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El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47

http://www.advancesindifferenceequations.com/content/2011/1/47

RESEARCH Open Access

Stability of a nonlinear non-autonomous

fractional order systems with different delays and

non-local conditions

1* 2Ahmed El-Sayed and Fatma Gaafar

* Correspondence: Abstract

amasayed5@yahoo.com

1

Faculty of Science, Alexandria In this paper, we establish sufficient conditions for the existence of a unique solution

University, Alexandria, Egypt for a class of nonlinear non-autonomous system of Riemann-Liouville fractional

Full list of author information is

differential systems with different constant delays and non-local condition is. Theavailable at the end of the article

stability of the solution will be proved. As an application, we also give some

examples to demonstrate our results.

Keywords: Riemann-Liouville derivatives, nonlocal non-autonomous system,

timedelay system, stability analysis

1 Introduction

Here we consider the nonlinear non-local problem of the form

αD x (t) = f (t,x (t),...,x (t)) +g (t,x (t −r ),...,x (t −r )),t ∈ (0,T), T < ∞, (1)i i 1 n i 1 1 n n

x(t)= (t)for t < 0 and lim (t)=0, (2)

−t→0

1−αI x(t)| =0, (3)t=0

awhere D denotes the Riemann-Liouville fractional derivative of order aÎ (0, 1), x(t)

=(x (t), x (t), ..., x (t))’,where ‘ denote the transpose of the matrix, and f, g :[0, T]×1 2 n i i

nR ® R are continuous functions,F(t)=(j(t)) are given matrix and O is the zeroi n×1

matrix, r ≥ 0, j = 1, 2, ..., n, are constant delays.j

Recently, much attention has been paid to the existence of solution for fractional

differential equations because they have applications in various fields of science and

engineering. We can describe many physical and chemical processes, biological systems,

etc., by fractional differential equations (see [1-9] and references therein).

In this work, we discuss the existence, uniqueness and uniform of the solution of

stability non-local problem (1)-(3). Furthermore, as an application, we give some

examples to demonstrate our results.

For the earlier work we mention: De la Sen [10] investigated the non-negative

solution and the stability and asymptotic properties of the solution of fractional differential

dynamic systems involving delayed dynamics with point delays.

© 2011 El-Sayed and Gaafar; licensee Springer. 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.El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 2 of 8

http://www.advancesindifferenceequations.com/content/2011/1/47

El-Sayed [11] proved the existence and uniqueness of the solution

u(t)oftheproblem

c α c βD u(t)+C D u(t −r)= Au(t)+Bu(t −r), 0 ≤ β ≤ α ≤ 1,a a

u(t) = g(t), t ∈ [a −r,a], r > 0

by the method of steps, where A, B, C are bounded linear operators defined on a

Banach space X.

El-Sayed et al. [12] proved the existence of a unique uniformly stable solution of the

non-local problem

n n

αD x (t)= a (t)x (t)+ b (t)x (t −r)+h (t), t > 0,i ij j ij j j i

j=1 j=1

βx(t)= (t)for t < 0, lim (t)= O and I x(t)| = O, β ∈ (0,1).t=0

−t→0

Sabatier et al. [6] delt with Linear Matrix Inequality (LMI) stability conditions for

fractional order systems, under commensurate order hypothesis.

Abd El-Salam and El-Sayed [13] proved the existence of a unique uniformly stable

solution for the non-autonomous system

c α 0D x(t)= A(t)x(t)+f(t), x(0) = x , t > 0,a

c αwhere D is the Caputo fractional derivatives (see [5-7,14]), A(t)and f(t) are contin-a

uous matrices.

Bonnet et al. [15] analyzed several properties linked to the robust control of

fractional differential systems with delays. They delt with the BIBO stability of both

retarded and neutral fractional delay systems. Zhang [16] established the existence of a

unique solution for the delay fractional differential equation

αD x(t) = A x(t)+A x(t −r)+f(t), t > 0, x(t) = φ(t), t ∈ [−r,0],0 1

by the method of steps, where A , A are constant matrices and studied the finite0 1

time stability for it.

2 Preliminaries

Let L [a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a1

b<b < ∞ with the norm .||x|| = |x(t)|dtL1 a

Definition 1 The fractional (arbitrary) order integral of the function f(t)Î L [a, b]of1

+order aÎ R is defined by (see [5-7,14,17])

α−1t (t −s)αI f(t)= f(s)ds,a (α)a

where Γ (.) is the gamma function.

