International Journal of Differential Equations

International Journal of Differential Equations / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 632434 | 7 pages | https://doi.org/10.1155/2014/632434

Existence of Solutions for Two-Point Boundary Value Problem of Fractional Differential Equations at Resonance

Academic Editor: Patricia J. Y. Wong
Received19 Feb 2014
Accepted21 Jul 2014
Published05 Aug 2014

Abstract

We establish the existence results for two-point boundary value problem of fractional differential equations at resonance by means of the coincidence degree theory. Furthermore, a result on the uniqueness of solution is obtained. We give an example to demonstrate our results.

1. Introduction

Fractional differential equations have been studied extensively. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications such as physics, chemistry, phenomena arising in engineering, economy, and science; see, for example, [15].

Recently, more and more authors have paid their attentions to the boundary value problems of fractional differential equations; see [621]. Moreover, there have been many works related to the existence of solutions for boundary value problems at resonance; see [1221]. It is considerable that there are many papers that have dealt with the solutions of multipoint boundary value problems of fractional differential equations at resonance (see, e.g., [12, 16]).

In [12], Bai and Zhang considered a three-point boundary value problem of fractional differential equations with nonlinear growth given by where , , , , is Riemann-Liouville fractional derivative, and are given functions.

In [13], Hu et al. have studied a two-point boundary value problem for fractional differential equation at resonance where , is Caputo fractional derivative, and satisfies Carathéodory conditions.

As far as we know, there are few works on the existence of two-point boundary value problems of the fractional differential equations at resonance. Motivated by the works above, we discuss the existence and uniqueness of solutions for the following nonlinear fractional differential equation: where , , is Riemann-Liouville fractional derivative, and is continuous function.

More precisely, we use the coincidence degree theorem due to Mawhin [22]. The rest of this paper is organized as follows. In Section 2, we give some necessary notations, definitions, and lemmas. In Section 3, we study the existence of solutions of (3) by the coincidence degree theory. Finally, an example is given to illustrate our results in Section 4.

The two-point boundary value problem (3) happens to be at resonance in the sense that the associated linear homogeneous boundary value problem has as a nontrivial solution.

2. Preliminaries

In this section, we present the necessary definitions and lemmas from fractional calculus theory. These definitions and properties can be found in the literature. For more details see [13].

Definition 1 (see [1]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 2 (see [1]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined on .

Lemma 3 (see [1]). Let , ; then where , .

Lemma 4 (see [1]). If , and . If the fractional derivatives and exist, then

Lemma 5 (see [1]). The relation is valid in following cases , , and .

Now let us recall some notations about the coincidence degree continuation theorem.

Let , be real Banach spaces, let be a Fredholm map of index zero, and let , be continuous projectors such that , , and , . It follows that is invertible. We denote the inverse of this map by . If is an open bounded subset of , the map will be called -compact on if is bounded and is compact.

Theorem 6. Let be a Fredholm operator of index zero and be -compact on . Suppose that the following conditions are satisfied:(1) for each ;(2) for each ;(3) , where is a continuous projection as above with and is any isomorphism.Then the equation has at least one solution in .

3. Main Results

In this section, we will prove the existence results for (3).

We use the Banach space with the norm . For , , we define a linear space By means of the functional analysis theory, we can prove that is a Banach space with the norm .

Define to be the linear operator from to with and We define by Then the problem (3) can be written by .

Lemma 7. The mapping is a Fredholm operator of index zero.

Proof. It is clear that Let , so there exists a function which satisfies . By (11) and Lemma 3, we have By , we can obtain . Hence Then, we have Taking into account , we obtain On the other hand, suppose satisfy . Let , we can easily prove .
Thus, we conclude that Consider the linear operators defined by Take ; then We can see .
For in the type , obviously, and . That is to say, . If , we have ; then . As a result , and we get .
Note that . Then is a Fredholm mapping of index zero.

We can define the operators , where For , So we have .

Note that Since , it is easy to say that and . So we have . If , then . We can derive from . Then For , where .

We define by .

For , we have For , we have . And for , the coefficients in the expressions are all equal to zero. Thus, we obtain This shows that . Again for each , where .

Lemma 8. Assume is an open bounded subset such that ; then map is -compact on

Proof. By the continuity of , we can get that and are bounded. So, in view of the Arzela-Ascoli theorem, we need only to prove that is equicontinuous. From the continuity of , there exists a constant , such that , for all , .
For , , we have Furthermore, we have where . Since and are uniformly continuous on , we can get that is compact. The proof is completed.

To obtain our main results, we need the following conditions.(H1)There exist functions , , such that for all , , (H2)There exists a constant such that for every , if for all , then (H3)There exists a constant such that, for each , satisfying . We have either at least one of the following: or (H4) , where , .

Lemma 9. is bounded.

