Journal of Applied MathematicsVolume 2011 (2011), Article ID 352341, 22 pagesdoi:10.1155/2011/352341

Research Article

Positive Solution of Singular Fractional Differential Equation in Banach Space

Jianxin Cao1,2 and Haibo Chen¹

¹Department of Mathematics, Central South University, Changsha, Hunan 410075, China²Faculty of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China

Received 17 June 2011; Accepted 9 September 2011

Academic Editor: J. Biazar

Copyright © 2011 Jianxin Cao and Haibo Chen. Jianxin Cao and . 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.

We investigated a singular multipoint boundary value problem for fractional differential equation in Banach space. The nonlinear term is positive and singular at and (or) . Employing regularization, sequential techniques, and diagonalization methods, we obtained some new existence results of positive solution.

1. Introduction

Recently, fractional differential equations have been investigated extensively. The motivation for those works rises from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, chemistry, aerodynamics, and electrodynamics of the complex medium. For examples and details, see [1–5] and the references therein.

Prompted by the application of multipoint boundary value problem (BVP for short) to applied mathematics and physics, these problems have provoked a great deal of attention by many authors. Here, for fractional differential equations, we refer the reader to [6–12]. Rehman and Khan [7] studied the problem where , , , , with , and represents the standard Riemann-Liouville fractional derivative. The existence and uniqueness of solutions were obtained, by means of Schauder fixed-point theorem and Banach contraction principle. Importantly, they gave the Green function of the multipoint BVP (1.1). But they have not proved the positivity of Green function, so the existence of positive solution is unobtainable. However, only positive solutions are useful for many applications, as some physicists pointed out.

The authors of [13–17] investigated singular problem for fractional differential equations with bounded domain. In particular, Agarwal et al. [13] considered the following Dirichlet problem: where , . satisfies the Carathéodory conditions and is singular at . In order to overcome the singularity, they used regularization and sequential techniques for the existence of a positive solution.

When the domain where the problem is considered is unbounded, there are few papers about BVP for fractional differential equations in literatures. This situation has changed recently. One can find some works, for example, see [18–22].

In [21], the following BVP: was studied. Using the equicontinuity on any compact intervals and the equiconvergence at infinity of a bounded set, the authors proved that the corresponding operator was completely continuous, then the existence of solutions was obtained by the Leray-Schauder nonlinear alternative theorem.

Let be a real Banach space. is a cone in which defines a partial ordering in by if and only if . is said to be normal if there exists a positive constant such that implies , where denotes the zero element of , and the smallest is called the normal constant of (it is clear that ). If and , we write . Let . So, if and only if . For details on cone theory, see [23].

In this paper, we are concerned with the existence of positive solution of a BVP for fractional differential equation with bounded domain or with unbounded domain Here, , , , , are real numbers, and is the standard Riemann-Liouville fractional derivative. And is singular at and and satisfies other conditions which will be specified later. In addition, is the Carathéodory function.

We say that satisfies the Carathéodory conditions on , if(i) is measurable for all ,(ii) is continuous for ,(iii)for each compact set , there is a function such that

No contribution exists, as far as we know, concerning the existence of positive solution of the problems (1.4)-(1.5) and (1.6)-(1.7). In the present paper, we consider, firstly, the case of bounded domain, that is, BVP (1.4)-(1.5), and give some existence results by means of regularization process combined with fixed-point theorem due to Krasnosel'skii. Then we investigate the BVP (1.6)-(1.7). As we know, is noncompact. In order to overcome these difficulties, based on the results of BVP (1.4)-(1.5), we use diagonalization process to establish the existence of positive solutions for BVP (1.6)-(1.7). Let us mention that this method was widely used for integer-order differential equations, see, for instance, [5, 22]. Using diagonalization process, Agarwal et al. [20] have considered a class of boundary value problems involving Riemann-Liouville fractional derivative on the half line. And Arara et al. [19] continued this study by considering a BVP with the Caputo fractional derivative.

The remainder of this paper is organized as follows. In Section 2, we introduced some notations, definitions, and preliminary facts about the fractional calculus, which are used in the next two sections. In Section 3, the case with bounded domain is considered. In Section 4, we discuss the existence of a positive solution for the BVP (1.6)-(1.7). We end this paper with giving an example to demonstrate the application of our results in Section 5.

2. Preliminaries

Now, we introduce the Riemann-Liouville fractional- (arbitrary)-order integral and derivative as follows.

Definition 2.1. The fractional- (arbitrary)-order integral of the function of is defined by

Definition 2.2. The Riemann-Liouville fractional derivative of order for a function given in the interval is defined by provided that the right hand side is point wise defined. Here, and means the integral part of the number , and is the Euler gamma function.

The following properties of the fractional calculus theory are well known, see, for example, [2, 4]:(i) for a.e. , where , ,(ii) if and only if , where are arbitrary constants, , ,(iii), , ,(iv) and for , .

More details on fractional derivatives and their properties can be found in [2, 4].