Definition 2 The Caputo fractional (arbitrary) order derivatives of order a, n <a <n

+ 1, of the function f(t) is defined by (see [5-7,14]),

t1c α n−α n n−α−1D f(t)= I D f(t)= (t −s) f(s)ds, t ∈ [a,b],a a (n − α) aEl-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 3 of 8

http://www.advancesindifferenceequations.com/content/2011/1/47

Definition 3 The Riemann-liouville fractional (arbitrary) order derivatives of order a,

n <a <n + 1 of the function f (t) is defined by (see [5-7,14,17])

n n td 1 dα n−α n−α−1D f(t)= I f(t)= (t −s) f(s)ds, t ∈ [a,b],a an ndt (n − α) dt a

The following theorem on the properties of fractional order integration and

differentiation can be easily proved.

+

Theorem 1 Let a, bÎ R . Then we have

α β α+βα(i) I : L → L , and if f(t)Î L then .1 1 I I f(t)= I f(t)a 1 a aa

α nlimI = I(ii) , n = 1,2,3,... uniformly.a aα→n

−α(t −a)c α α(iii) D f(t)= D f(t) − f(a), aÎ (0,1), f (t) is absolutely continuous.

(1 − α)

df

c α α(iv) , aÎ (0,1), f (t) is absolutely continuous.lim D f(t)= = lim D f(t)a

α→1 dt α→1

3 Existence and uniqueness

Let X=(C (I), || . || ), where C (I) is the class of all continuous column n-vectorsn 1 n

n −Ntfunction. For xÎ C [0, T], the norm is defined by ||x|| = sup {e |x (t)|},1 in t∈[0,T]i=1

where N>0.

n

Theorem 2 Let f , g :[0, T]× R ® R be continuous functions and satisfy thei i

Lipschitz conditions

n

|f (t,u ,...,u ) −f (t,v ,...,v ) ≤ h |u −v |,i 1 n i 1 n ij j j

j=1

n

|g (t,u ,...,u ) −g (t,v ,...,v )|≤ k |u −v |,i 1 n i 1 n ij j j

j=1

n n n nand h = |h | = max |h |, k = |k | = max |k |.i ∀j ij i ∀j iji=1 i=1 i=1 i=1

Then there exists a unique solution ×Î X of the problem (1)-(3).

Proof Let tÎ (0, T). Then equation (1) can be written as

d 1−αI x (t)= f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ).i i 1 n i 1 1 n n

dt

Integrating both sides, we obtain

t

1−α 1−αI x (t)−I x (t)| = {f (t,x (t),...,x (t))+g (t,x (t−r ),...,x (t−r ))}ds.i i t=0 i 1 n i 1 1 n n

0

From (3), we get

t

1−αI x (t)= {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}ds.i i 1 n i 1 1 n n

0

a

Operating by I on both sides, we obtain

α+1Ix (t) = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}.i i 1 n i 1 1 n n

El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 4 of 8

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Differentiating both side is, we get

αx (t) = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}, i=1,2,...,n. (4)i i 1 n i 1 1 n n

Now let F : X® X, defined by

αFx = I {f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}.i i 1 n i 1 1 n n

then

α|Fx −Fy | = |I {f (t,x (t),...,x (t)) −f (t,y (t),...,y (t))i i i 1 n i 1 n

+g (t,x (t −r ),...,x (t −r )) −g (t,y (t −r ),...,y (t −r ))}|i 1 1 n n i 1 1 n n

α−1t (t −s)

≤ |f (s,x (s),...,x (s)) −f (s,y (s),...,y (s))|dsi 1 n i 1 n

(α)0

α−1t (t −s)

+ |g (s,x (s −r ),...,x (s −r )) −g (s,y (s −r ),...,y (s −r ))|dsi 1 1 n n i 1 1 n n

(α)0

α−1 nt (t −s)

≤ h |x (s) −y (s)|dsij j j

(α)0

j=1

nt α−1(t −s)

+ k |x (s −r ) −y (s −r )|dsij j j j j

(α)0 j=1

and

n t α−1 (t −s)−Nt −N(t−s) −Nse |Fx −Fy|≤ h e |x (s) −y (s)|dsei i i j j

(α)0j=1

n α−1t (t −s) −N(t−s+r ) −N(s−r )j j+k e e |x (s −r ) −y (s −r)|dsi j j j j

(α)rjj=1

n t α−1 (t −s)−Nt −N(t−s)≤ h sup{e |x (t) −y (t)|} e dsi j j

(α)t 0j=1

n α−1t (t −s)−Nt −Nr −N(t−s)j+k sup{e |x (t) −y (t)|}e e dsi j j

(α)t rjj=1

n Nt α−1 −u 1 u e−Nt≤ h sup{e |x (t) −y (t)|} dui j j αN (α)t 0j=1

n −Nr N(t−r ) α−1 −uj je u e−Nt+k sup{e |x (t) −y (t)|} dui j j αN (α)t 0j=1

nh ki i −Nt≤ ||x −y|| + sup{e |x (t) −y (t)|}1 j jα αN N t

j=1

h +ki i

≤ ||x −y||1αN

and

n n h +ki i−Nt||Fx −Fy|| = supe |Fx −Fy|≤ ||x −y||1 i i 1αNti=1 i=1

h+k

≤ ||x −y|| .1αN

h+kNow choose Nlargeenoughsuchthat < 1,sothemap F : X® X is a contrac-αN

tion and hence, there exists a unique column vector xÎ X which is the solution of the

integral equation (4).El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 5 of 8

http://www.advancesindifferenceequations.com/content/2011/1/47

Now we complete the proof by proving the equivalence between the integral

equation (4) and the non-local problem (1)-(3). Indeed:

