Open Access

Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions

Advances in Difference Equations20132013:43

https://doi.org/10.1186/1687-1847-2013-43

Received: 7 July 2012

Accepted: 31 January 2013

Published: 25 February 2013

Abstract

In this article, we establish sufficient conditions for the existence, uniqueness and stability of solutions for nonlinear fractional differential equations with delays and integral boundary conditions.

MSC:34A08, 34A30, 34D20.

Keywords

Riemann-Liouvile derivativesnonlinear fractional differential equationdelayintegral boundary conditionsstability

1 Introduction

Fractional differential equations is a generalization of ordinary differential equations and integration to arbitrary non-integer orders. The origin of fractional calculus goes back to Newton and Leibniz in the seventeenth century. Fractional differential equations appear naturally in a number of fields such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, control theory, etc. An excellent account of the study of fractional differential equations can be found in [111] and the references therein. Boundary value problems for fractional differential equations have been discussed in [1222]. By contrast, the development of stability for solutions of fractional differential equations is a bit slow. El-Sayed, Gaafar and Hamadalla [23] discuss the existence, uniqueness and stability of solutions for the non-local non-autonomous system of fractional order differential equations with delays
D α x i ( t ) = j = 1 n a i j ( t ) x j ( t ) + j = 1 n b i j ( t ) x j ( t r j ) + h i ( t ) , t > 0 ,

where D α denotes the Riemann-Liouville derivative of order α.

We consider nonlinear fractional differential equations with delay and integral boundary conditions of the form
D α x ( t ) = j = 1 n a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) , t > 0 ,
(1.1)
x ( t ) = ϕ ( t ) for  t < 0 and lim t 0 ϕ ( t ) = 0 ,
(1.2)
I 1 α x ( t ) | t = 0 = 0 ,
(1.3)

where α ( 0 , 1 ) , f : R + × R 2 R are continuous functions, a j ( t ) , ϕ ( t ) are given continuous functions, τ j 0 , j = 1 , 2 , , n are constants.

In this article our aim is to show the existence of a unique solution for (1.1)-(1.3) and its uniform stability.

2 Preliminaries

In this section, we introduce notation, definitions and preliminary facts which are used throughout this paper.

Definition 2.1 The fractional integral of order α > 0 of a function f : R + R of order α R + is defined by
I α f ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s ) d s ,

provided the right-hand side exists pointwise on R + . Γ is the gamma function.

For instance, I α f exists for all α > 0 when f C 0 ( R + ) L loc 1 ( R + ) ; note also that when f C 0 ( R 0 + ) , then I α f C 0 ( R 0 + ) and moreover I α f ( 0 ) = 0 .

Definition 2.2 The Riemann-Liouville fractional derivative of order α ( 0 , 1 ) of a function f : R + R is given by
D α f ( t ) = d d t I 1 α f ( t ) = 1 Γ ( 1 α ) d d t 0 t ( t s ) α f ( s ) d s .
Definition 2.3 Let f : R + × R 2 R be continuous functions and satisfy the Lipschitz conditions
| f ( t , x , y j ) f ( t , u , v j ) | k | x u | + k j | y j v j | , k > 0 , k j > 0 , j = 1 , 2 , , n

for all x , y j , u , v j R .

3 Existence of a unique solution for nonlinear fractional differential equations (1.1)-(1.3)

Let X be the class of all continuous functions defined on R + with the norm
x = sup t R + { e N t | x ( t ) | } , x X .
Theorem 3.1 Let f : R × R 2 R be continuous and satisfy the Lipschitz condition: if
j = 1 n a j ( k + k j e N τ j ) N α < 1 ,

where a j = max t R + { | a j ( t ) | } , then nonlinear fractional differential equations (1.1)-(1.3) have a unique positive solution.

