RESEARCH Open Access

Stability of a nonlinear non-autonomousfractional order systems with different delays andnon-local conditionsAhmed El-Sayed1* and Fatma Gaafar2

* Correspondence:amasayed5@yahoo.com1Faculty of Science, AlexandriaUniversity, Alexandria, EgyptFull list of author information isavailable at the end of the article

Abstract

In this paper, we establish sufficient conditions for the existence of a unique solutionfor a class of nonlinear non-autonomous system of Riemann-Liouville fractionaldifferential systems with different constant delays and non-local condition is. Thestability of the solution will be proved. As an application, we also give someexamples to demonstrate our results.

Keywords: Riemann-Liouville derivatives, nonlocal non-autonomous system, time-delay system, stability analysis

1 IntroductionHere we consider the nonlinear non-local problem of the form

Dαxi(t) = fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn)), t ∈ (0, T), T < ∞, (1)

x(t) = �(t) for t < 0 and limt→0−

�(t) = 0, (2)

I1−αx(t)|t=0 = 0, (3)

where Da denotes the Riemann-Liouville fractional derivative of order a Î (0, 1), x(t)

= (x1(t), x2(t), ..., xn(t))’, where ‘ denote the transpose of the matrix, and fi, gi : [0, T] ×

Rn ® R are continuous functions, F(t) = (ji(t))n × 1 are given matrix and O is the zero

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

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

ferential equations because they have applications in various fields of science and engi-

neering. 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 sta-

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

ples to demonstrate our results.

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

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

dynamic systems involving delayed dynamics with point delays.

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© 2011 El-Sayed and Gaafar; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

El-Sayed [11] proved the existence and uniqueness of the solution u(t) of the pro-

blem

cDαa u(t) + CcDβ

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

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

Dαxi(t) =n∑

j=1

aij(t)xj(t) +n∑

j=1

bij(t)xj(t − rj) + hi(t), t > 0,

x(t) = �(t) for t < 0, limt→0−

�(t) = O and Iβx(t)|t=0 = O, β ∈ (0, 1).

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

cDαa x(t) = A(t)x(t) + f (t), x(0) = x0, t > 0,

where cDαa is the Caputo fractional derivatives (see [5-7,14]), A(t) and f(t) are contin-

uous matrices.

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

tional 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

Dαx(t) = A0x(t) + A1x(t − r) + f (t), t > 0, x(t) = φ(t), t ∈ [−r, 0],

by the method of steps, where A0, A1 are constant matrices and studied the finite

time stability for it.

2 PreliminariesLet L1[a, b] be the space of Lebesgue integrable functions on the interval [a, b], 0 ≤ a

<b < ∞ with the norm ||x||L1 =∫ b

a |x(t)|dt.

Definition 1 The fractional (arbitrary) order integral of the function f(t) Î L1[a, b] of

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

Iαa f (t) =∫ t

a

(t − s)α−1

�(α)f (s)ds,

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]),

cDαa f (t) = In−α

a Dnf (t) =1

�(n− α)

∫ t

a(t − s)n−α−1f (s)ds, t ∈ [a, b],

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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])

Dαa f (t) =

dn

dtnIn−αa f (t) =

1�(n− α)

dn

dtn

∫ t

a(t − s)n−α−1f (s)ds, t ∈ [a, b],

The following theorem on the properties of fractional order integration and differen-

tiation can be easily proved.

Theorem 1 Let a, b Î R+. Then we have

(i) Iαa : L1 → L1, and if f(t) Î L1 then Iαa Iβa f (t) = Iα+βa f (t).

(ii) limα→n

Iαa = Ina, n = 1,2,3,... uniformly.

(iii) cDαf (t) = Dαf (t)− (t − a)−α

�(1− α)f (a), a Î (0,1), f (t) is absolutely continuous.

(iv) limα→1

cDαa f (t) =

dfdt�= lim

α→1Dα f (t), a Î (0,1), f (t) is absolutely continuous.

