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Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System

Abstract

Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.

1. Introduction

In the past decades, fractional differential equations have been intensively studied. This is due to the rapid development of the theory of fractional differential equations itself and the applications of such construction in various sciences such as physics, mechanics, chemistry, and engineering [1, 2]. For the basic theory of fractional differential equations, we refer the readers to [3–7].

Recently, many researchers have devoted their attention to studying the existence of (positive) solutions of boundary value problems for differential equations with fractional order [8–23]. We mention that the fractional order involved is generally in with the exception that in [12, 23] and in [8, 17]. Though there have been extensive study on systems of fractional differential equations, not much has been done for boundary value problems for systems of fractional differential equations [18–20].

On the other hand, we know that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays [24–31].

As a result, in this paper, we consider the following nonlinear system of fractional order differential equations with delays,

(1.1)

where is the standard Riemann-Liouville fractional derivative of order   for some integer , for , for , and is a nonlinear function from to . The purpose is to establish sufficient conditions on the existence of positive solutions to (1.1) by using some fixed point theorems and some properties of the Green function. By a positive solution to (1.1) we mean a mapping with positive components on such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional differential equations when for all and . Therefore, the obtained results generalize and include some existing ones.

The remaining part of this paper is organized as follows. In Section 2, we introduce some basics of fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4.

2. Preliminaries

We first introduce some basic definitions of fractional derivative for the readers' convenience.

Definition 2.1 (see [3, 32]).

The fractional integral of order   of a function is defined as

(2.1)

provided that the integral exists on , where is the Gamma function.

Note that has the semigroup property, that is,

(2.2)

Definition 2.2 (see [3, 32]).

The Riemann-Liouville derivative of order of a function is given by

(2.3)

provided that the right-hand side is pointwise defined on , where .

It is well known that if then . Furthermore, if and then for .

The following results on fractional integral and fractional derivative will be needed in establishing our main results.

Lemma 2.3 (see [10]).

Let . Then solutions to the fractional equation can be written as

(2.4)

where , .

Lemma 2.4 (see [10]).

Let . Then

(2.5)

for some , .

Now, we cite the fixed point theorems to be used in Section 3.

Lemma 2.5 (the Banach contraction mapping theorem [33]).

Let be a complete metric space and let be a contraction mapping. Then has a unique fixed point.

Lemma 2.6 (see [16, 34]).

Let be a closed and convex subset of a Banach space . Assume that is a relatively open subset of with and is completely continuous. Then at least one of the following two properties holds:

(i) has a fixed point in ;

(ii)there exists and with .

Lemma 2.7 (the Krasnosel'skii fixed point theorem [33, 35]).

Let be a cone in a Banach space . Assume that and are open subsets of with and . Suppose that is a completely continuous operator such that either

(i) for and for

or

(ii) for and for .

Then has a fixed point in .

3. Existence of Positive Solutions

Throughout this paper, we let . Then is a Banach space, where

(3.1)

In this section, we always assume that .

Lemma 3.1.

System (1.1) is equivalent to the following system of integral equations:

(3.2)

where

(3.3)

Proof.

It is easy to see that if satisfies (3.2) then it also satisfies (3.2). So, assume that is a solution to (1.1). Integrating both sides of the first equation of (1.1) of order with respect to gives us

(3.4)

for , . It follows that

(3.5)

for , . This, combined with the boundary conditions in (1.1), yields

(3.6)

Similarly, one can obtain

(3.7)
(3.8)

  . Then it follows from (3.8) and the boundary condition that

(3.9)

Therefore, for ,

(3.10)

This completes the proof.

The following two results give some properties of the Green functions .

Lemma 3.2.

For is continuous on and for .

Proof.

Obviously, is continuous on . It remains to show that for . It is easy to see that for . We only need to show that for . For , let

(3.11)
(3.12)

Then

(3.13)

Note that and for . It follows that and hence for .

Therefore, for and the proof is complete.

Lemma 3.3.

  1. (i)

    If , then for .

  2. (ii)

    If , then for .

Proof.

  1. (i)

    Obviously, for . Now, for , we have

    (3.14)

where is the function defined by (3.11). It follows that for . In summary, we have proved (i).

  1. (ii)

    Again, one can easily see that for . When , we have in this case that

    (3.15)

which implies that for . To summarize, we have proved (ii) and this completes the proof.

Now, we are ready to present the main results.

Theorem 3.4.

