Open Access

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

Boundary Value Problems20092008:437453

https://doi.org/10.1155/2008/437453

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

We are interested in the following -dimension nonlinear fractional differential system:
(1.1)
that is subject to the boundary conditions
(1.2)

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
(1.3)

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
(1.4)

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
(1.5)
with the boundary conditions
(1.6)
(1.7)

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
(2.1)
provided that the integral exists, where is the Euler gamma function defined by
(2.2)
for which, the reduction formula
(2.3)
the Dirichlet formula
(2.4)

hold.

Definition 2.2.

The fractional derivative of order of a continuous function can be written as
(2.5)

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:
(2.6)

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
(2.7)

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:
(2.8)

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
(2.9)

3. Related Lemmas

Let with the norm
(3.1)
where , with
(3.2)

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

Define the cone by
(3.3)

Lemma 3.1.

If , then , .

Proof.

For , we have
(3.4)

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:
(3.5)
for , where
(3.6)
Denote , we can see
(3.7)

. For the Green functions , , we can obtain

(i) for , for , , where
(3.8)
here, is the unique solution of the equation
(3.9)

(ii) and .

Lemma 3.2.

is completely continuous.

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.

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
(3.10)
Notice that
(3.11)
one gets
(3.12)
where , we can see that is an equicontinuous set. Now, we proof that is uniformly bounded. For any ,
(3.13)

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
(4.1)
such that
(4.2)

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

Proof.

Lemma 3.2 indicates that is completely continuous.

For , let
(4.3)
Define , then . For , . Thus, and :
(4.4)

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
(4.5)
where
(4.6)
Choose
(4.7)

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
(5.1)

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

For simplicity, we denote
(5.2)

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
(5.3)

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 .

In fact, for , , , and then , , . In view of (H1), we have
(5.4)

Then and , that is, .

Step 2.

To check the condition (C1) in Lemma 2.6, we choose , . It is easy to see that
(5.5)
Consequently, . For any , from (H2), one gets
(5.6)

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
(5.7)

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

Therefore, the operator has three points with
(5.8)
Then the BVP (1.1)-(1.2) has three positive solutions such that
(5.9)

Example 5.2.

Consider the problem
(5.10)
where
(5.11)
Here, we have , , , . By choosing and the definition of and , , , one gets
(5.12)
Taking , and , we have
(5.13)
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
(5.14)

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.

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

(1)
Department of Applied Mathematics, Beijing Institute of Technology

References

  1. 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
  2. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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

Copyright

© A. Yang andW. Ge. 2008

This article is published under license to BioMed Central Ltd. 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.