3 Existence and uniquenessLet X = (Cn(I), || . ||1), where Cn (I) is the class of all continuous column n-vectors

function. For x Î Cn [0, T], the norm is defined by ||x||1 =∑n

i=1 supt∈[0,T]{e−Nt|xi(t)|},where N > 0.

Theorem 2 Let fi, gi : [0, T] × Rn ® R be continuous functions and satisfy the

Lipschitz conditions

|fi(t, u1, . . . , un)− fi(t, v1, . . . , vn) ≤n∑

j=1

hij|uj − vj|,

|gi(t, u1, . . . , un)− gi(t, v1, . . . , vn)| ≤n∑

j=1

kij|uj − vj|,

and h =∑n

i=1 |hi| =∑n

i=1 max∀j|hij|, k =∑n

i=1 |ki| =∑n

i=1 max∀j|kij|.Then there exists a unique solution × Î X of the problem (1)-(3).

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

ddt

I1−αxi(t) = fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn).

Integrating both sides, we obtain

I1−αxi(t)− I1−αxi(t)|t=0 =∫ t

0{fi(t, x1(t), . . . , xn(t))+gi(t, x1(t− r1), . . . , xn(t− rn))}ds.

From (3), we get

I1−αxi(t) =∫ t

0{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}ds.

Operating by Ia on both sides, we obtain

Ixi(t) = Iα+1{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}.

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Differentiating both side is, we get

xi(t) = Iα{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}, i = 1, 2, . . . , n. (4)

Now let F : X ® X, defined by

Fxi = Iα{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}.

then

|Fxi − Fyi| = |Iα{fi(t, x1(t), . . . , xn(t))− fi(t, y1(t), . . . , yn(t))

+ gi(t, x1(t − r1), . . . , xn(t − rn))− gi(t, y1(t − r1), . . . , yn(t − rn))}|

≤∫ t

0

(t − s)α−1

�(α)|fi(s, x1(s), . . . , xn(s))− fi(s, y1(s), . . . , yn(s))|ds

+∫ t

0

(t − s)α−1

�(α)|gi(s, x1(s− r1), . . . , xn(s− rn))− gi(s, y1(s− r1), . . . , yn(s− rn))|ds

≤∫ t

0

(t − s)α−1

�(α)

n∑j=1

hij|xj(s)− yj(s)|ds

+∫ t

0

(t − s)α−1

�(α)

n∑j=1

kij|xj(s− rj)− yj(s− rj)|ds

and

e−Nt|Fxi − Fyi| ≤ hi

n∑j=1

∫ t

0

(t − s)α−1

�(α)e−N(t−s)e−Ns|xj(s)− yj(s)|ds

+ ki

n∑j=1

∫ t

rj

(t − s)α−1

�(α)e−N(t−s+rj)e−N(s−rj)|xj(s− rj)− yj(s− rj)|ds

≤ hi

n∑j=1

supt{e−Nt|xj(t)− yj(t)|}

∫ t

0

(t − s)α−1

�(α)e−N(t−s)ds

+ ki

n∑j=1

supt{e−Nt|xj(t)− yj(t)|}e−Nrj

∫ t

rj

(t − s)α−1

�(α)e−N(t−s)ds

≤ hi

n∑j=1

supt{e−Nt|xj(t)− yj(t)|} 1

Nα

∫ Nt

0

uα−1e−u

�(α)du

+ ki

n∑j=1

supt{e−Nt|xj(t)− yj(t)|} e

−Nrj

Nα

∫ N(t−rj)

0

uα−1e−u

�(α)du

≤ hi

Nα||x− y||1 +

ki

Nα

n∑j=1

supt{e−Nt|xj(t) − yj(t)|}

≤ hi + ki

Nα||x − y||1

and

||Fx− Fy||1 =n∑

i=1

supt

e−Nt|Fxi − Fyi| ≤n∑

i=1

hi + ki

Nα||x − y||1

≤ h + kNα

||x− y||1.

Now choose N large enough such that h+kNα < 1, so the map F : X ® X is a contrac-

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

integral equation (4).