Suppose that there exist functions , , 2, , such that

(3.16)

for , ,. If

(3.17)
(3.18)

then (1.1) has a unique positive solution.

Proof.

Let

(3.19)

It is easy to see that is a complete metric space. Define an operator on by

(3.20)

where and

(3.21)

Because of the continuity of and , it follows easily from Lemma 3.2 that maps into itself. To finish the proof, we only need to show that is a contraction. Indeed, for , by (3.16) we have

(3.22)

This, combined with Lemma 3.3 and (3.17) and (3.18), immediately implies that is a contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5.

The following result can be proved in the same spirit as that for Theorem 3.4.

Theorem 3.5.

For , suppose that there exist nonnegative function and nonnegative constants such that and

(3.23)

for , . If

(3.24)

then (1.1) has a unique positive solution.

Theorem 3.6.

For , suppose that there exist nonnegative real-valued functions such that

(3.25)

for almost every and all . If

(3.26)

then (1.1) has at least one positive solution.

Proof.

Let and be defined by (3.19) and (3.20), respectively. We first show that is completely continuous through the following three steps.

Step 1.

Show that is continuous. Let be a sequence in such that . Then is bounded in . Since is continuous, it is uniformly continuous on any compact set. In particular, for any , there exists a positive integer such that

(3.27)

for and . Then, for , we have

(3.28)

for and . Therefore,

(3.29)

which implies that is continuous.

Step 2.

Show that maps bounded sets of into bounded sets. Let be a bounded subset of . Then is bounded. Since is continuous, there exists an such that

(3.30)

It follows that, for and ,

(3.31)

Immediately, we can easily see that is a bounded subset of .

Step 3.

Show that maps bounded sets of into equicontinuous sets. Let be a bounded subset of . Similarly as in Step 2, there exists such that

(3.32)

Then, for any and and ,

(3.33)

Now the equicontituity of on follows easily from the fact that is continuous and hence uniformly continuous on .

Now we have shown that is completely continuous. To apply Lemma 2.6, let

(3.34)

Fix and define

(3.35)

We claim that there is no such that for some . Otherwise, assume that there exist and such that . Then

(3.36)

If , then

(3.37)

Similarly, we can have if . To summarize, , a contradiction to . This proves the claim. Applying Lemma 2.6, we know that has a fixed point in , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete.

As a consequence of Theorem 3.6, we have the following.

Corollary 3.7.

If all , , are bounded, then (1.1) has at least one positive solution.

To state the last result of this section, we introduce

(3.38)

Theorem 3.8.

Suppose that there exist and positive constants with such that

(i)  for

and

(ii), for ,

where . Then (1.1) has at least a positive solution.

Proof.

Let be defined by (3.19) and . Obviously, is a cone in . From the proof of Theorem 3.6, we know that the operator defined by (3.20) is completely continuous on . For any , it follows from Lemma 3.3 and condition (ii) that

(3.39)

that is,

(3.40)

On the other hand, for any , it follows from Lemma 3.3 and condition (i) that, for ,

(3.41)

if , whereas

(3.42)

if . In summary,

(3.43)

Therefore, we have verified condition (ii) of Lemma 2.7. It follows that has a fixed point in , which is a positive solution to (1.1). This completes the proof.

4. Examples

In this section, we demonstrate the feasibility of some of the results obtained in Section 3.

Example 4.1.

Consider

(4.1)

Here

(4.2)

One can easily see that (3.16) is satisfied with

(4.3)

Moreover,

(4.4)

and hence

(4.5)

It follows from Theorem 3.4 that (4.1) has a unique positive solution on .

Example 4.2.

Consider

(4.6)

Here

(4.7)

where

(4.8)

Hence, and satisfy (3.25). Moreover, simple calculations give us

(4.9)

Then and

(4.10)

Choose , and

(4.11)

Then, for and , we have

(4.12)

for and , we have

(4.13)

By now we have verified all the assumptions of Theorem 3.8. Therefore, (4.6) has at least one positive solution satisfying .

References

  1. Debnath L: Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003, (54):3413-3442.

    Article  Google Scholar 

  2. Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007.

    Book  Google Scholar 

  3. 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 

  4. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3337-3343. 10.1016/j.na.2007.09.025

    Article  MathSciNet  MATH  Google Scholar 

  5. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge, UK; 2009.

