Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 585639, 9 pages

http://dx.doi.org/10.1155/2013/585639

Research Article

Positive Solutions of a Two-Point Boundary Value Problem for Singular Fractional Differential Equations in Banach Space

Bo Liu and Yansheng Liu

Department of Mathematics, Shandong Normal University, Jinan 250014, China

Received 29 May 2013; Accepted 11 July 2013

Academic Editor: William P. Ziemer

Copyright © 2013 Bo Liu and Yansheng Liu. 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.

This paper investigates the existence of positive solutions to a two-point boundary value problem (BVP) for singular fractional differential equations in Banach space and presents a number of new results. First, by constructing a novel cone and using the fixed point index theory, a sufficient condition is established for the existence of at least two positive solutions to the approximate problem of the considered singular BVP. Second, using Ascoli-Arzela theorem, a sufficient condition is obtained for the existence of at least two positive solutions to the considered singular BVP from the convergent subsequence of the approximate problem. Finally, an illustrative example is given to support the obtained new results.

1. Introduction

Fractional differential equations have been widely investigated recently due to its wide applications [1–3] in biology, physics, medicine, control theory, and so forth. As a matter of fact, fractional derivatives provide a more excellent tool for the description of memory and hereditary properties of various materials and processes than integer derivatives. As an important issue for the theory of fractional differential equations, the existence of positive solutions to kinds of boundary value problems (BVPs) has attracted many scholars’ attention, and lots of excellent results have been obtained [4–11] by means of fixed point theorems, upper and lower solutions technique, and so forth.

It is noted that as a special class of fractional differential equations, the singular fractional differential equations with kinds of boundary values have been studied in a series of recent works [7, 12, 13]. In [7], Jiang et al. studied a singular nonlinear semipositone fractional differential system with coupled boundary conditions and presented some sufficient conditions for the existence of a positive solution by using the fixed point theory in cone and constructing some available integral operators together with approximating technique. Zhang et al. [13] considered a class of two-point BVP for singular fractional differential equations with a negatively perturbed term and established some results on the multiplicity of positive solutions by using the approximating technique. In [12], Agarwal et al. investigated the existence of positive solutions for a two-point singular fractional boundary value problem and proposed some existence criteria by using sequential techniques. It should be pointed out that the nonlinearities of [7, 13] are singular at , while the nonlinearity of [12] is singular at . To our best knowledge, there are fewer results on two-point BVPs for singular fractional differential equations with the nonlinearity being singular at both and . Motivated by this, we consider the following two-point BVP of singular fractional differential equations in Banach space:

where is a real number, , is continuous, denotes the null element in the Banach space with the norm , is the standard Riemann-Liouville fractional derivative, and may be singular at and . Firstly, we establish a sufficient condition for the existence of at least two positive solutions to the approximate problem of BVP (1) by constructing a novel cone and using the fixed point index theory. Secondly, using Ascoli-Arzela theorem, we obtained a sufficient condition for the existence of at least two positive solutions to BVP (1) from the convergent subsequence of the approximate problem. Finally, we give an illustrative example to support the obtained new results.

The main features of this paper are as follows. (i) A class of fractional-order two-point boundary value problems with the nonlinearity being singular at both and is firstly studied in this paper, which generalizes the existing singular fractional differential equations [7, 12, 13] and has wider applications. (ii) A sequential-based method is proposed for singular fractional differential equations with the nonlinearity being singular at both and , which enriches the theory of fractional differential equations.

The rest of this paper is organized as follows. Section 2 contains the definition of Riemann-Liouville fractional derivative and some notation. The main results are presented in Section 3, which is followed by an illustrative example in Section 4.

2. Preliminaries

We first recall some well-known results about Riemann-Liouville derivative. For details, please refer to [14, 15] and the references therein.

Definition 1. The Riemann-Liouville fractional integral of order of a function is given by

provided the right side is pointwise defined on .

Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is given by

where is the smallest integer greater than or equal to , provided that the right side is pointwise defined on .

One can easily obtain the following properties from the definition of Riemann-Liouville derivative.