El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47http://www.advancesindifferenceequations.com/content/2011/1/47

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

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

Since x Î Cn and I1-a x(t) Î Cn(I), and fi, gi Î C(I) then I1-a fi(t), I1-a gi(t) Î C(I).

Operating by I1-a on both sides of (4), we get

I1−αxi(t) = I1−αIα{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}= I{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))}.

Differentiating both sides, we obtain

DI1−αxi(t) = DI{fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn))},

which implies that

Dαxi(t) = fi(t, x1(t), . . . , xn(t)) + gi(t, x1(t − r1), . . . , xn(t − rn)), t > 0,

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

Now we prove that limt→0+ xi = 0. Since fi(t, x1(t), ..., xn(t)), gi(t, x1(t - r1), ..., xn(t -

rn)) are continuous on [0, T] then there exist constants li, Li, mi, Mi such that li ≤ fi(t,

x1(t), ..., xn(t)) ≤ Li and mi ≤ gi(t, x1(t - r1) ), ..., xn(t - rn)) ≤ Mi, and we have

Iαfi(t, x1(t), . . . , xn(t)) =∫ t

0

(t − s)α−1

�(α)fi(s, x1(s), . . . , xn(s))ds,

which implies

li

∫ t

0

(t − s)α−1

�(α)ds ≤ Iαfi(t, x1(t), . . . , xn(t)) ≤ Li

∫ t

0

(t − s)α−1

�(α)ds ⇒

litα

�(α + 1)≤ Iαfi(t, x1(t), . . . , xn(t)) ≤ Litα

�(α + 1)

and

limt→0+

Iαfi(t, x1(t), . . . , xn(t)) = 0.

Similarly, we can prove

limt→0+

Iαgi(t, x1(t − r1), . . . , xn(t − rn)) = 0.

Then from (4),limt→0+ xi(t) = 0. Also from (2), we have limt→0− �(t) = O.

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

(1)-(3) takes the form

xi(t) =

⎧⎪⎨⎪⎩

φi(t), t < 00, t = 0∫ t

0(t−s)α−1

�(α) {fi(s, x1(s), . . . , xn(s)) + gi(s, x1(s− r1), . . . , xn(s− rn))}ds, t > 0.

4 StabilityIn 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) = (x1(t), x2(t), ..., xn(t))’

and x̃(t) = (x̃1(t), x̃2(t), . . . , x̃n(t))′ with the initial conditions (2)-(3) and

El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47http://www.advancesindifferenceequations.com/content/2011/1/47

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||x(t) − x̃(t)||1 < εrespectively, one has ||�(t) − �̃(t)||1 ≤ δ, then ||x(t) − x̃(t)||1 < ε

for all t ≥ 0.

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

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

and {Iβ x̃(t)|t=0 = 0, x̃(t) = �̃(t), t < 0 and limt→0 �̃(t) = O}, respectively. Then for t > 0,

we have from (4)

|xi − x̃i| = |Iα{fi(t, x1(t), . . . , xn(t))− fi(t, x̃1(t), . . . , x̃n(t))

+ gi(t, x1(t − r1), . . . , xn(t − rn))− gi(t, x̃1(t − r1), . . . , x̃n(t − rn))}|

≤∫ t

0

(t − s)α−1

�(α)| fi(s, x1(s), . . . , xn(s))− fi(s, y1(s), . . . , yn(s))|ds

+∫ t

0

(t − s)α−1

�(α)|gi(s, x1(s− r1), . . . , xn(s− rn))− gi(s, x̃1(s− r1), . . . , x̃n(s− rn))|ds

