The Existence of Positive Solution to a Nonlinear Fractional Differential Equation with Integral Boundary Conditions

Abstract

The expression and properties of Green's function for a class of nonlinear fractional differential equations with integral boundary conditions are studied and employed to obtain some results on the existence of positive solutions by using fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.

1. Introduction

Fractional calculus is an area having a long history, its infancy dates back to three hundred years, the beginnings of classical calculus. It had attracted the interest of many old famous mathematicians, such as L'Hospital, Leibniz, Liouville, Riemann, Grünward, Letnikov, and so forth [1, 2]. As the old mathematicians expected, in recent several decades fractional differential equations have been found to be a powerful tool in more and more fields, such as materials, physics, mechanics, and engineering [1–5]. For the basic theory and recent development of the subject, we refer the reader to a text by Lakshmikantham et al. [6]. For more details and examples, see [7–24] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages, and many aspects of this theory need to be explored.

In [13], Bai and Lü used the fixed point theorems to show the existence and multiplicity of positive solutions to the nonlinear fractional boundary value problem

(11)

where , is the standard Riemann-Liouville differentiation, and is a given continuous function.

In [15], Zhang showed the existence and multiplicity of positive solutions of the fractional boundary value problem

(12)

where is a real number, and is the Caputo's fractional derivative. The function is continuous on .

Recently, Ahmad and Nieto [11] investigated some existence results for a nonlinear fractional integrodifferential equation with integral boundary conditions

(13)

where is the Caputo fractional derivative, , for ,

(14)

, , are real numbers, and is a Banach space.

Being directly inspired by [11, 13, 15], we intend in this paper to study the following boundary value problems of fractional order differential equation

(15)

where , and may be singular at or/and at is the standard Riemann-Liouville differentiation, is nonnegative, and .

In the case of , for all , boundary value problem (1.5) reduces to the problem studied by Kaufmann and Mboumi [19]. In [19], the authors used the fixed point theorems to show sufficient conditions for the existence of at least one and at least three positive solutions to problem (1.5). For the case of , boundary value problem (1.5) is related to a boundary value problems of integer-order differential equation. Feng et al. [25] considered the existence and multiplicity of positive solutions to boundary value problem (1.5) by applying the fixed point theory in a cone for strict set contraction operators.

The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, we present the expression and properties of Green's function associated with boundary value problem (1.5). In Section 3, we give some preliminaries about operator. In particular, we state fixed point theory in cones. In Section 4, the main results of boundary value problem (1.5) will be stated and proved. In Section 5, we offer some interesting discussion of the associated boundary value problem (1.5). Finally, conclusions in Section 6 close the paper.

The fractional differential equations-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to fractional differential equations. The readers who are unfamiliar with this area can consult for example [1–6] for details.

Definition 1.1 (see [4]).

The integral

(16)

where is called Riemann-Liouville fractional integral of order .

Definition 1.2 (see [4]).

For a function given in the interval , the expression

(17)

where , denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order .

Lemma 1.3 (see [13]).

Assume that with a fractional derivative of order that belongs to . Then

(18)

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

2. Expression and Properties of Green's Function

In this section, we present the expression and properties of Green's function associated with boundary value problem (1.5).

Lemma 2.1.

Assume that . Then for any , the unique solution of boundary value problem

(21)

is given by

(22)

where

(23)

(24)

(25)

Proof.

By Lemma 1.3, we can reduce the equation of problem (2.1) to an equivalent integral equation

(26)

By , there is , and

(27)

By (2.7) and , we have

(28)

which yields that

(29)

Therefore, the unique solution of BVP (2.1) is

(210)

where is defined by (2.4).

Multiplying (2.10) with and integrating it, we can see

(211)

Therefore,

(212)

Substituting (2.12) into (2.10), we obtain

(213)

where , , and are defined by (2.3), (2.4), and (2.5), respectively. The proof is complete.

From (2.3), (2.4), and (2.5), we can prove that , and have the following properties.

Proposition 2.2.

The function defined by (2.4) satisfies the following.

• (i) is continuous for all , ,  for all ;

• (ii) ,  for all , .

Proof.

• (i) It is obvious that is continuous on . For ,

  

  (214)

So, by (2.4), we have

(215)

Similarly, for , we have .

• (ii) Since , for given , , we have

  

  (216)

(217)

Therefore, from (2.17) and the definition of , for given , , we have

(218)

On the other hand, it is clear that

(219)

Therefore, we have

(220)

Let

(221)

Proposition 2.3.

If , then one has

• (i) is continuous for all , , for all ;

• (ii) , for all , .

Proof.

From the properties of , and the definition of , we can prove that the results of Proposition 2.3 hold.

