Abstract and Applied Analysis

Volume 2014 (2014), Article ID 314083, 8 pages

http://dx.doi.org/10.1155/2014/314083

Research Article

The Existence of Solutions for Four-Point Coupled Boundary Value Problems of Fractional Differential Equations at Resonance

Yumei Zou,¹ Lishan Liu,² and Yujun Cui³

¹Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China

²School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

³Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

Received 18 December 2013; Accepted 14 February 2014; Published 23 March 2014

Academic Editor: Xinguang Zhang

Copyright © 2014 Yumei Zou 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.

A four-point coupled boundary value problem of fractional differential equations is studied. Based on Mawhin’s coincidence degree theory, some existence theorems are obtained in the case of resonance.

1. Introduction

In this paper, we are concerned with the following four-point coupled boundary value problem for nonlinear fractional differential equation. Consider

where , and are the standard Riemann-Liouville differentiation and integration, , , , and

The subject of fractional calculus has gained considerable popularity and importance because of its intensive development of the theory of fractional calculus itself and its varied applications in many fields of science and engineering. As a result, the subject of fractional differential equations has attracted much attention; see [1–11] for a good overview.

At the same time, we notice that coupled boundary value problems, which arise in the study of reaction-diffusion equations and Sturm-Liouville problems, have wide applications in various fields of sciences and engineering, for example, the heat equation [12–14] and mathematical biology [15, 16]. In [17], Asif and Khan used the Guo-Krasnosel’skii fixed-point theorem to show the existence of positive solutions to the nonlinear differential system with coupled four-point boundary value conditions

where , , and are two given continuous functions.

In [18], the authors considered the existence of positive solutions of four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation

where is a parameter, satisfy , , is a real number and , and is Riemann-Liouville’s fractional derivative.

Recently, Cui and Sun [19] showed the existence of positive solutions of singular superlinear coupled integral boundary value problems for differential systems

where are bounded linear functionals on given by

with being functions of bounded variation with positive measures.

A key assumption in the above papers is that the case studied is not at resonance; that is, the associated fractional (or ordinary) linear differential operators are invertible. In this paper, instead, we are interested in the resonance case due to the critical condition (2). Boundary value problems at resonance have been studied by several authors including the most recent works [20–31]. In this paper, we establish new results on the existence of a solution for the couple boundary value problems at resonance. Our method is based on the coincidence degree theorem of Mawhin [32, 33].

Now, we briefly recall some notations and an abstract existence result.

Let be real Banach spaces and let be a Fredholm operator of index zero. and are continuous projectors such that

It follows that is invertible. We denote the inverse of the mapping by (generalized inverse operator of ). If is an open bounded subset of such that , the mapping will be called -compact on if is bounded and is compact.

Theorem 1 (see [32, 33]). Let be a Fredholm operator of index zero and let be -compact on . Assume that the following conditions are satisfied.

(i) for every .(ii) for every .(iii), where is a projector as above with .

Then the equation has at least one solution in .

For convenience, let us set the following notations:

2. Preliminaries and Lemmas

In this section, first we provide recall of some basic definitions and lemmas of the fractional calculus, which will be used in this paper. For more details, we refer to books [1, 2, 4].

Definition 2 (see [1, 4]). The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side is pointwise defined on .

Definition 3 (see [1, 4]). 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 .

We use the classical Banach space with the norm and with the norm . We also use the space defined by

and the Banach space ()

with the norm .

Lemma 4 (see [1]). Let , . Assume that with a fractional integration of order that belongs to . Then the equality

holds almost everywhere on .

In the following lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that for .

Lemma 5 (see [1]). Assume that ; then

(i)let . If and exist, then

(ii)if , then

 satisfies at any point on for and ;(iii)let . Then holds on ;(iv)note that, for , , one has

Remark 6. If and satisfies and , then . In fact, with Lemma 4, one has

Combined with , there is . So

Lemma 7 (see [34]). is a sequentially compact set if and only if is uniformly bounded and equicontinuous. Here to be uniformly bounded means that there exists such that for every

and to be equicontinuous means that for all, and for all , , , and , the following holds:

We also use the following two Banach spaces with the norm

and with the norm

Let the linear operator with

be defined by

where and are defined by

Let the nonlinear operator be defined by

where are defined by

Then four-point coupled boundary value problems (1) can be written as

Lemma 8. Let be the linear operator defined as above. If (2) holds, then

Proof. Let and let . Then by Lemma 5, we have , , , and . So

For every , if , then

Considering that , and , we can obtain that and . It yields the following:

Let ; then there is such that ; that is, , and , . By Lemma 4,

and by the couple boundary conditions, we have

It yields the following:

On the other hand, suppose that satisfy (36). Let and , and then , , and

Therefore, (30) holds.

Lemma 9. If (2) holds, then is a Fredholm operator of index zero and . Furthermore, the linear operator can be defined by

Also

Proof. Define operator as follows:

where is defined by

It is easy to see that ; that is, is a continuous linear projector. Furthermore, . For , set . Then and . It follows from and that . So we have

Now, , and so is a Fredholm operator of index 0.

Let be continuous linear operator defined by

Obviously, is a linear projector and

It is easy to know that .

Define by

Since

then

In fact, if , then

By Lemma 4, for ,

( since and since ). Hence,

The proof is complete.