    MATH  Google Scholar 

  6. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2677-2682. 10.1016/j.na.2007.08.042

    Article  MathSciNet  MATH  Google Scholar 

  7. Podlubny I: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.

    Google Scholar 

  8. Bai C: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (24):-10.

    Google Scholar 

  9. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(2):916-924. 10.1016/j.na.2009.07.033

    Article  MathSciNet  MATH  Google Scholar 

  10. 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.2005.02.052

    Article  MathSciNet  MATH  Google Scholar 

  11. Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7-8):2391-2396. 10.1016/j.na.2009.01.073

    Article  MathSciNet  MATH  Google Scholar 

  12. El-Shahed M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstract and Applied Analysis 2007, 2007:-8.

    Google Scholar 

  13. Jafari H, Daftardar-Gejji V: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Applied Mathematics and Computation 2006, 180(2):700-706. 10.1016/j.amc.2006.01.007

    Article  MathSciNet  MATH  Google Scholar 

  14. Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (3):-11.

    MathSciNet  Google Scholar 

  15. Kosmatov N: A singular boundary value problem for nonlinear differential equations of fractional order. Journal of Applied Mathematics and Computing 2009, 29(1-2):125-135. 10.1007/s12190-008-0104-x

    Article  MathSciNet  MATH  Google Scholar 

  16. Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers & Mathematics with Applications 2010, 59(3):1363-1375.

    Article  MathSciNet  MATH  Google Scholar 

  17. Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(11):5545-5550. 10.1016/j.na.2009.04.045

    Article  MathSciNet  MATH  Google Scholar 

  18. 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.2008.03.001

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. International Journal of Differential Equations 2010, 2010:-12.

    Google Scholar 

  20. Yang A, Ge W: Positive solutions for boundary value problems of N -dimension nonlinear fractional differential system. Boundary Value Problems 2008, 2008:-15.

    Google Scholar 

  21. Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia B 2006, 26(2):220-228. 10.1016/S0252-9602(06)60044-1

    Article  MATH  Google Scholar 

  22. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, (36):-12.

    Google Scholar 

  23. Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Communications in Nonlinear Science and Numerical Simulation 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017

    Article  MathSciNet  MATH  Google Scholar 

  24. Babakhani A: Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay. Abstract and Applied Analysis 2010, 2010:-16.

    Google Scholar 

  25. Babakhani A, Enteghami E: Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients. Abstract and Applied Analysis 2009, 2009:-12.

    Google Scholar 

  26. Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008, 338(2):1340-1350. 10.1016/j.jmaa.2007.06.021

    Article  MathSciNet  MATH  Google Scholar 

  27. Deng W, Li C, Lü J: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 2007, 48(4):409-416. 10.1007/s11071-006-9094-0

    Article  MathSciNet  MATH  Google Scholar 

  28. Hu L, Ren Y, Sakthivel R: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 2009, 79(3):507-514. 10.1007/s00233-009-9164-y

    Article  MathSciNet  MATH  Google Scholar 

  29. Maraaba TA, Jarad F, Baleanu D: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Science in China. Series A 2008, 51(10):1775-1786. 10.1007/s11425-008-0068-1

    Article  MathSciNet  MATH  Google Scholar 

  30. Mophou GM, N'Guérékata GM: A note on a semilinear fractional differential equation of neutral type with infinite delay. Advances in Difference Equations 2010, 2010:-8.

    Google Scholar 

  31. Zhang X: Some results of linear fractional order time-delay system. Applied Mathematics and Computation 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069

    Article  MathSciNet  MATH  Google Scholar 

  32. Podlubny I: Fractional Differential Equations. Academic Press, San Diego, Calif, USA; 1993.

    Google Scholar 

  33. Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.

    Book  Google Scholar 

  34. Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. Journal de Mathématiques Pures et Appliquées 1991, 70(2):153-196.

    MathSciNet  MATH  Google Scholar 

  35. Krasnosel'skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan, New York, NY, USA; 1964:xi + 395.

    Google Scholar 

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Acknowledgment

Supported partially by the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, the Foundation of Science and Technology Department of Hunan Province under Grant no. 2009RS3019 and the Subject Lead Foundation of University of South China no. 2007XQD13. Research was partially supported by the Natural Science and Engineering Re-search Council of Canada (NSERC) and the Early Researcher Award (ERA) Pro-gram of Ontario.

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Ouyang, Z., Chen, Y. & Zou, S. Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System. Bound Value Probl 2011, 475126 (2011). https://doi.org/10.1155/2011/475126

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