Theorem 2.4.

If , the function defined by (2.3) satisfies the following.

• (i) is continuous for all , , for all ;

• (ii) (, for all , where

  

  (222)

Proof.

• (i) From Propositions 2.2 and 2.3, we obtain that is continuous for all , , for all .

• (ii) From Proposition 2.2 and (2.3), we have

  

  (223)

Remark 2.5.

From (i) of Theorem 2.4, we obtain that there exists such that

(224)

where .

3. Preliminaries

In this section, we give some preliminaries for discussing the existence of positive solutions of boundary value problem (1.5).

Let . The basic space used in this paper is . It is well known that is a real Banach space with the norm defined by . Let

(31)

where .

On the basis of Lemma 3.3 below we will establish in Section 4 the existence of positive solution to the problem (1.5). Here we make the following hypotheses:

(H ₁ ), on any subinterval of (0,1) and ;

(H ₂ ) and uniformly with respect to on ;

(H ₃ ), where is defined by (2.21).

Define by

(32)

where is defined by (2.3).

Lemma 3.1.

Let (H₁)–(H₃) hold. Then boundary value problems (1.5) has a solution if and only if is a fixed point of .

Proof.

From Lemma 2.1, we can prove the results of this Lemma.

Lemma 3.2.

Let (H₁)–(H₃) hold. Then and is completely continuous.

Proof.

For any , by (3.2), we can obtain . Next by similar proof of Lemma 3.1 in [12] and Ascoli-Arzela theorem one can prove is completely continuous. So it is omitted.

Lemma 3.3 (see [26]).

Let and be two bounded open sets in a real Banach space , such that and . Let operator be completely continuous, where is a cone in . Suppose that one of the following two conditions is satisfied.

• (i) There exists such that , for all ,; , for all , .

• (ii) There exists such that , for all , ; , for all , .

Then, has at least one fixed point in .

4. Existence of Positive Solutions

In this section, we apply Lemma 3.3 to establish the existence of positive solutions for boundary value problems (1.5).

Theorem 4.1.

Suppose (H₁)– (H₃) and satisfies the following conditions.

(H₄)There exists such that ;

(H₅)There exists such that .

Then boundary value problems (1.5) has at least one positive solution.

Proof.

For applying Lemma 3.3, we construct a function via

(41)

Obviously, is a nonnegative continuous function, that is, , and .

Suppose that there is such that

(42)

if not, then the conclusion holds. The condition (H ₄ ) and imply that there exist , such that

(43)

Let , and choose . We now show that

(44)

In fact, if there exist , such that , then (4.4) implies that . On the other hand, . So we can choose , then , . Therefore,

(45)

Consequently, for any , (2.24) and (4.3) imply

(46)

that is, , . Noticing the definition of , we have

(47)

which is a contradiction to the definition of . Hence, (4.4) holds.

Now turning to (H ₅ ), there exist , , for , , such that . Letting , then

(48)

Choosing such that

(49)

where . Now we prove that

(410)

If not, then there exist such that . By (4.8) and (ii) of Theorem 2.4, then for any , we have

(411)

So , that is,

(412)

which is a contradiction to (4.9). So, (4.10) holds.

By (ii) of Lemma 3.3, (4.4) and (4.10) yield that has a fixed point , . Thus it follows that boundary value problems (1.5) has at least one positive solution with . The proof is complete.

5. Discussion

In this section, we offer some interesting discussion associated with boundary value problems (1.5).

Since the proof of the main theorem (Theorem 4.1) in this paper is independent of the expression form of and only dependent on its continuity and nonnegativity, there are similar conclusions by analogous methods for boundary value problems (1.5) subject to other boundary value conditions, respectively, the following.

• (i) We have

  

  (51)

  then

  

  (52)

  where

  

  (53)

  Obviously is continuous on , and it is easy to see that by , where

  

  (54)

• (ii) We have

  

  (55)

  then

  

  (56)

  where

  

  (57)

  Obviously is continuous on , and it is easy to see that , by , where is defined by (2.21).

• (iii) We have

  

  (58)

  then

  

  (59)

  where

  

  (510)

Obviously is continuous on , and it is easy to see that by , where is defined in (5.4).

6. Conclusions

In this paper, by using the fixed point theorem of cone, we have investigated the existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and have obtained some easily verifiable sufficient criteria which extend previous results. It is worth mentioning that there are still many problems that remain open in this vital field other than the results obtained in this paper: for example, whether or not we can study the fractional differential equations with integral boundary conditions at resonance (see, e.g., [27]), and whether or not we can give a unified approach applicable to many BVPs (see, e.g., [28–31]). More efforts are still needed in the future.