3. Main Results

In this section, we will use Theorem 1 to prove the existence of solutions to BVP (1). To obtain our main theorem, we use the following assumptions.

(H1)There exist functions such that for all ,

(H2)There exists a constant such that, for , if for all , then or .(H3)There exists a constant such that either, for each ,

 or, for each ,

Theorem 10. Suppose (2) and ()–() hold. Then (1) has at least one solution in Y, provided that

Proof. Set

Take . Since , so and ; hence,

Thus, from (H2), there exist such that

Noticing that

so

Thus

For all , . Considering Lemma 9, we get . Together with (39), we have

From (60) and (61), we have

From (62), we discuss various cases.

Case  1 (). From (H1), we have

which yield

Thus, is bounded.

Case  2  (). From (H1), we have

which yield

Thus, is bounded. Let

For and , so , , . Noticing that , then we get , and thus and . From (H2), we get , and thus is bounded.

We define the isomorphism by

If the first part of (H3) is satisfied, then let

For ,

If , then . Otherwise, if , in view of (H3) and , one has

which contradict . Thus is bounded.

If the second part of (H3) holds, then define the set

and here is as above. Similar to the above argument, we can show that is bounded too.

In the following, we will prove that all conditions of Theorem 1 are satisfied. Let be a bounded open subset of such that . By standard arguments, we can prove that is compact, and thus is -compact on . Then by the above argument we have

(i), for every ,(ii) for .

Finally, we will prove that (iii) of Theorem 1 is satisfied. Let . According to the above argument, we know that

Thus, by the homotopy property of degree,

Then by Theorem 1, has at least one solution in so that BVP (1) has a solution in . The proof is complete.

Conflict of Interests

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

Acknowledgments

The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the presentation of the paper. The project was supported by the National Natural Science Foundation of China (11371221, 61304074), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province (201303074), and Foundation of SDUST.

References

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 Science B.V., Amsterdam, The Netherlands, 2006.

V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.

J. J. Nieto and J. Pimentel, “Positive solutions of a fractional thermostat model,” Boundary Value Problems, vol. 2013, article 5, 2013.

I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.

W. Wu and X. Zhou, “Eigenvalue of fractional differential equations with P-Laplacian operator,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 137890, 8 pages, 2013.

X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013.

X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013.

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.

X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012.

X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.

X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.

K. Deng, “Blow-up rates for parabolic systems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 47, no. 1, pp. 132–143, 1996.

K. Deng, “Global existence and blow-up for a system of heat equations with non-linear boundary conditions,” Mathematical Methods in the Applied Sciences, vol. 18, no. 4, pp. 307–315, 1995.

L. Zhigui and X. Chunhong, “The blow-up rate for a system of heat equations with nonlinear boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 34, no. 5, pp. 767–778, 1998.

D. G. Aronson, “A comparison method for stability analysis of nonlinear parabolic problems,” SIAM Review, vol. 20, no. 2, pp. 245–264, 1978.

M. Pedersen and Z. Lin, “Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition,” Applied Mathematics Letters, vol. 14, no. 2, pp. 171–176, 2001.

N. A. Asif and R. A. Khan, “Positive solutions to singular system with four-point coupled boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 848–861, 2012.

C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 13, p. 17, 2012.

Y. Cui and J. Sun, “On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system,” Electronic Journal of Qualitative Theory of Differential Equations, no. 41, pp. 1–13, 2012.

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.

Y. Cui, “Solvability of second-order boundary-value problems at resonance involving integral conditions,” Electronic Journal of Differential Equations, no. 45, pp. 1–9, 2012.

Z. Hu and W. Liu, “Solvability for fractional order boundary value problems at resonance,” Boundary Value Problems, vol. 2011, article 20, 2011.

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.

N. Kosmatov, “Multi-point boundary value problems on time scales at resonance,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 253–266, 2006.

G. Wang, W. Liu, S. Zhu, and T. Zheng, “Existence results for a coupled system of nonlinear fractional 2m-point boundary value problems at resonance,” Advances in Difference Equations, vol. 2011, article 80, 2011.

J. R. L. Webb, “Remarks on nonlocal boundary value problems at resonance,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 497–500, 2010.

J. R. L. Webb and M. Zima, “Multiple positive solutions of resonant and non-resonant non-local fourth-order boundary value problems,” Glasgow Mathematical Journal, vol. 54, no. 1, pp. 225–240, 2012.

X. Zhang, M. Feng, and W. Ge, “Existence result of second-order differential equations with integral boundary conditions at resonance,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 311–319, 2009.

X. Zhang, C. Zhu, and Z. Wu, “Solvability for a coupled system of fractional differential equations with impulses at resonance,” Boundary Value Problems, vol. 2013, artice 80, 2013.

Z. Zhao and J. Liang, “Existence of solutions to functional boundary value problem of second-order nonlinear differential equation,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 614–634, 2011.

Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,” Advances in Difference Equations, vol. 2013, article 25, 2013.

J. Mawhin, “Topological degree and boundary value problems for nonlinear differential equations,” in Topological Methods for Ordinary Differential Equations, P. M. Fitzpertrick, M. Martelli, J. Mawhin, and R. Nussbaum, Eds., vol. 1537 of Lecture Notes in Mathematics, pp. 74–142, Springer, Berlin, Germany, 1993.

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979.

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.