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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 585639, 9 pages
Positive Solutions of a Two-Point Boundary Value Problem for Singular Fractional Differential Equations in Banach Space
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.
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.
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 , 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.  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 , 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  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.
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 ). 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 ). 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 ). 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 ). 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 ). Let , and there exists a such that for all ; then , .
Lemma 8 (Ascoli-Arzela theorem ). is relative compact if and only if is equicontinuous, and for any , is a relatively compact set in .
Lemma 9 (see ). 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 .
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
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
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 , 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.
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 .
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).
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