Proposition 3 (see [15]). Let ; if one assumes that , then, the fractional differential equation has , as unique solutions, where is the smallest integer greater than or equal to .

Proposition 4 (see [15]). Assume that with a fractional derivative of order that belongs to . Then,

for some , , where is the smallest integer greater than or equal to .

The following lemmas will be used in the proof of the main results.

Lemma 5 (see [16]). If is bounded and equicontinuous, then

where and denote the Kuratowski noncompactness measure of bounded sets in and , respectively, , and is the Banach space of all continuous functions with the norm .

Lemma 6 (see [16]). Let be a cone in , and let . Let be a strict set contraction. Assume that there exist a and such that for any and . Then =0.

Lemma 7 (see [16]). Let , and there exists a such that for all ; then , .

Lemma 8 (Ascoli-Arzela theorem [16]). is relative compact if and only if is equicontinuous, and for any , is a relatively compact set in .

Lemma 9 (see [17]). Let be bounded open set. is condensing. If there exists , such that , where , then .

3. Main Results

Let . Then, one can see that is a normal solid cone of . Define and . Let denote the dual cone of . We consider BVP (1) in . is called a solution to BVP (1), if satisfies (1). In addition, we call a positive solution to BVP (1), if .

For convenience, let us list the following assumptions.

(H1) and

where , , and .(H2) for all , and there exists such that , where

(H3) is uniformly continuous with respect to on , where and .(H4) There exists a constant , such that

where , and .(H5) There exists with and , such that

uniformly in , where .(H6) There exist with and , such that

uniformly on .

According to [18], BVP (1) is equivalent to

where

Consider the operator associated with the singular boundary value problem (1), which is defined by

Set . As in [19], we have

Define

Then, one can see that is a fixed point of the operator if and only if is a solution of BVP (1).

Choose with . We consider the following approximate problem of (15):

Denote

It is easy to check that is a cone in . Let . By (H1) and (H2), we can conclude that

which implies that the operator is well defined. In addition, from the definition of , we can prove that

By (16), we have . Thus .

In the following, we first investigate the existence of two positive solutions to the approximate problem (16) and then establish the existence criterion for the existence of two positive solutions to BVP (1) by using the sequential-based technique and Ascoli-Arzela theorem.

Lemma 10. Let (H1) and (H2) be satisfied. Then for any , the operator is a continuous bounded operator from into .

Proof. By (H1), we can easily see that is bounded from to .

Next, we prove that is continuous.

Let as . From (H1), we have

Hence

which together with the dominated convergence theorem imply that

We now show that

In fact, if (23) is not true, then there exist a positive number and a subsequence such that

Since is relatively compact, there is a subsequence of which converges to some . Without loss of generality, we assume that itself converges to , that is,

By virtue of (22) and (25), we have , which contradicts with (24). Hence (23) holds, and the continuity of is proved.

Lemma 11. Let (H1)–(H4) be satisfied. Then for any , the operator is a strict set contraction from to .

Proof. By virtue of Lemma 10, we know that is bounded and equicontinuous on . Thus, from Lemma 5, one can see that

where .

Set

Based on (H1) and (H2), we have

where . Hence

where denotes the Hausdorff metrics, which implies that

Since

by Lemmas 5 and 7, we have

where . By (H4), we can obtain

Consequently, the operator is a strict set contraction from into .

Theorem 12. Let (H1)–(H6) hold. Then, there exists such that the operator has two fixed points in and , respectively, for arbitrary sufficiently large positive integer , where is given in (H2).

Proof. For any given , we have . Since , there exists a such that

Then by (H5), there exists such that

holds for and . Letting , we prove that for any , as . In fact, suppose that this is false. Then there exist , such that . It is obvious that

Thus

This implies that

which is in contradiction with

Combining with Lemma 6, we obtain that

Now, we show that . By Lemma 9, we need only to prove that for and . In fact, if there exist and some such that , then

Therefore, by (H1) and (H2), we have

which is a contradiction. Consequently,

In the following, we choose

By (H6), there exists such that holds for .

