Skip to content

Advertisement

  • Research
  • Open Access

Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders

Advances in Difference Equations20112011:10

https://doi.org/10.1186/1687-1847-2011-10

  • Received: 11 November 2010
  • Accepted: 14 June 2011
  • Published:

Abstract

In this article, we study the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders

where 2 < α ≤ 3, 3 < β ≤ 4, , are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] × [0, +∞) → [0, +∞) are given continuous functions, f(t, 0) ≡ 0, g(t, 0) ≡ 0. Our analysis relies on fixed point theorems on cones. Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. As an application, examples are presented to illustrate the main results.

Keywords

  • Positive solution
  • coupled system
  • fractional Green's function
  • fixed point theorem

1 Introduction

Fractional differential equations have been of great interest recently. It is caused by the both intensive development of the theory of fractional calculus itself and applications, see [16]. Recently, there are a large number of papers dealing with the existence of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, Adomian decomposition method, etc.), see [721]. The articles [1321] considered boundary value problems for fractional differential equations.

Yu and Jiang [20] examined the existence of positive solutions for the following problem

where 2 < α ≤ 3 is a real number, f C([0,1] × [0, +∞); (0, +∞)) and is the Riemann-Liouville fractional differentiation. Using the properties of the Green function, they obtained some existence criteria for one or two positive solutions for singular and nonsingular boundary value problems by means of the Krasnosel'skii fixed point theorem and a mixed monotone method.

Xu et al. [21] considered the existence of positive solutions for the following problem

where 3 < α ≤ 4 is a real number, f C([0, 1] × [0, +∞), (0, +∞)) and is the Riemann-Liouville fractional differentiation. Using the properties of the Green function, they gave some multiple positive solutions for singular and nonsingular boundary value problems, and also they gave uniqueness of solution for singular problem by means of Leray-Schauder nonlinear alternative, a fixed point theorem on cones and a mixed monotone method.

On the other hand, the study of coupled systems involving fractional differential equations is also important as such systems occur in various problems, see [2230].

Bai and Fang [24] considered the existence of positive solutions of singular coupled system

where 0 < s, p < 1, and f, g : [0, 1) × [0, +∞) → [0, +∞) are two given continuous functions, , and D s , D p are two standard Riemann-Liouville fractional derivatives. They established the existence results by a nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem on a cone.

Su [25] discussed a boundary value problem for a coupled differential system of fractional order

where 1 < α, β ≤ 2, μ, ν > 0, α - ν ≥ 1, β - μ ≥ 1, f, g : [0, 1] × × are given functions and D is the standard Riemann-Liouville fractional derivative. By means of Schauder fixed point theorem, an existence result for the solution was obtained.

From the above works, we can see a fact, although the coupled systems of fractional boundary value problems have been investigated by some authors, coupled systems due to mixed fractional orders are seldom considered. Motivated by all the works above, in this article we investigate the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders
(1.1)

where 2 < α ≤ 3, 3 < β ≤ 4, , are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] × [0, +∞) → [0, +∞) are given continuous functions, f(t, 0) ≡ 0, g(t, 0) ≡ 0. Our analysis relies on fixed point theorems on cones. Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. Finally, we present some examples to demonstrate our results.

The article is organized as follows. In Sect. 2, we shall give some definitions and lemmas to prove our main results. In Sect. 3, we establish existence results of at least one or two positive solutions for boundary value problem (1.1) by fixed point theorems on cones. In Sect. 4, examples are presented to illustrate the main results.

2 Preliminaries

For the convenience of readers, we give some background materials from fractional calculus theory to facilitate analysis of problem (1.1). These materials can be found in the recent literature, see [20, 21, 3133].

Definition 2.1 [31] The Riemann-Liouville fractional derivative of order α > 0 of a continuous function f : (0, +∞) → is given by

where n = [α]+1, [α] denotes the integer part of number α, provided that the right side is pointwise defined on (0, +∞).

Definition 2.2 [31] The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, +∞) → is given by

provided that the right side is pointwise defined on (0, +∞).

