Open Access

Positive Solutions for Boundary Value Problems of -Dimension Nonlinear Fractional Differential System

Boundary Value Problems20092008:437453

Received: 27 October 2008

Accepted: 18 December 2008

Published: 5 January 2009


We study the boundary value problem for a kind -dimension nonlinear fractional differential system with the nonlinear terms involved in the fractional derivative explicitly. The fractional differential operator here is the standard Riemann-Liouville differentiation. By means of fixed point theorems, the existence and multiplicity results of positive solutions are received. Furthermore, two examples given here illustrate that the results are almost sharp.


Fractional DerivativeFractional CalculusFractional Differential EquationReal Banach SpaceTopological Degree

1. Introduction

We are interested in the following -dimension nonlinear fractional differential system:
that is subject to the boundary conditions

where is the standard Riemann-Liouville fractional derivative of order , and ,

Recently, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, and so on. For an extensive collection of such results, we refer the readers to the monographs by Samko et al. [1], Podlubny [2], Miller and Ross [3], and Kilbas et al. [4].

Some basic theory for the initial value problems of FDE involving Riemann-Liouville differential operator has been discussed by Lakshmikantham [57], El-Sayed et al. [8, 9], Diethelm and Ford [10], and Bai [11], and so on. Also, there are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE boundary value problems (in short BVPs) by using techniques of topological degree theory. For example, Su [12] considered the BVP of the coupled system

By using the Schauder fixed point theorem, one existence result was given.

In [13], Bai and Lü obtained positive solutions of the two-point BVP of FDE

by means of Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem. is the standard Riemann-Liouville fractional derivative.

Zhang discussed the existence of solutions of the nonlinear FDE
with the boundary conditions

in [14, 15], respectively. Since conditions (1.6) and (1.7) are nonzero boundary values, the Riemann-Liouville fractional derivative is not suitable. Therefore, the author investigated the BVPs (1.5)-(1.6) and (1.5)–(1.7) by involving in the Caputo fractional derivative .

From above works, we can see a fact, although the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, higher-dimension fractional equation systems are seldom considered. Su in [12] studied the two-dimension system, however, the Schauder fixed point theorem cannot ensure the solutions to be positive. Since only positive solutions are useful for many applications, we investigate the existence and multiplicity of positive solutions for BVP (1.1)-(1.2) in this paper. In addition, two examples are given to demonstrate our results.

2. Preliminaries

For the convenience of the reader, we first recall some definitions and fundamental facts of fractional calculus theory, which can be found in the recent literatures [14].

Definition 2.1.

The fractional integral of order of a function is given by
provided that the integral exists, where is the Euler gamma function defined by
for which, the reduction formula
the Dirichlet formula


Definition 2.2.

The fractional derivative of order of a continuous function can be written as

where denotes the integer part of , provided that the right side is pointwise defined on .

Remark 2.3.

The following properties are useful for our discussion:

In the following, we present the useful lemmas which are fundamental in the proof of our main results.

Lemma 2.4 (see [16]).

Let be a convex subset of a normed linear space and be an open subset of with . Then every compact continuous map has at least one of the following two properties:

(A1) has a fixed point;

(A2)there is an with for some .

Definition 2.5.

The map is said to be a nonnegative continuous concave functional on a cone of a real Banach space provided that is continuous and

for all and .

Let and be nonnegative continuous convex functionals on the cone , be a nonnegative continuous concave functional on . Then for positive real numbers and , one defines the following convex sets:

The assumptions below about the nonnegative continuous convex functionals , will be used as follows:

(B1)there exists such that for all ;

(B2) for all .

Lemma 2.6 (see [17]).

Let be a cone in a real Banach space , , and . Assume that and are nonnegative continuous convex functionals satisfying (B1) and (B2), is a nonnegative continuous concave functional on such that for all and is a completely continuous operator. Suppose

(C1) , , for ;

(C2) , , for all ;

(C3) for all with .