Proof For t > 0 , equation (1.1) can be written as
d d t I 1 α x ( t ) = j = 1 n a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) .
Integrating both sides of the above equation, we obtain
I 1 α x ( t ) I 1 α x ( t ) | t = 0 = j = 1 n 0 t a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) d s
then
I 1 α x i ( t ) = j = 1 n 0 t a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) d s .
Applying the operator by I α on both sides,
I x ( t ) = j = 1 n I α + 1 a j ( t ) f ( t , x ( t ) , x ( t τ j ) )
differentiating both sides, we obtain
x ( t ) = j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) .
(3.1)
Now, let F : X X be defined by
F x = j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) .
Then
| F x ( t ) F y ( t ) | = | j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) j = 1 n I α a j ( t ) f ( t , y ( t ) , y ( t τ j ) ) | = | j = 1 n 0 t ( t s ) α 1 Γ ( α ) { a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) a j ( s ) f ( s , y ( s ) , y ( s τ j ) ) } d s | j = 1 n 0 t ( t s ) α 1 Γ ( α ) | a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) a j ( s ) f ( s , y ( s ) , y ( s τ j ) ) | d s j = 1 n a j 0 t ( t s ) α 1 Γ ( α ) { k | x ( s ) y ( s ) | + k j | x ( s τ j ) y ( s τ j ) | } d s j = 1 n a j k 0 t ( t s ) α 1 Γ ( α ) | x ( s ) y ( s ) | d s + j = 1 n a j k j 0 τ j ( t s ) α 1 Γ ( α ) | x ( s τ j ) y ( s τ j ) | d s + j = 1 n a j k j τ j t ( t s ) α 1 Γ ( α ) | x ( s τ j ) y ( s τ j ) | d s .
By conditions (1.2), we have
| F x ( t ) F y ( t ) | j = 1 n a j k 0 t ( t s ) α 1 Γ ( α ) | x ( s ) y ( s ) | d s + j = 1 n a j k j τ j t ( t s ) α 1 Γ ( α ) | x ( s τ j ) y ( s τ j ) | d s
and
e N t | F x ( t ) F y ( t ) | j = 1 n a j k 0 t ( t s ) α 1 Γ ( α ) e N ( t s ) e N s | x ( s ) y ( s ) | d s + j = 1 n a j k j τ j t ( t s ) α 1 Γ ( α ) e N ( t s + τ j ) e N ( s τ j ) | x ( s τ j ) y ( s τ j ) | d s j = 1 n a j k sup t R + { e N t | x ( t ) y ( t ) | } 0 t ( t s ) α 1 Γ ( α ) e N ( t s ) d s + j = 1 n a j k j 0 t τ j ( t θ τ j ) α 1 Γ ( α ) e N ( t θ ) e N θ | x ( θ ) y ( θ ) | d θ j = 1 n a j k sup t R + { e N t | x ( t ) y ( t ) | } 0 N t u α 1 e u Γ ( α ) d u + j = 1 n a j k j sup t R + { e N t | x ( t ) y ( t ) | } 0 t τ ( t θ τ j ) α 1 Γ ( α ) e N ( t θ ) d θ j = 1 n a j k N α x y + j = 1 n a j k j sup t R + { e N t | x ( t ) y ( t ) | } 0 t τ j u α 1 Γ ( α ) e N u e N τ j d u j = 1 n a j k N α x y + j = 1 n a j k j sup t R + { e N t | x ( t ) y ( t ) | } e N τ j N α 0 N ( t τ j ) u α 1 e u Γ ( α ) d u j = 1 n a j k N α x y + j = 1 n a j k j sup t R + { e N t | x ( t ) y ( t ) | } e N τ j N α j = 1 n a j k N α x y + j = 1 n a j k j e N τ j N α sup t R + e N t | x ( t ) y ( t ) | j = 1 n a j ( k + k j e N τ j ) N α x y .

Now, choose N large enough such that j = 1 n a j ( k + k j e N τ j ) N α < 1 . So, the map F : X X is a contraction and it has a fixed point x = F x , and hence there exists a unique x X which is a solution of integral equation (3.1).

We now prove the equivalence between integral equation (3.1) and nonlinear fractional differential equations (1.1)-(1.3). Indeed, since x X and I 1 α x ( t ) C ( X ) , applying the operator I 1 α on both sides of (3.1), we obtain
I 1 α x ( t ) = j = 1 n I 1 α I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) = j = 1 n I a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) .
Differentiating both sides,
D I 1 α x ( t ) = j = 1 n D I a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) ,
we get
D α x ( t ) = j = 1 n a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) , t > 0 ,
which proves the equivalence of (3.1) and (1.1). We want to prove that lim t 0 + x = 0 . Since a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) are continuous on [ 0 , T ] , there exist constants m, M such that m a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) M . We have
I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) = 1 Γ ( α ) 0 t ( t s ) α 1 a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) d s ,
which implies
m 0 t ( t s ) α 1 Γ ( α ) d s I α f ( t , x ( t ) , x ( t τ j ) ) M 0 t ( t s ) α 1 Γ ( α ) d s , n m 0 t ( t s ) α 1 Γ ( α ) d s j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) n M 0 t ( t s ) α 1 Γ ( α ) d s ,
which in turn implies
n m t α Γ ( α + 1 ) j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) n M t α Γ ( α + 1 )
and
lim t 0 + j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) = 0 .

