The Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations

Abstract

We consider the existence of positive solutions for the nonlinear fractional differential equations boundary value problem where is a real number, is the Riemann-Liouville fractional derivative of order , and is a given continuous function. Our analysis relies on the fixed point index theory in cones.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, or polymer rheology; see [1–5]. The interest of the study of fractional-order differential equations lies in the fact that fractional-order models are more accurate than integer-order models; that is, there are more degrees of freedom in the fractional-order models. Recently, there are some papers dealing with the existence of solutions (or positive solution) of nonlinear initial value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, lower and upper solution method, Adomian decomposition method, ect.); see [6–15].

The famous viscous liquid flow problems in the fields of integer-order differential equations can be described by third-order ordinary differential equation boundary value problem where is continuous [16–18]. However, there are only a few exisitng contributions, as far as we know, in the field of fractional-order differential equation. In this paper, we discuss the existence of positive solution for the nonlinear fractional differential equations boundary value problem (BVP) where is a real number, is the Riemann-Liouville fractional derivative, and is a continuous function.

For a more general case, specially, where is continuous with , and is the Riemann-Liouville fractional derivative; El-Shahed [20] obtained the existence and nonexistence of positive solutions by employing the well-known Guo-Krasnoselskii fixed point theorem of cone extension or compression. The purpose of this paper is to extend this result. Our argument is based on the fixed point index theory, which is more precise than the fixed point theorem of cone extension or compression. We will employ the theory of fixed point index in cones to present some more extensive conditions on guaranteeing the existence of positive solution of the BVP (2). As far as we know, the method of the fixed point index theory is firstly applied to BVP (2).

This paper is organized as follows. In Section 2, we introduce some basic definitions and properties, preliminary results that will be used to prove our main results. In Section 3, we obtain the existence of the positive solutions for BVP (2) by using the fixed point index theory.

2. Preliminaries

In this section, we introduce some preliminary facts which are used throughout this paper. For details, see [19].

Definition 1 (see [19]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where is Gamma function and , denotes the integer part of number , provided that the right side is pointwise defined on .

Definition 2 (see [19]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on .

Lemma 3. Let , if ; then the fractional differential equation has unique solutions , , , where is the smallest integer greater than or equal to .

Lemma 4. Assume that with a fractional derivative of order that belongs to . Then for some , , where is the smallest integer greater than or equal to .
In the following, we present the Green’s function of fractional differential equation boundary value problem.

Lemma 5 5 (see [20]). Let and . The linear fractional differential equation boundary value problem has a unique solution where

Lemma 6 (see [20]). Let be Green’s function related to problem (8), which is given by the expression (10). Then, for all , the following properties are fulfilled:(1), ;(2), ;(3), ;(4) is a continuous function, .
Let be the Banach space endowed with the norm . We define the operator by where is the Green’s function defined in (10).
It is clear, form Lemma 5, that the nontrivial fixed points of operator coincide with the positive solutions of BVP (2).
Let . Define a cone by

Lemma 7. is completely continuous.

Proof. From the continuity and the nonnegativeness of functions and on their domains of definition, we have that if , then and for all ; from properties (2) and (3) of Lemma 6, for all , Hence, .
Next, we show that is uniformly bounded.
Let be bounded, which is to say, there exists a positive constant such that for all . Define now Then, for all , it is satisfied that That is, the set is bounded in .
Finally, we show that is equicontinuous.
For each , we have As consequence, for all , , we have Hence the set is equicontinuous.
Now, from the Arzela-Ascoli Theorem, we conclude that is relatively compact. Hence, is a completely continuous operator.

Define an operator by Clearly, also is a completely continuous linear operator and .

Lemma 8. The operator defined by (18) satisfies

Proof . Let . For every , by the definition of , Hence, This implies that The proof is completed.

Hereafter, we use to denote the spectral radius of the operator .

Lemma 9. Suppose that is defined by (18); then the spectral radius .

Proof. Set . Then, by (18) and the positivity of , we have and Inductively, we obtain that Consequently, So, By this and Gelfand’s formula of spectral radius we have The proof of Lemma 9 is completed.