For the sake of convenience, we introduce the following assumptions:, , and there exists a positive constant such that for all , fulfills the estimate, where , , are nonincreasing, and, for any , while are nondecreasing and for a.e. , and for all , is relatively compact.

Remark 2.3. It follows from (2.4) that under condition , and .

In the sequel, denote the Banach space of functions which are Lebesgue integrable with the norm

We give now some auxiliary lemmas in scalar space, which will take an important role throughout the paper.

Lemma 2.4. Suppose that and that holds, then the unique solution of linear BVP , a.e. with the boundary condition (1.5) is given by where

Proof. The proof is similar to that of [7, Lemma 2.2], so we omit it.

Lemma 2.5. Suppose that holds, then defined as (2.11) has the following properties: (i) is uniformly continuous about in ,(ii) for all and , where (iii) if .

Proof. From (2.3), it is easy to verify (i) and (ii). We now show that (iii) is true. Firstly, if , then (2.11) gives Then, From (2.14) and (2.3), we deduce from the Lagrange mean value theorem that In view of and , one can obtain for that Analogously, if , one has It follows from (2.16) and (2.17) that The proof is complete.

3. Existence Results for BVP (1.4)-(1.5)

In this section, we discuss the uniqueness, existence, and continuous dependence of positive solution for problem (1.4)-(1.5). To this end, we introduce some auxiliary technical lemmas.

Let equipped with the norm , then is a real Banach space (see [24]).

Since the nonlinear term is singular at and , we use the following regularization process. For each , define by the formula where is a given element of and .

Remark 3.1. The function defined by (3.1) satisfies , . And conditions and imply

Remark 3.2. The function defined by (3.1) satisfies .

Define operator by the following:

Lemma 3.3. Suppose that holds, then where

Proof. The proof is similar to that of [7, Lemma 2.2], and we omit it.

Lemma 3.4. Suppose that holds, then defined as (3.7) has the following properties: (i) is uniformly continuous about in ,(ii) for all and , where (iii) if .

Proof. The proof of this Lemma is similar to that of Lemma 2.5. Hence it is omitted.

Lemma 3.5. Suppose that and following condition hold: there exist positive constants such that and , then has a unique fixed point.

Proof. Obviously, defined by the formula (3.1) satisfy also the condition . By (3.5) and (3.6), it is easy to show that , then Banach contraction principle implies that the operator has a unique fixed point, which completes this proof.

The following fixed-point result of cone compression type is due to Krasnosel'skii, which is fundamental to establish another auxiliary existence result (Lemma 3.8).

Lemma 3.6 (see, e.g., [23, 25]). Let be a Banach space, and let be a cone in . Let be bounded open balls of centered at the origin with . Suppose that is a completely continuous operator such that hold, then has a fixed point in .

Lemma 3.7. Let hold, then and is a completely continuous operator.

Proof. Firstly, let , because is positive. It follows from Lemma 2.5 (i) and (ii) that and for . Similarly, from Lemma 3.4 (i) and (ii) we can get that and for . To summarize, .Secondly, we prove that is a continuous operator. Let be a convergent sequence and , then and , where is a positive constant. In view of , we have . Since by (3.2), (3.3), the Lebesgue dominated convergence theorem gives Now, from (3.12), Lemma 2.5(ii), Lemma 3.4(ii) and from the inequalities (cf. (3.5), (3.6)) we have that , which proves that is a continuous operator.Thirdly, let be bounded in and let for all , where is a positive constant. We are in position to prove that is bounded. Keeping in mind , there exists such that then (cf. (3.5)) and (cf. (3.6)) for and all . Therefore, is bounded in .Fourthly, by and (3.5), it is easy to show that is relatively compact.Finally, let . From Lemma 2.5(i) and the functions being uniformly continuous on , for any arbitrary , there exists a positive number , such that when , one has and , then (cf. (3.14)) the inequality holds. Hence the set of functions is equicontinuous on .Therefore, by the Arzelá-Ascoli theorem, is relatively compact in . We have proved that is a completely continuous operator.

Lemma 3.8. Suppose that hold, then the operator has at least a fixed point.

Proof. By Lemma 3.7, is completely continuous. In order to apply Lemma 3.6, we construct two bounded open balls and prove that the conditions (3.10) are satisfied with respect to .Firstly, let , where and is defined as in . It follows from Lemma 2.5, , Remark 3.1 and from the definition of that . Then . Immediately: Secondly, (3.3) and Lemma 2.5(ii) imply that, for , because are nondecreasing as stated in . Analogously, by (3.3) and Lemma 3.4(ii), one can get that for Let and . Hence for , we have the following inequality Since by , there exists a sufficiently large number such that Let , then (cf. (3.21) and (3.22)) Applying Lemma 3.6, we conclude from (3.18) and (3.23) that has a fixed point in .

Lemma 3.9. Suppose that hold, then the sequences and are relatively compact in , where be a fixed point of operator defined by (3.5).