Proof. For , and . By (12), ; that is, By the integral mean value theorem, there exits a constant such that Form (H2), we can get .
Again for , and . From (29), we have Now by Lemma 4 That is, From (25) and (38), we have Furthermore, it follows from (40) and (H1) that By the definition and (H4), it is easy to see that and are bounded. So, is bounded.

Lemma 10. is bounded.

Proof. Let , so we have , . For , By the integral mean value theorem, there exits a constant such that From (H2), it follows that . Hence, is bounded.

Lemma 11. is bounded.

Proof. Let , so we have , . If , then . If , we have .
If and , then It follows that Then we get which, together with (H3), implies . Here, is bounded.

Remark 12. If the other parts of (H3) hold, then the set is bounded.

Theorem 13. Suppose (H1)–(H4) hold; then the problem (3) has at least one solution in .

Proof. Let be a bounded open set of , such that . It follows from Lemma 8, is -compact on . By Lemmas 9, 10, and 11, we get the following:(1) , for every ;(2) for every ;(3)let , where is the identical operator. Via the homotopy property of degree, we obtain that Applying Theorem 6, we conclude that has at least one solution in .

Under the stronger conditions imposed on , we can prove the uniqueness of solutions to the (3) studied above.

Theorem 14. Suppose the conditions (H1) in the theorem are replaced by the following conditions.(H1)′There exist positive constants , , such that, for all , , one has (H1)′′There exist constants , , such that for all , , one has Then, the BVP (3) has a unique solution, provided that

Proof. Let , , and ; then the condition (H1) is satisfied. According to Theorem 13, BVP (3) has at least one solution. Suppose , are two solutions of (3); then Note that , so satisfy the equation According to , we have By the integral mean value theorem, there exists , such that By (H1)′′, we have We can have Thus, we can obtain According to (25), (38), and (58), we have From the definition of and the assumption (51), we have , so that .

4. Example

Let us consider the following boundary value problems: Corresponding to the problem (3), we have that and Moreover, We can get that the condition (H1) holds; that is, , , and . Taking , , we can calculate that (H2)–(H4) hold.

Hence, by Theorem 13, we obtain that (60) has at least one solution.

Conflict of Interests

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

Authors’ Contribution

All authors typed, read, and approved the final paper.

Acknowledgments

Research was supported by the National Natural Science Foundation of China (11371364) and 2013 Science and Technology Research Project of Beijing Municipal Education Commission (KM201310016001).

References

  1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. View at: MathSciNet
  2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fraction al Differential Equations, John Wiley & Sons, 1993. View at: MathSciNet
  3. I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet
  4. W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995. View at: Publisher Site | Google Scholar
  5. F. Mainardi, Ed., Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997. View at: Publisher Site | MathSciNet
  6. R. P. Agarwal, D. O'Regan, and S. Staně, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57–68, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  7. Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  8. X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. B. Ahmad and A. Alsaedi, “Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations,” Fixed Point Theory and Applications, vol. 2010, Article ID 364560, pp. 1–17, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  11. G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  12. Z. Bai and Y. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1719–1725, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. Z. Hu, W. Liu, and T. Chen, “Two-point boundary value problems for fractional differential equations at resonance,” Bulletin of the Malaysian Mathematical Sciences Society 2, vol. 36, no. 3, pp. 747–755, 2013. View at: Google Scholar | MathSciNet
  14. Z. Hu, W. Liu, and T. Chen, “Existence of solutions for a coupled system of fractional differential equations at resonance,” Boundary Value Problems, vol. 2012, article 98, 13 pages, 2012. View at: Google Scholar | MathSciNet
  15. N. Xu, W. Liu, and L. Xiao, “The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance,” Boundary Value Problems, vol. 2012, article 65, 10 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  16. W. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1987–1994, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  17. N. Kosmatov, “Multi-point boundary value problems on an unbounded domain at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2158–2171, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  18. N. Kosmatov, “A boundary value problem of fractional order at resonance,” Electronic Journal of Differential Equations, vol. 135, pp. 1–10, 2010. View at: Google Scholar | MathSciNet
  19. Z. Bai, “Solvability for a class of fractional m-point boundary value problem at resonance,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1292–1302, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  20. Y. Zhang, Z. Bai, and T. Feng, “Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance,” Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1032–1047, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  21. Y. Zhang and Z. Bai, “Existence of solutions for nonlinear fractional three-point boundary value problems at resonance,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 417–440, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. J. Mawhin, “Topological degree and boundary value problems for nonlinear differential equations,” in Topological Methods for Ordinary Differential Equations, vol. 1537 of Lecture Notes in Mathematics, pp. 74–142, Springer, Berlin, Germany, 1993. View at: Publisher Site | Google Scholar

Copyright © 2014 Lei Hu 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.

1033 Views | 510 Downloads | 1 Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19.