Now, since the operator is a completely continuous linear operator, by the well-known Krein-Rutman theorem ([21], Theorem 19.3), the operator has the maximum positive real eigenvalue ; then there exists a eigenfunction such that .

Set ; then . Thus, is the minimum positive real eigenvalue of the linear equation (8).

To prove the existence of at least one positive solution of BVP (2), we will find the nonzero fixed point of (defined in (11)) by using the fixed point index theory in cones.

We recall some concepts and conclusions on the fixed point index in cones in [21, 22], which will be used in the argument later. Let be a Banach space and let be a closed convex cone in . Assume that is a bounded open subset of with boundary and . Let be a completely continuous mapping. If for every , then the fixed point index is well defined. One important fact is that if , then has a fixed point in .

For , let , and , which is the relative boundary of in . The following two lemmas are needed in our argument.

Lemma 10 (see [22]). Let be a completely continuous mapping. If there exists an such that then the fixed point index .

Lemma 11 (see [22]). Let be a completely continuous mapping. If then the fixed point index .

Lemma 12 (see [22]). Let be a completely continuous mapping and it satisfies that for every . If , , then the fixed point index .

3. Main Results

In this section we show the existence of positive solutions of BVP (2) by using the fixed point index theory in cones.

Theorem 13. Assume is continuous and satisfies the following conditions.(F1)There exist and , such that (F2)There exist and , such that Then, the BVP (2) has at least one positive solution.

Proof. Let be the positive eigenfunction of corresponding to ; thus , where is defined by (18).
Choose , where is the constant in assumption (F1). For every , from assumption (F1), we have Namely,
Suppose that has no fixed point on (otherwise, the proof is completed). Now we show that If it is not true, there exist and (if , the proof is completed) such that Then, That is, Let It is easy to see that and . Taking into account the positivity of the Green’s function and definition of the operator , it is easy to know that is a nondecreasing linear operator, so Therefore by (33) which contradicts the definition of . Hence (34) holds and we have from Lemma 10 that
On the other hand, we choose . Now we show that if is large enough, then From (F2), ; then there exist , such that .
Let , . Then is a bounded linear operator and .
Let In the following, we prove that is bounded.
For any , we have Thus,
Since , therefore ; it is easy to get that the inverse operator exists and It follows from that . Hence we have and is bounded. Let ; then, by Lemma 11, we have Now by the additivity of fixed point index, (41) and (48), we have Therefore has a fixed point in , which is a positive solution of BVP (2).
The proof of Theorem 13 is completed.

For convenience, we set .

Theorem 14. Assume that is continuous and satisfies the following conditions.(F3)There exist and , such that (F4)There exist and , such that Then, BVP (2) has at least one positive solution.

Proof. Let , , where is defined in (18). Then, is a bounded linear operator; we have Choose , where is the constant in assumption (F3). For every , from assumption (F3), we have Namely, , .
Suppose that has no fixed point on (otherwise, the proof is completed). Now we show that If it is not true, there exist and such that Then, , . Therefore, we have Thus, . Using Gelfand’s theorem, we have which contradicts (52). Hence (54) holds; from Lemma 11, we have
On the other hand, from (F4), we have We choose . For every , by the definitions of and , we have and Hence, form (59), it follows that Now, by (11) and (61), we have This means that Hence, by Lemma 12, we have that Now by the additivity of fixed point index, (58) and (64), we have Therefore has a fixed point in , which is a positive solution of BVP (2).
The proof of Theorem 14 is completed.

Example 15. Consider the following fractional differential equations boundary value problem: where , are constants. We assume that is large enough such that . Then it is easy to verify that satisfies conditions (F1) and (F2) of Theorem 13. Hence, by Theorem 13, (66) has at least one positive solution.

Example 16. Consider the fractional differential equations boundary value problem where , are constants. Let . We can easily see that satisfies conditions (F3) and (F4) of Theorem 14. By Theorem 14, (68) has at least one positive solution.

Conflict of Interests

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

Acknowledgments

This research is supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).