 3249–3256, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1251–1268, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 10, pp. 4676–4688, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Jiang, “Eigenvalue interval for multi-point boundary value problems of fractional differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4570–4575, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - G. Wang, S. K. Ntouyas, and L. Zhang, “Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument,”
*Advances in Difference Equations*, vol. 2011, article 2, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Bai, “Eigenvalue intervals for a class of fractional boundary value problem,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3253–3257, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,”
*Applied Mathematics and Computation*, vol. 218, no. 17, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sun, Y. Zhao, Z. Han, and J. Liu, “Eigenvalue problem for a class of nonlinear fractional differential equations,”
*Annals of Functional Analysis*, vol. 4, no. 1, pp. 25–39, 2013. View at MathSciNet - D. J. Guo and V. Lakshmikantham,
*Nonlinear Problems in Abstract Cones*, vol. 5, Academic Press, Boston, Mass, USA, 1988. View at MathSciNet