Then from (3.1) lim t 0 + x ( t ) = 0 and from (1.2), we have lim t 0 + ϕ ( t ) = 0 . □

Now, for t ( , T ] , T < , the solution of nonlinear fractional differential equations (1.1)-(1.3) takes the form
x ( t ) = { ϕ ( t ) , t < 0 , 0 , t = 0 , j = 1 n 1 Γ ( α ) 0 t ( t s ) α 1 a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) d s , t > 0 .

4 Stability of a unique solution for nonlinear fractional differential equations (1.1)-(1.3)

In this section, we study the stability of the solution of nonlinear fractional differential equations (1.1)-(1.3).

The x ˜ ( t ) is a solution of the nonlinear fractional differential equations
( P ˜ ) { D α x ( t ) = j = 1 n a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) , t > 0 , x ( t ) = ϕ ˜ ( t ) for  t < 0 and lim t 0 ϕ ˜ ( t ) = 0 , I 1 α x ˜ ( t ) | t = 0 = 0 .

Definition 4.1 The solution of nonlinear fractional differential equation (1.1) is stable if for any ϵ > 0 , there exists δ > 0 such that for any two solutions x ( t ) and x ˜ ( t ) of nonlinear fractional differential equations (1.1)-(1.3) and P ˜ respectively, one has ϕ ( t ) ϕ ˜ ( t ) δ , then x ( t ) x ˜ ( t ) < ϵ for all t 0 .

Theorem 4.2 The solution of nonlinear fractional differential equations (1.1)-(1.3) is uniformly stable.

Proof Let x ( t ) and x ˜ ( t ) be the solutions of nonlinear fractional differential equations (1.1)-(1.3) and P ˜ respectively, then for t > 0 , from (3.1), we have
| x ( t ) x ˜ ( t ) | = | j = 1 n I α a j ( t ) f ( t , x ( t ) , x ( t τ j ) ) j = 1 n I α a j ( t ) f ( t , x ˜ ( t ) , x ˜ ( t τ j ) ) | j = 1 n 0 t ( t s ) α 1 Γ ( α ) | a j ( s ) f ( s , x ( s ) , x ( s τ j ) ) a j ( s ) f ( s , x ˜ ( s ) , x ˜ ( s τ j ) ) | d s j = 1 n a j k 0 t ( t s ) α 1 Γ ( α ) | x ( s ) x ˜ ( s ) | d s + j = 1 n a j k j 0 τ j ( t s ) α 1 Γ ( α ) | ϕ ( s τ j ) ϕ ˜ ( s τ j ) | d s + j = 1 n a j k j τ j t ( t s ) α 1 Γ ( α ) | x ( s τ j ) x ˜ ( s τ j ) | d s
and
e N t | x ( t ) x ˜ ( t ) | j = 1 n a j k 0 t ( t s ) α 1 Γ ( α ) e N ( t s ) e N s | x ( s ) x ˜ ( s ) | d s + j = 1 n a j k j 0 τ j ( t s ) α 1 Γ ( α ) e N ( t s + τ j ) e N ( s τ j ) | ϕ ( s τ j ) ϕ ˜ ( s τ j ) | d s + j = 1 n a j k j τ j t ( t s ) α 1 Γ ( α ) e N ( t s + τ j ) e N ( s τ j ) | x ( s τ j ) x ˜ ( s τ j ) | d s j = 1 n a j k N α x ( t ) x ˜ ( t ) 0 N t u α 1 e u Γ ( α ) d u + j = 1 n a j k j sup t R + { e N t | ϕ ( t ) ϕ ˜ ( t ) | } τ j 0 ( t θ τ j ) α 1 Γ ( α ) e N ( t θ ) d θ + j = 1 n a j k j sup t R + { e N t | x ( t ) x ˜ ( t ) | } 0 t τ j ( t θ τ j ) α 1 Γ ( α ) e N ( t θ ) d θ j = 1 n a j k N α x ( t ) x ˜ ( t ) + j = 1 n a j k j sup t R + { e N t | ϕ ( t ) ϕ ˜ ( t ) | } e N τ j N α N ( t τ j ) N t u α 1 e N u Γ ( α ) d u + j = 1 n a j k j sup t R + { e N t | x ( t ) x ˜ ( t ) | } e N τ j N α 0 N ( t τ ) u α 1 e u Γ ( α ) d u j = 1 n a j k N α x ( t ) x ˜ ( t ) + j = 1 n a j k j e N τ j N α sup t R + { e N t | x ( t ) x ˜ ( t ) | } + j = 1 n a j k j e N τ j N α sup t R + { e N t | ϕ ( t ) ϕ ˜ j ( t ) | } j = 1 n a j ( k + k j e N τ j ) N α x ( t ) x ˜ ( t ) + j = 1 n a j k j e N τ j N α ϕ ( t ) ϕ ˜ ( t ) .
Then
[ 1 j = 1 n a j ( k + k j e N τ j ) N α ] x ( t ) x ˜ ( t ) j = 1 n a j k j e N τ j N α ϕ ( t ) ϕ ˜ ( t )
and
x ( t ) x ˜ ( t ) j = 1 n a j k j e N τ j N α [ 1 j = 1 n a j ( k + k j e N τ j ) N α ] 1 ϕ ( t ) ϕ ˜ ( t ) ;