1-a 1-a 1-aSince xÎ C and I x(t)Î C (I), and f, g Î C(I)then I f(t), I g(t)Î C(I).n n i i i i

1-aOperating by I on both sides of (4), we get

1−α 1−α αI x (t)= I I {f (t,x (t),...,x (t)) +g (t,x (t −r ),...,x (t −r ))}i i 1 n i 1 1 n n

= I{f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))}.i 1 n i 1 1 n n

Differentiating both sides, we obtain

1−αDI x (t) = DI{f (t,x (t),...,x (t))+g (t,x (t −r ),...,x (t −r ))},i i 1 n i 1 1 n n

which implies that

αD x (t) = f (t,x (t),...,x (t)) +g (t,x (t −r ),...,x (t −r )), t > 0,i i 1 n i 1 1 n n

which completes the proof of the equivalence between (4) and (1).

Now we prove that + .Since f(t, x (t), ..., x (t)), g(t, x (t-r ), ..., x (t-lim x =0t→0 i i 1 n i 1 1 n

r )) are continuous on [0, T] then there exist constants l, L, m, M such that l ≤ f(t,n i i i i i i

x (t), ..., x (t)) ≤ L and m ≤ g(t, x (t - r ) ), ..., x (t-r )) ≤ M, and we have1 n i i i 1 1 n n i

α−1t (t −s)αI f (t,x (t),...,x (t)) = f (s,x (s),...,x (s))ds,i 1 n i 1 n

(α)0

which implies

t α−1 t α−1(t −s) (t −s)αl ds ≤ I f (t,x (t),...,x (t)) ≤ L ds ⇒i i 1 n i

(α) (α)0 0

α αl t L ti iα≤ I f (t,x (t),...,x (t)) ≤i 1 n

(α+1) (α+1)

and

αlim I f (t,x (t),...,x (t)) = 0.i 1 n

+t→0

Similarly, we can prove

αlim I g (t,x (t −r ),...,x (t −r )) = 0.i 1 1 n n

+t→0

Then from (4),lim + x (t) = 0. Also from (2), we have lim − (t) = O.t→0 i t→0

Now for tÎ (-∞, T], T < ∞, the continuous solution x(t)Î (-∞, T] of the problem

(1)-(3) takes the form

⎧

⎪φ (t), t < 0i⎨

0, t=0x (t)=i α−1⎪ t (t−s)⎩ {f (s,x (s),...,x (s))+g (s,x (s −r ),...,x (s −r ))}ds, t > 0.i 1 n i 1 1 n n0 (α)

4 Stability

In this section we study the stability of the solution of the non-local problem (1)-(3)

Definition 5 The solution of the non-autonomous linear system (1) is stable if for

any ε > 0, there exists δ > 0 such that for any two solutions x(t)=(x (t), x (t), ..., x (t))’1 2 n

and x˜(t) = (x˜ (t),x˜ (t),...,x˜ (t)) with the initial conditions (2)-(3) and1 2 nEl-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 6 of 8

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˜||x(t) −x˜(t)|| < ε ||x(t) −x˜(t)|| < εrespectively, one has ||(t) − (t)|| ≤ δ,then1 11

for all t ≥ 0.

Theorem 3 The solution of the problem (1)-(3) is uniformly stable.

Proof Let x(t)and x˜(t) be two solutions of the system (1) under conditions (2)-(3)

β ˜ ˜and , respectively. Then for t>0,{I x˜(t)| =0,x˜(t) = (t),t < 0 andlim (t) = O}t=0 t→0

we have from (4)

α|x −x˜ | = |I {f (t,x (t),...,x (t)) −f (t,x˜ (t),...,x˜ (t))i i i 1 n i 1 n

+g (t,x (t −r ),...,x (t −r )) −g (t,x˜ (t −r ),...,x˜ (t −r ))}|i 1 1 n n i 1 1 n n

α−1t (t −s)

≤ | f (s,x (s),...,x (s)) −f (s,y (s),...,y (s))|dsi 1 n i 1 n(α)0

α−1t (t −s)