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 2.1 [31] Let α > 0. If we assume u C(0, 1) ∩ L(0, 1), then the fractional differential equation

has u(t) = c 1 t α - 1+ c 2 t α - 2+ + c n t α - n , c i , i = 1, 2,..., n, as unique solutions, where n is the smallest integer greater than or equal to α.

Lemma 2.2 [31] Assume that u C(0, 1) ∩ L(0, 1) with a fractional derivative of order α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then

where n is the smallest integer greater than or equal to α.

In the following, we present the Green function of fractional differential equation boundary value problem.

Lemma 2.3 [20] Let h 1 C[0, 1] and 2 < α ≤ 3. The unique solution of problem
(2.1)
(2.2)
is
where
(2.3)

Here G 1 (t, s) is called the Green function of boundary value problem (2.1) and (2.2).

Lemma 2.4 [20] The function G 1(t, s) defined by (2.3) satisfies the following conditions:

(A1) G 1 (t, s) = G 1(1 - s, 1 - t), for t, s (0, 1);

(A2) t α - 1(1 - t)s(1 - s) α - 1≤ Γ(α)G 1 (t, s) ≤ (α - 1)s(1 - s) α - 1, for t, s (0, 1);

(A3) G 1 (t, s) > 0, for t, s (0, 1);

(A4) t α - 1(1 - t)s(1 - s) α - 1≤ Γ(α)G 1 (t, s) ≤ (α - 1)(1 - t) t α - 1, for t, s (0, 1).

Remark 2.1 Let q 1(t) = t α - 1(1 - t), k 1(s) = s(1 - s) α - 1. Then
Lemma 2.5 [21] Let h 2 C[0, 1] and 3 < β ≤ 4. The unique solution of problem
(2.4)
(2.5)
is
where
(2.6)

Here G 2(t, s) is called the Green function of boundary value problem (2.4) and (2.5).

Lemma 2.6 [21] The function G 2(t, s) defined by (2.6) satisfies the following conditions:

(B1) G 2(t, s) = G 2(1 - s, 1 - t), for t, s (0, 1);

(B2) (β - 2)t β - 2(1 - t)2 s 2(1 - s) β - 2≤ Γ(β)G 2(t, s) ≤ M 0 s 2(1 - s) β - 2, for t, s (0, 1);

(B3) G 2(t, s) > 0, for t, s (0, 1);

(B4) (β - 2)s 2(1 - s) β - 2 t β - 2(1 - t)2 ≤ Γ(β)G 2(t, s) ≤ M 0 t β - 2(1 - t)2, for t, s (0, 1),

here M 0 = max{β - 1, (β - 2)2}.

Remark 2.2 Let q 2(t) = t β-2(1 - t)2, k 2(s) = s 2(1 - s) β - 2. Then

The following two lemmas are fundamental in the proofs of our main results.

Lemma 2.7 [32] Let E be a Banach space, and let P E be a cone in E. Assume Ω1, Ω2 are open subsets of E with , and let S : PP be a completely continuous operator such that, either

(D1) ||Sw|| ≤ ||w||, w P ∩ ∂Ω1, ||Sw|| ≥ ||w||, w P ∩ ∂Ω2, or

(D2) ||Sw|| ≥ ||w||, w P ∩ ∂Ω1, ||Sw|| ≤ ||w||, w P ∩ ∂Ω2.

Then S has a fixed point in .

Lemma 2.8 [33] Let E be a Banach space, and let P E be a cone in E. Assume Ω1, Ω2 and Ω3 are open subsets of E with , and let be a completely continuous operator such that

(E1) ||Sw|| ≥ ||w||, w P ∩ ∂Ω1;

(E2) ||Sw|| ≤ ||w||, Sww, w P ∩ ∂Ω2;

(E3) ||Sw|| ≥ ||w||, w P ∩ ∂Ω3.

Then S has two fixed points w 1 and w 2 in with and .

3 Main results

In this section, we establish the existence results of positive solutions for boundary value problem (1.1).