Proof. Let be a fixed point of operator , that is, And consider (cf. (3.6)) By Lemma 2.5(iii), Lemma 3.4(iii), and Remark 3.1, we have also Hence (cf. (3.4)), for a.e. , and all . Therefore, by (3.26), (3.27), Lemma 2.5(ii), Lemma 3.4(ii), and Remark 3.1, for , , where In particular, where are defined in the proof of Lemma 3.8. Since by , there exists a constant such that for each , Immediately, (cf. (3.33)) Hence, the sequences and are uniformly bounded.We will take similar discussions as in Lemma 3.7 to show that and are equicontinuous on . Let , then we have Using (3.28), (3.35), one can get From Lemma 2.5 (i), Lemma 3.4 (i), and the functions being uniformly continuous on , choosing an arbitrary , there exists a positive number . When , we can get , , , and . Therefore (cf. (3.36) and (3.37)) the inequalities hold, where , are defined as (3.31) and (3.32), respectively. As a result, and are equicontinuous on .Finally, we prove that and are relatively compact. Because is a Banach space, we need only to show that and are completely bounded. For all , by the Remark 3.2, there exists a sufficiently large positive integer , such that if , where .Hence, by (3.5) and (3.6), we have and , for , where This implies that and have an -net constituted by finite elements ( and , resp.) of , that is, completely bounded.Therefore, and are relatively compact in by the Arzelá-Ascoli theorem.Using above results, we now give the existence of positive solution of singular problem (1.4)-(1.5).

Theorem 3.10. Suppose that hold, then problem (1.4)-(1.5) has a positive solution and Moreover, is continuous and , where is a constant as in (3.35).

Proof. From Lemmas 3.8 and 3.9, the operator has a fixed point satisfying (3.26), (3.27), (3.35). And and are relatively compact in . Hence, is relatively compact in . And therefore, there exist and a subsequence of such that in . Consequently, is positive and continuous. Moreover satisfies (3.47), . And Keeping in mind (3.35) holding, where is a positive constant, it follows from inequalities (3.4) and (3.26) and from Lemma 2.5(ii) that for a.e. and all , . Hence, by the Lebesgue dominated convergence theorem, we have for . Now, passing to the limit as in we have Consequently, is a positive solution of BVP (1.4)-(1.5) by Lemma 2.4.

By Lemmas 3.5 and 3.9, and Theorem 3.10, we give the following unique result without proof.

Theorem 3.11. Suppose that hold, then problem (1.4)-(1.5) has a unique positive solution and Moreover, is continuous and , where is a constant as in (3.35).

4. Existence Results for BVP (1.6)-(1.7)

We now give the existence of positive solution of BVP (1.6)-(1.7) by using diagonalization process.

Theorem 4.1. Suppose that hold, then BVP (1.6)-(1.7) has a positive solution , and is also positive.

Proof. Firstly, choose then consider the BVP, subject to Theorem 3.10 guarantees that BVP (4.2)-(4.3) has a positive continuous solution . And for any , where is a constant defined similarly to .Secondly, we apply the following diagonalization process. For , let Here, is defined in (4.1). Notice that with Also for and , we get where are similarly defined as in (2.11), but all of should be replaced by . Then for , we have Thus, when , similarly to (3.38), hold for an arbitrary , where is a suitable positive number and are defined similarly to as in (3.31) and (3.32), respectively. By using , we know that, for a.e. , is relatively compact, where . Therefore, and are relatively compact. The Arzelá-Ascoli theorem guarantees that there is a subsequence of and a function with in as through . Obviously, is positive. Let , noticing that Similarly to above argumentation, we have that there is a subsequence of and a function with in as through . Obviously, is positive. Note that on since . Let . Proceed inductively to obtain for a subsequence of and a function with in as through . Also, is positive. Let .Define a function as follows. Fix , and Let with , then define . Hence .Again fix and Let with . Then for we get Let through to obtain that is, We can use this method for each and for each . Hence, for each . Consequently, the constructed function is a solution of (1.6)-(1.7). This completes the proof of the theorem.

Remark 4.2. In [21], the authors considered the BVP (1.3). Under some suitable conditions, they obtained the existence result of unbounded solution. In nature, BVP (1.3) is a special form of BVP (1.6)-(1.7). In that scalar situation, , , , , , and are not singular, then our Theorem 4.1 includes the result from [21]. But our approach is different from those of [21].

Proceeding the similar arguments above, we list the following unique result for BVP (1.6)-(1.7), and the proof will be omitted.

Theorem 4.3. Suppose that hold, then BVP (1.6)-(1.7) has a unique positive solution , and is also positive.

5. Application

We end this paper with giving an example to demonstrate the application of our existence result.

Example 5.1. Consider the following BVP in scalar space: where , . We will apply Theorem 4.1 with , , , , , , . Clearly holds because for , . And also holds for , , and . Hence, Theorem 4.1 guarantees the existence of positive solution of (5.1).

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