+ |g (s,x (s −r ),...,x (s −r )) −g (s,x˜ (s −r ),...,x˜ (s −r ))|dsi 1 1 n n i 1 1 n n(α)0

nt α−1(t −s)

≤ h |x (s) −x˜ (s)|dsij j j

(α)0

j=1

nt α−1(t −s)

+ k |x (s −r ) −x˜ (s −r )|dsij j j j j

(α)0

j=1

and

n t α−1 (t −s)−Nt −N(t−s) −Nse |x −x˜ |≤ h e e |x (s) −x˜ (s)|dsi i i j j

(α)0j=1

n α−1r j (t −s) −N(t−s+r ) −N(s−r )j j ˜+k e e |φ (s −r ) − φ (s −r)|dsi j j j j

(α)0j=1

n α−1t (t −s) −N(t−s+r ) −N(s−r )j j+k e e |x (s −r ) −x˜ (s −r )|dsi j j j j

(α)rjj=1

Nt α−1 −uh u ei

≤ ||x (t) −x˜ (t)|| duj j 1αN (α)0

n Nt−Nr α−1 −u je u e−Nt ˜+k sup{e |φ (t) − φ(t)|} dui j j αN (α)t N(t−r )jj=1

n N(t−r )−Nr j α−1 −u je u e−Nt+k sup{e |x (t) −x˜ (t)|} dui j j αN (α)t 0j=1

nh ki i −Nr −Ntj≤ ||x (t) −x˜ (t)|| + e sup{e |x (t) −x˜ (t)|}j j 1 j jα αN N t

j=1

nki −Nr −Ntj ˜+ e sup{e |ϕ (t) − φ (t)|}j jαN t

j=1

h +k ki i i ˜˜≤ ||x −x|| + || − || .1 1α αN N

Then we have,

n n h +k ki i i ˜||x −x˜|| ≤ ||x −x˜|| + || − ||1 1 1α αN N

i=1 i=1

h+k k

˜≤ ||x −x˜|| + || − ||1 1α αN N

−1h+k k k h+k˜i.e. 1 − ||x −x˜|| ≤ || − || ˜and ||x −x˜|| ≤ 1 − || − ||1 1 1 1α α α αN N N NEl-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47 Page 7 of 8

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−1

k h+k˜Therefore, for δ >0 s.t. , we can find s.t.|| − || < δ ε = 1 − δ1 α αN N

˜ which proves that the solution x(t) is uniformly stable.||x −x|| ≤ ε1

5 Applications

Example 1 Consider the problem

n n

αD x (t)= a (t)x (t)+ g (t,x (t −r), t > 0i ij j ij j j

j=1 j=1

x(t)= (t)fort < 0and lim (t)= O

−t→0

1−αI x(t)| = O,t=0

n (g (t,x (t −r ),...,x (t −r ))) =( g (t,x (t −r ))where A(t)=(a (t)) andij n×n i 1 1 n n ij j jj=1

are given continuous matrix, then the problem has a unique uniformly stable solution

xÎ X on (-∞, T], T < ∞

Example 2 Consider the problem

n n

αD x (t)= f (t,x (t))+ b (t)x (t −r ), t > 0i ij j ij j j

j=1 j=1

x(t)= (t)for t < 0and lim (t)= O

−t→0

1−αI x(t)| = O,t=0

n (f (t,x (t),...,x (t))) =( f (t,x (t)))where B(t)=(b (t)) , and i 1 n ij j are given con-ij n×n j=1

tinuous matrices, then the problem has a unique uniformly stable solution xÎ X on

(-∞, T], T < ∞

Example 3 Consider the problem (see [12])

n n

αD x (t)= a (t)x (t)+ b (t)x (t −r)+h (t), t > 0i ij j ij j j i

j=1 j=1

x(t)= (t)for t < 0and lim (t)= O

−t→0

1−αI x(t)| = O,t=0

where A(t)=(a (t)) B(t)=(b (t)) ,and H(t)=(h(t)) are given continuousij n×n ij n×n i n×1

matrices, then the problem has a unique uniformly stable solution xÎ X on (-∞, T], T

< ∞.

Author details

1 2Faculty of Science, Alexandria University, Alexandria, Egypt Faculty of Science, Damanhour University, Damanhour,

Egypt

Authors’ contributions section

All authors contributed equally to the manuscript and read and approved the final draft.

Competing interests

The authors declare that they have no competing interests.

Received: 1 March 2011 Accepted: 27 October 2011 Published: 27 October 2011

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doi:10.1186/1687-1847-2011-47

Cite this article as: El-Sayed and Gaafar: Stability of a nonlinear non-autonomous fractional order systems with

different delays and non-local conditions. Advances in Difference Equations 2011 2011:47.

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