≤∫ t

0

(t − s)α−1

�(α)

n∑j=1

hij|xj(s)− x̃j(s)|ds

+∫ t

0

(t − s)α−1

�(α)

n∑j=1

kij|xj(s− rj)− x̃j(s− rj)|ds

and

e−Nt|xi − x̃i| ≤ hi

n∑j=1

∫ t

0

(t − s)α−1

�(α)e−N(t−s)e−Ns|xj(s)− x̃j(s)|ds

+ ki

n∑j=1

∫ rj

0

(t − s)α−1

�(α)e−N(t−s+rj)e−N(s−rj)|φj(s− rj)− φ̃j(s− rj)|ds

+ ki

n∑j=1

∫ t

rj

(t − s)α−1

�(α)e−N(t−s+rj)e−N(s−rj)|xj(s− rj)− x̃j(s− rj)|ds

≤ hi

Nα||xj(t)− x̃j(t)||1

∫ Nt

0

uα−1e−u

�(α)du

+ ki

n∑j=1

supt{e−Nt|φj(t)− φ̃j(t)|} e

−Nrj

Nα

∫ Nt

N(t−rj)

uα−1e−u

�(α)du

+ ki

n∑j=1

supt{e−Nt|xj(t) − x̃j(t)|} e

−Nrj

Nα

∫ N(t−rj)

0

uα−1e−u

�(α)du

≤ hi

Nα||xj(t)− x̃j(t)||1 +

ki

Nα

n∑j=1

e−Nrj supt{e−Nt|xj(t) − x̃j(t)|}

+ki

Nα

n∑j=1

e−Nrj supt{e−Nt|ϕj(t)− φ̃j(t)|}

≤ hi + ki

Nα||x− x̃||1 +

ki

Nα||�− �̃||1.

Then we have,

||x− x̃||1 ≤n∑

i=1

hi + ki

Nα||x− x̃||1 +

n∑i=1

ki

Nα||�− �̃||1

≤ h + kNα

||x − x̃||1 +k

Nα||�− �̃||1

i.e.(

1− h + kNα

)||x − x̃||1 ≤ k

Nα||�− �̃||1and ||x− x̃||1 ≤ k

Nα

(1− h + k

Nα

)−1

||�− �̃||1

El-Sayed and Gaafar Advances in Difference Equations 2011, 2011:47http://www.advancesindifferenceequations.com/content/2011/1/47

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Therefore, for δ > 0 s.t.||�− �̃||1 < δ, we can find ε = kNα

(1− h+k

Nα

)−1δ s.t.

||x − x̃||1 ≤ εwhich proves that the solution x(t) is uniformly stable.

5 ApplicationsExample 1 Consider the problem

Dαxi(t) =n∑

j=1

aij(t)xj(t) +n∑

j=1

gij(t, xj(t − rj), t > 0

x(t) = �(t)fort < 0and limt→0−

�(t) = O

I1−αx(t)|t=0 = O,

where A(t) = (aij(t))n×n and (gi(t, x1(t − r1), . . . , xn(t − rn)))′ = (∑n

j=1 gij(t, xj(t − rj))′

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

x Î X on (-∞, T], T < ∞

Example 2 Consider the problem

Dαxi(t) =n∑

j=1

fij(t, xj(t)) +n∑

j=1

bij(t)xj(t − rj), t > 0

x(t) = �(t)for t < 0and limt→0−

�(t) = O

I1−αx(t)|t=0 = O,

where B(t) = (bij(t))n×n, and (fi(t, x1(t), . . . , xn(t)))′ = (∑n

j=1 fij(t, xj(t)))′ are given con-

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

(-∞, T], T < ∞

Example 3 Consider the problem (see [12])

Dαxi(t) =n∑

j=1

aij(t)xj(t) +n∑

j=1

bij(t)xj(t − rj) + hi(t), t > 0

x(t) = �(t) for t < 0and limt→0−

�(t) = O

I1−αx(t)|t=0 = O,

where A(t) = (aij(t))n×n B(t) = (bij(t))n×n, and H(t) = (hi(t))n×1 are given continuous

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

< ∞.

Author details1Faculty of Science, Alexandria University, Alexandria, Egypt 2Faculty of Science, Damanhour University, Damanhour,Egypt

Authors’ contributions sectionAll authors contributed equally to the manuscript and read and approved the final draft.

Competing interestsThe 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:10.1186/1687-1847-2011-47Cite this article as: El-Sayed and Gaafar: Stability of a nonlinear non-autonomous fractional order systems withdifferent delays and non-local conditions. Advances in Difference Equations 2011 2011:47.

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