Let

We now claim that for and . As a matter of fact, if this is not true, then there exists such that , that is, , which implies that

which is a contradiction. Therefore, by Lemma 6, we can obtain that

This together with (40), (43), and (47) implies that

Thus, there exist and such that and . This completes the proof.

Theorem 13. Let (H1)–(H6) be satisfied. Then BVP (1) has at least two positive solutions in .

Proof. From Theorem 12, there exists an integer , such that the operator , has two fixed points in and . Denote . Obviously is uniformly bounded. Next, we show that is equicontinuous.

Firstly, we prove that

On one hand, by the absolute continuity of integration [19], for any , there exists such that .

On the other hand, from (H2), we set

and . For , we have

which implies that (49) holds. Similarly, one can prove that (50) holds.

Next, we show that is equicontinuous for , .

From (H3), we define

By the absolute continuity of integration, for and the previous , there exists such that

Since is bounded for , there exists , such that

For , one can see that

Similarly, for , we have

Since

is bounded on , and as , there exists such that

We choose . For , we have

This together with (49) and (50) implies that is equicontinuous for .

Now, we show that is relatively compact. By Lemma 7, we have

which together with Lemma 5 and (H4) implies that

Thus, . It follows from Lemma 8 that there is a convergent subsequence of . Without loss of generality, we assume that itself converges to some . Then, the dominated convergence theorem and (16) imply that

Thus, the singular BVP (1) has a positive solution . Similarly, has a convergent subsequence, which converges to . Then, is also a positive solution to BVP (1).

Finally, we show . We only need to prove that the operator has no fixed point in .

In fact, if it is not true, then we assume that is a fixed point of the operator in . Then

with .

By (H1) and (H2), one can see that

which is a contradiction. The proof of this theorem is completed.

Remark 14. It is noted that the singularity of at is overcome by (H2), while the singularity at is handled by solving the approximate problem and using the sequential-based technique.

4. Example

Consider the following BVP:

where . Then, BVP (66) has at least two positive solutions.

Proof. We consider the problem (66) in -dimensional Euclidean space with the norm . Then, BVP (66) has the form of (1) with

It is easy to see that is singular at and . Set and . One can easily see that (H1) holds.

We choose and . Considering that

we can easily check that (H2)–(H6) hold. From Theorem 13, BVP (66) has at least two positive solutions and which satisfy .

Acknowledgments

The authors wish to thank referees and Dr. Haitao Li for their valuable suggestions. The project was supported by the National Natural Science Foundation of China (11171192), Graduate Educational Innovation Foundation of Shandong Province (SDYY1005), and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF025).

References

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.

A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.

R. P. Agarwal and D. O'Regan, “Singular boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 27, pp. 645–656, 1996.

R. P. Agarwal and D. O'Regan, “Nonlinear superlinear singular and nonsingular second order boundary value problems,” Journal of Differential Equations, vol. 143, no. 1, pp. 60–95, 1998.

H. H. Alsulami, S. K. Ntouyas, and B. Ahmad, “Existence results for a Riemann-Liouvilletype fractional multivalued problem with integral boundary conditions,” Journal of Function Spaces and Applications, vol. 2013, Article ID 676045, 7 pages, 2013.

J. Jiang, L. Liu, and Y. Wu, “Positive solutions to singular fractional differential system with coupled boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 11, pp. 3061–3074, 2013.

H. Li, X. Kong, and C. Yu, “Existence of three non-negative solutions for a three-point boundary-value problem of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 88, pp. 12–1, 2012.

A. Shi and S. Zhang, “Upper and lower solutions method and a fractional differential equation boundary value problem,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 30, pp. 1–13, 2009.

Z. L. Wei, “Positive solutions of singular boundary value problems for negative exponent Emden-Fowler equations,” Acta Mathematica Sinica. Chinese Series, vol. 41, no. 3, pp. 655–662, 1998 (Chinese).

S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006.

R. P. Agarwal, D. O'Regan, and S. Staněk, “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.

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. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gordon and Breach, Yverdon, Switzerland, 1993.

D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

D. Guo, Nonlinera Function Analysis, Shandong Science and Technology Publishing House, Jinan, China, 1985.

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.

D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 2, pp. 710–719, 2010.