Consider the following coupled system of integral equations:
(3.1)

Lemma 3.1 Suppose that f, g : [0, 1] × [0, +∞) → [0, +∞) are continuous. Then (u, v) C[0, 1] × C[0, 1] is a solution of (1.1) if and only if (u, v) C[0, 1] × C[0, 1] is a solution of system (3.1).

This proof is similar to that of Lemma 3.3 in [25], so is omitted.

From (3.1), we can get the following integral equation
Let Banach space E = C[0, 1] be endowed with the norm ||u|| = max0≤t≤ 1|u(t)|. De ne the cone P E by
We define an operator T : PE as follows

Lemma 3.2 T : PP is completely continuous.

Proof. For u P, 0 ≤ t ≤ 1, by Lemma 2.4,

Thus we have T (P) P.

The operator T : PP is continuous in view of continuity of G(t, s), f(t, u), and g(t, u). For any bounded set M, T (M) is uniformly bounded and equicontinuous. This proof is similar to that of Lemma 2.1.1 in [20], so is omitted. By means of Arzela-Ascoli Theorem, T : PP is completely continuous. This completes the proof.

We consider the following hypotheses in what follows.

(A 1) ;

(A 2) ;

(A 3) ;

(A 4) ;

(A 5) f(t, u) and g(t, u) are two increasing functions with respect to u, and there exists N > 0 such that

where n 1 = max0≤t,s≤ 1 G 1(t, s), n 2 = max0≤t,s≤ 1 G 2(t, s).

Theorem 3.1 Assume that hypotheses (A 1) and (A 2) hold. Then the boundary value problem (1.1) has at least one positive solution (u, v).

Proof. By hypothesis (A 1), we see that there exists p 1 (0, 1) such that
(3.2)
where λ 1, λ 2 > 0 and satisfy
(3.3)
For u P with , we have
then by (3.2) and (3.3), we get
Hence, if we choose , then
(3.4)
From hypothesis (A 2), there exist positive constants μ 1, μ 2, C 1, and C 2 such that
(3.5)
where μ 1 and μ 2 satisfy
(3.6)
For u P and l (0, 1), then by (3.5) and (3.6), we have
where
so,
Thus, if we set p 2 > max{p 1, C 3} and Ω2 = {u E : ||u|| < p 2}, then
(3.7)

Now, from (3.4), (3.7), and Lemma 2.7, we guarantee that T has a fix point , and clearly (u, v) is a positive solution of (1.1). The proof is completed.

Theorem 3.2 Assume that hypotheses (A 3) and (A 4) hold. Then the boundary value problem (1.1) has at least one positive solution (u, v).

Proof. By hypothesis (A 3), we see that there exists p (0, 1) such that
(3.8)
where η 1, s 2 > 0 and satisfy
(3.9)
From g(t, 0) ≡ 0 and the continuity of g, then there exists p 3 (0, 1) such that
For u P with ||u|| = p 3, we have
for l (0, 1), by (3.8) and (3.9), we get
Hence, if we choose Ω3 = {u E : ||u|| < p 3}, then
(3.10)
From hypothesis (A 4), there exist positive constants δ 1, δ 2, C 4, and C 5 such that
(3.11)
where δ 1 and δ 2 satisfy
(3.12)
Then by (3.11) and (3.12), we have
where
Thus, if we set p 4 > max{2p 3, 2C 6} and Ω4 = {u E : ||u|| < p 4}, then
(3.13)

Now, from (3.10), (3.13), and Lemma 2.7, we guarantee that T has a fix point , and clearly (u, v) is a positive solution of (1.1). The proof is completed.

Theorem 3.3 Assume that hypotheses (A 2), (A 3), and (A 5) hold. Then the boundary value problem (1.1) has at least two positive solutions (u 1, v 1) and (u 2, v 2).