Then has at least three fixed points with

3. Related Lemmas

Let with the norm
where , with

where is the standard sup norm of the space . Throughout, we denote and . Then is a Banach space (see [12]).

Define the cone by

Lemma 3.1.

If , then , .


For , we have

That is, , .

It is well known that the solution for the system BVP (1.1)-(1.2) is equivalent to the fixed point of the following integral system:
for , where
Denote , we can see

. For the Green functions , , we can obtain

(i) for , for , , where
here, is the unique solution of the equation

(ii) and .

Lemma 3.2.

is completely continuous.


We divide the proof into three steps.

Step 1.

. In fact, for any , since for and for , for . Moreover, implies that .

Step 2.

is continuous on , which is valid due to the continuity of the function .

Step 3.

We will show that is relatively compact. For any given bounded set , there exists such that for all . We take For , let be such that , we have
Notice that
one gets
where , we can see that is an equicontinuous set. Now, we proof that is uniformly bounded. For any ,

where . That is, is uniformly bounded. Thus, is relatively compact. By means of the Arzela-Ascoli theorem, is completely continuous.

4. The Existence of One Positive Solution

Theorem 4.1.

If there exist , satisfying
such that

Then the BVP (1.1)-(1.2) has at least one positive solution.


Lemma 3.2 indicates that is completely continuous.

For , let
Define , then . For , . Thus, and :

indicate that , and then . Take in Lemma 2.4, for any , does not hold. Hence, the operator has at least a fixed point, then the BVP (1.1)-(1.2) has at least one positive solution.

Example 4.2.

Consider the problem

It is easy to check that (4.1) holds. Thus, by Theorem 4.1, the BVP (4.5) has at least one positive solution. In fact, is such a solution.

5. The Existence of Triple Positive Solutions

Let the nonnegative continuous convex functionals , and the nonnegative continuous concave functional be defined on the cone by

Obviously, and satisfy (B1) and (B2), for all .

For simplicity, we denote

Theorem 5.1.

Assume that there exist constants such that for . Suppose

(H1) , ;

(H2) , ;

(H3) , ;

(H4) ,

Then the BVP (1.1)-(1.2) has at least three positive solutions , and such that


Lemma 3.2 has showed that is completely continuous. Now, we will verify that all the conditions of Lemma 2.6 are satisfied. The proof is based on the following steps.

Step 1.

We will show that (H1) implies .

In fact, for , , , and then , , . In view of (H1), we have

Then and , that is, .

Step 2.

To check the condition (C1) in Lemma 2.6, we choose , . It is easy to see that
Consequently, . For any , from (H2), one gets

then we can obtain .

Step 3.

It is similar to Step 1 that we can prove by condition (H3), that is, (C2) in Lemma 2.6 holds.

Step 4.

We verify that (C3) in Lemma 2.6 is satisfied. For with , we have

Thus, , (C3) in Lemma 2.6 is satisfied.

Therefore, the operator has three points with
Then the BVP (1.1)-(1.2) has three positive solutions such that

Example 5.2.

Consider the problem
Here, we have , , , . By choosing and the definition of and , , , one gets
Taking , and , we have
that is, satisfies the conditions (H1)–(H4) of Theorem 5.1. Similarly, we can show that satisfies (H1)–(H4). Thus, by Theorem 5.1, the BVP (5.10) has at least three positive solutions , and such that

Remark 5.3.

The particular case has been studied by [12] for the existence of one solution, our paper generalizes [12] for the obtaining of one and three positive solutions. For , we develop [1315] by the nonlinear terms involved in the -order Riemann-Liouville derivative explicitly.



This work is supported by National Natural Science Foundation of China (NNSF) (10671012) and the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) of China (20050007011).

Authors’ Affiliations

Department of Applied Mathematics, Beijing Institute of Technology, Beijing, China


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© A. Yang andW. Ge. 2008

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