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

- Aijun Yang
^{1}Email author and - Weigao Ge
^{1}

**Received: **27 October 2008

**Accepted: **18 December 2008

**Published: **5 January 2009

## Abstract

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.

## 1. Introduction

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].

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

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

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 [1–4].

Definition 2.1.

hold.

Definition 2.2.

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

Remark 2.3.

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:

(A2)there is an with for some .

Definition 2.5.

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

(B1)there exists such that 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

## 3. Related Lemmas

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

Lemma 3.1.

Proof.

. For the Green functions , , we can obtain

Lemma 3.2.

Proof.

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.

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.

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

Proof.

Lemma 3.2 indicates that is completely continuous.

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.

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

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

Theorem 5.1.

Assume that there exist constants such that for . Suppose

Proof.

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 .

Step 2.

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.

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

Example 5.2.

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 [13–15] by the nonlinear terms involved in the -order Riemann-Liouville derivative explicitly.

## Declarations

### Acknowledgments

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

## References

- Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach, New York, NY, USA; 1993:xxxvi+976.MATHGoogle Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication*. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar - Lakshmikantham V, Vatsala AS:
**Basic theory of fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(8):2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V:
**Theory of fractional functional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3337-3343. 10.1016/j.na.2007.09.025MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Vatsala AS:
**General uniqueness and monotone iterative technique for fractional differential equations.***Applied Mathematics Letters*2008,**21**(8):828-834. 10.1016/j.aml.2007.09.006MathSciNetView ArticleMATHGoogle Scholar - El-Sayed AMA, El-Mesiry AEM, El-Saka HAA:
**On the fractional-order logistic equation.***Applied Mathematics Letters*2007,**20**(7):817-823. 10.1016/j.aml.2006.08.013MathSciNetView ArticleMATHGoogle Scholar - El-Sayed AMA, El-Maghrabi EM: Stability of a monotonic solution of a non-autonomous multidimensional delay differential equation of arbitrary (fractional) order. Electronic Journal of Qualitative Theory of Differential Equations 2008, (16):1-9.Google Scholar
- Diethelm K, Ford NJ:
**Analysis of fractional differential equations.***Journal of Mathematical Analysis and Applications*2002,**265**(2):229-248. 10.1006/jmaa.2000.7194MathSciNetView ArticleMATHGoogle Scholar - Bai C:
**Positive solutions for nonlinear fractional differential equations with coefficient that changes sign.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(4):677-685. 10.1016/j.na.2005.04.047MathSciNetView ArticleMATHGoogle Scholar - Su X:
**Boundary value problem for a coupled system of nonlinear fractional differential equations.***Applied Mathematics Letters*2009,**22**(1):64-69. 10.1016/j.aml.2008.03.001MathSciNetView ArticleMATHGoogle Scholar - Bai Z, Lü H:
**Positive solutions for boundary value problem of nonlinear fractional differential equation.***Journal of Mathematical Analysis and Applications*2005,**311**(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar - Zhang S:
**Existence of solution for a boundary value problem of fractional order.***Acta Mathematica Scientia*2006,**26**(2):220-228. 10.1016/S0252-9602(06)60044-1MathSciNetView ArticleMATHGoogle Scholar - Zhang S:
**Positive solutions for boundary-value problems of nonlinear fractional differential equations.***Electronic Journal of Differential Equations*2006,**2006**(36):1-12.MathSciNetGoogle Scholar - Mawhin J:
*Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics*.*Volume 40*. American Mathematical Society, Providence, RI, USA; 1979:v+122.View ArticleGoogle Scholar - Bai Z, Ge W:
**Existence of three positive solutions for some second-order boundary value problems.***Computers & Mathematics with Applications*2004,**48**(5-6):699-707. 10.1016/j.camwa.2004.03.002MathSciNetView ArticleMATHGoogle Scholar

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