Proof. Set B N = {u E : ||u|| < N}. From (A 5), for u P ∩ ∂B N , then we have
Thus, ||Tu|| < ||u||, u P ∩ ∂B N . By (A 2) and (A 3), we can get

So, we can choose p 2, p 3, and N such that p 3 < N < p 2 and satisfy the above three inequalities. By Lemma 2.8, we guarantee that T has two fix points and . Then the boundary value problem (1.1) at least two positive solutions (u 1, v 1) and (u 2, v 2). This completes the proof.

In fact, from (3.1), we can also obtain the following integral equation
Define the cone P' E by
We define an operator T': P' → E as follows
For v P', 0 ≤ t ≤ 1, by Lemma 2.6,

Thus we have T' (P') P'.

The operator T': P' → P' is continuous in view of continuity of G(t, s), f(t, u), and g(t, u). For any bounded set M', T' (M') is uniformly bounded and equicontinuous. This proof is similar to that of Lemma 3.1 in [21], so is omitted. By means of Arzela-Ascoli Theorem, T': P' → P' is completely continuous.

Remark 3.1 Theorems 3.1 and 3.2 also hold for the boundary value problem (1.1).

Proof. This proof is similar to that of Theorems 3.1 and 3.2, so is omitted.

Theorem 3.4 If conditions (A 5) in the Theorem 3.3 is replaced by

f(t, u) and g(t, u) are two increasing functions with respect to u, and there exists N' > 0 such that

where n 1 = max0 ≤ t,s ≤1 G 1(t, s), n 2 = max0 ≤ t,s ≤1 G 2(t, s).

Then the conclusion of Theorem 3.3 also holds.

Proof. This proof is similar to that of Theorem 3.3, so is omitted.

Remark 3.2 In this article, conditions f(t, 0) ≡ 0 and g(t, 0) ≡ 0 are too strong for the boundary value problem (1.1). So, we will give some new existence criteria for the boundary value problem (1.1) without conditions f(t, 0) ≡ 0 and g(t, 0) ≡ 0 in a new paper.

4 Examples

In this section, we will present examples to illustrate the main results.

Example 4.1 Consider the system of nonlinear differential equations
(4.1)
Choose f(t, v) = v(v + t -1), g(t, u) = u(u + t -1). Then

So (A 1) and (A 2) hold. Thus, by Theorem 3.1, the boundary value problem (4.1) has a positive solution.

Example 4.2 Discuss the system of nonlinear differential equations
(4.2)
Choose , . Then

So (A 3) and (A 4) hold. Thus, by Theorem 3.2, the boundary value problem (4.2) has a positive solution.

Declarations

Acknowledgements

This research is supported by the Natural Science Foundation of China (11071143, 60904024, 11026112), China Postdoctoral Science Foundation funded project (200902564), and supported by Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003, Y2008A28), also supported by University of Jinan Research Funds for Doctors (XBS0843) and University of Jinan Innovation Funds for Graduate Students (YCX09014).

Authors’ Affiliations

(1)
School of Science, University of Jinan, Jinan, 250022, Shandong, PR China
(2)
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MI 65409-0020, USA
(3)
School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, PR China