therefore, for ϵ > 0 , we can find δ = ( j = 1 n a j k j e N τ j N α ) 1 [ 1 j = 1 n a j ( k + k j e N τ j ) N α ] ϵ such that ϕ ( t ) ϕ ˜ ( t ) < δ . Then x ( t ) x ˜ ( t ) ϵ , which proves that the solution x ( t ) is uniformly stable. □

Declarations

Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province (13JJ6068, 12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline in Hunan Province.

Authors’ Affiliations

(1)
Department of Mathematics and Computational Science, Hengyang Normal University

References

  1. Machado JT, Kiryakova V, Mainardi F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153. 10.1016/j.cnsns.2010.05.027MathSciNetView ArticleGoogle Scholar
  2. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. Theory and Applications of Fractional Differential Equations 2006.Google Scholar
  3. Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
  4. Agarwal RP, O’Regan D, Stanek S: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 2012, 285: 27-41. 10.1002/mana.201000043MathSciNetView ArticleGoogle Scholar
  5. Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034MathSciNetView ArticleGoogle Scholar
  6. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.Google Scholar
  7. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
  8. Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives. Gordon & Breach, New York; 1993.Google Scholar
  9. Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
  10. Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin; 2007.View ArticleGoogle Scholar
  11. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.Google Scholar
  12. Yang L, Chen H, Luo L, Luo Z: Successive iteration and positive solutions for boundary value problem of nonlinear fractional q -difference equation. J. Appl. Math. Comput. 2012. doi:10.1007/s12190-012-0622-4Google Scholar
  13. Yang L, Chen H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011. doi:10.1155/2011/404917Google Scholar
  14. 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 ArticleGoogle Scholar
  15. Wang G, Liu W: Existence results for a coupled system of nonlinear fractional 2 m -point boundary value problems at resonance. Adv. Differ. Equ. 2011. doi:10.1186/1687-1847-2011-44Google Scholar
  16. Caballero J, Harjani J, Sadarangani K: Positive solutions for a class of singular fractional boundary-value problems. Comput. Math. Appl. 2011, 62: 1325-1332. 10.1016/j.camwa.2011.04.013MathSciNetView ArticleGoogle Scholar
  17. Staněk S: The existence of positive solutions of singular fractional boundary-value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048MathSciNetView ArticleGoogle Scholar
  18. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36Google Scholar
  19. Ahmad B, Nieto JJ: Existence of solution for non-local boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009., 2009: Article ID 494720Google Scholar
  20. Liu S, Jia M, Tian Y: Existence of positive solutions for boundary-value problems with integral boundary conditions and sign changing nonlinearities. Electron. J. Differ. Equ. 2010., 2010: Article ID 163Google Scholar
  21. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576Google Scholar
  22. Zhang X: Some results of linear fractional order time-delay system. Appl. Math. Comput. 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069MathSciNetView ArticleGoogle Scholar
  23. El-Sayed AMA, Gaafar FM, Hamadalla EMA: Stability for a non-local non-autonomous system of fractional order differential equations with delays. Electron. J. Differ. Equ. 2010., 2010: Article ID 31Google Scholar

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