References

  1. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.Google Scholar
  2. Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.Google Scholar
  3. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York, Lindon, Toronto; 1999.Google Scholar
  4. Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivative. In Theory and Applications. Gordon and Breach, Yverdon; 1993.Google Scholar
  5. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problem. I Appl Anal 2001, 78: 153-192.MathSciNetView ArticleMATHGoogle Scholar
  6. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems. II Appl Anal 2002, 81: 435-493.MathSciNetView ArticleMATHGoogle Scholar
  7. Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J Math Anal Appl 1996, 204: 609-625. 10.1006/jmaa.1996.0456MathSciNetView ArticleMATHGoogle Scholar
  8. Li Q, Sun S: On the existence of positive solutions for initial value problem to a class of fractional differential equation. In Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation. World Academic Press Chongqing, II; 2010:886-889.Google Scholar
  9. Li Q, Sun S, Zhang M, Zhao Y: On the existence and uniqueness of solutions for initial value problem of fractional differential equations. J Univ Jinan 2010, 24: 312-315.Google Scholar
  10. Li Q, Sun S, Han Z, Zhao Y: On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations. 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao 2010, 452-457.View ArticleGoogle Scholar
  11. Zhang S: The existence of a positive solution for nonlinear fractional differential equation. J Math Anal Appl 2000, 252: 804-812. 10.1006/jmaa.2000.7123MathSciNetView ArticleMATHGoogle Scholar
  12. Jafari H, Gejji VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl Math Comput 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MathSciNetView ArticleMATHGoogle Scholar
  13. Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J Math Anal Appl 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
  14. Qiu T, Bai Z: Existence of positive solutions for singular fractional equations. Electron J Differ Equ 2008, 146: 1-9.MathSciNetMATHGoogle Scholar
  15. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron J Differ Equ 2006, 36: 1-12.View ArticleMathSciNetGoogle Scholar
  16. Zhang M, Sun S, Zhao Y, Yang D: Existence of positive solutions for boundary value problems of fractional differential equations. J Univ Jinan 2010, 24: 205-208.Google Scholar
  17. Zhao Y, Sun S: On the existence of positive solutions for boundary value problems of non-linear fractional differential equations. In Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation. World Academic Press Chongqing, II; 2010:682-685.Google Scholar
  18. Zhao Y, Sun S, Han Z, Zhang M: Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations. 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao 2010, 480-485.View ArticleGoogle Scholar
  19. Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun Nonlinear Sci Numer Simul 2011, 16: 2086-2097. 10.1016/j.cnsns.2010.08.017MathSciNetView ArticleMATHGoogle Scholar
  20. Yu Y, Jiang D: Multiple Positive Solutions for the Boundary Value Problem of A Nonlinear Fractional Differential Equation. Northeast Normal University; 2009.Google Scholar
  21. Xu X, Jiang D, Yuan C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal 2009, 71: 4676-4688. 10.1016/j.na.2009.03.030MathSciNetView ArticleMATHGoogle Scholar
  22. Ladaci S, Loiseau JL, Charef A: Fractional order adaptive high-gain controllers for a class of linear systems. Commun Nonlinear Sci Numer Simul 2008, 13: 707-714. 10.1016/j.cnsns.2006.06.009MathSciNetView ArticleMATHGoogle Scholar
  23. Yang A, Ge W: Positive solutions for boundary value problems of N-Dimension nonlinear fractional differential system. Bound Value Probl 2008, 2008: 1-15.MathSciNetView ArticleGoogle Scholar
  24. Bai C, Fang J: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl Math Comput 2004, 150: 611-621. 10.1016/S0096-3003(03)00294-7MathSciNetView ArticleMATHGoogle Scholar
  25. Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl Math Lett 2009, 22: 64-69. 10.1016/j.aml.2008.03.001MathSciNetView ArticleMATHGoogle Scholar
  26. Gejji VD: Positive solutions of a system of nonlinear fractional differential equations. J Math Anal Appl 2005, 302: 56-64. 10.1016/j.jmaa.2004.08.007MathSciNetView ArticleMATHGoogle Scholar
  27. Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int J Differ Equ 2010, 2010: 1-12.MathSciNetView ArticleMATHGoogle Scholar
  28. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput Math Appl 2009, 58: 1838-1843. 10.1016/j.camwa.2009.07.091MathSciNetView ArticleMATHGoogle Scholar
  29. Ahmad B, Alsaedi A: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl 2010, 2010: 1-17.MathSciNetView ArticleMATHGoogle Scholar
  30. Babakhani A: Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay. Abstr Appl Anal 2010, 2010: 1-16.MathSciNetView ArticleMATHGoogle Scholar
  31. Kilbas AA, Srivastava HH, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
  32. Krasnoselskii MA: Positive Solution of Operator Equation. Noordho, Groningen 1964.Google Scholar
  33. Kang P, Wei Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal 2009, 70: 444-451. 10.1016/j.na.2007.12.014MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Zhao et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement