New controllability results for fractional evolution equations in Banach spaces

NEW CONTROLLABILITY RESULTS FOR FRACTIONAL EVOLUTION EQUATIONS IN BANACH SPACES

Soumia Belarbi and Zoubir Dahmani

Abstract. In this paper, we use Caputo derivative to generate new sufficient conditions for the controllability of fractional evolution integro-differential equations in Banach spaces. These results are obtained using Banach contraction principle. Other results are also presented using Krasnoselskii theorem.

2000 Mathematics Subject Classification: 26A33, 93B05, 34K45. 1. Introduction

The fractional differential equations are emerged as a new branch of applied math-ematics by which many physical phenomena in various fields of science and engi-neering can be modeled. Significant development in this area has been achieved for the last two decades. For details, we refer to [6,7,10,11,13]. Moreover, the study of impulsive fractional integro-differential equations is also of great importance. The study of such equations is linked to their utility in simulating processes and phe-nomena subject to short-time perturbations during their evolution [3,5,12,13]. Many researchers have discussed controllability of impulsive systems in Banach space. For more details, we refer the reader to [1,2,4,8]. Motivated by the works [14,16], the aim of this paper is to establish the controllability results for fractional evolution integro-differential systems in Banach spaces by using the fractional calculus and fixed point theorems. So, let us consider the fractional non linear integro-differential equations

                       Dαx (t) = A (t) x (t) + Bu (t) + f  t, xt, t Z 0 a (t, s, xs) ds  , t 6= t_i, t ∈ J, 0 < α ≤ 1, x(s) + [g(xt1, ..., xtp)](s) = θ(s), s ∈ [−h, 0], ∆x|t=ti= Ii(x(t−_i )), i = 1, 2, ..., m, (1)

where Dα is the Caputo derivative, the variable x takes values in a Banach space X and the control function u is given in L2(J, V ); V as a Banach space and J := [0, b], Ω := {(t, s); 0 ≤ s ≤ t ≤ b}. We suppose that A is a closed linear densely defined operator in X, and B is a bounded linear operator from V into X.

Further, we suppose f : J × X × X → X, a : Ω × X → X, Ii : X →

X, ∆x|t=ti = x(t+_i ) − x(t−_i ), i = 1, ..., m, 0 = t0 < t1 < t2 < ... < tm < tm+1 = b,

g : [P C([−h, 0], X)]p → X are given functions. The function xt : (−h, 0] → X is

defined by xt(η) = x(t + η), for t ∈ [0, b], η ∈ [−h, 0].

We also take J0 := [0, t1], Ji = (ti, ti+1], i = 1, 2, ..., m, J0:= [0, b]\{t1, ..., tm},

P C([−h, b], X) := {x; x is a function from [−h, b] into X such that x(t) is continuous at t 6= ti, left continuous at t = ti and the right limit x(t+i ) exists for i = 1, 2, ..., m},

with norm

k x kP C:= sup t∈J

kx (t)k . It is to note that (1) is equivalent to

x (t) =                                                                    θ(0) − [g(xt1, ..., xtp)](0) + 1 Γ(α) X 0<ti<t ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 t Z ti (t − τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1  Bu (τ ) + f  τ, xτ, τ Z 0 a (τ, s, xs) ds    dτ +_Γ(α)1 t Z ti (t − τ )α−1  Bu (τ ) + f  τ, xτ, τ Z 0 a (τ, s, xs) ds    dτ + X 0<ti<t Ii x t−i

. (2) 2. Preliminaries

In the following, we give the necessary notation and basic definitions which will be used in this paper.

Definition 1. A real valued function f (t), t > 0 is said to be in the space Cµ, µ ∈ R

if there exists a real number p > µ such that f (t) = tpf1(t), where f1(t) ∈ C([0, ∞)).

Definition 2. A function f (t), t > 0 is said to be in the space C_µn, n ∈ N , if f(n)∈ Cµ.

Definition 3. The Riemann-Liouville fractional integral operator of order α ≥ 0, for a function f ∈ Cµ, (µ ≥ −1), is defined as

Jαf (t) = _Γ(α)1 t Z 0 (t − τ )α−1f (τ )dτ ; α > 0, t > 0 J0f (t) = f (t). (3)

The fractional derivative of f ∈ C₋₁n in the Caputo’s sense is defined as

Dαf (t) =        1 Γ(n−α) t Z 0 (t − τ )n−α−1f(n)(τ )dτ, n − 1 < α < n, n ∈ N∗, dn dtnf (t), α = n. (4) .

For more details, we refer to [9, 15].

For the controllability, we give the following definition [1,2]:

Definition 4. The system (1) is said to be controllable on the interval J if for every x1 ∈ X and [g(xt1, ..., xtp](s) ∈ P C([−h, b], X), there exists a control u ∈ L

2_{(J, V )}

such that the solution x(t) of (1) satisfies x(0) = x0 and x(b) = x1.

Let us now consider Br := {x ∈ X, ||x|| ≤ r, r > 0}. To study the

controllabil-ity problem, we assume the following conditions:

(H1) : A(t) is a bounded linear operator on X, for each t ∈ J and the function

t → A(t) is continuous in the uniform operator topology and max

(H2) : The linear operator W : L2(J ; V ) → X defined by W u (t) = 1 Γ (α) b Z 0 (b − τ )α−1Bu (τ ) dτ

has an inverse operator W−1, which takes values in L2(J ; V )/kerW and there exists a positive constant C > 0 such that k BW−1 k≤ C, for every x ∈ Br.

(H3) : The function f : J × X × X → X is continuous and there exist constants

β > 0,ß> 0, such that ||f (t, xt, ut) − f (t, yt, vt)|| ≤ β (||x − y|| + ku − vk) ; x, y, u, v ∈

X, t ∈ J, ß = maxt∈Jkf (t, 0, 0)k .

(H4) : For each (t, s) ∈ Ω, the function a : Ω × X → X is continuous and there

exist two constants ω > 0, ˆω > 0, such that

t Z 0 k a(t, s, xs) − a(t, s, ys) k ds ≤ ω k x − y k, x, y ∈ X, t, s ∈ J ˆ ω = max    t Z 0 k a(t, s, 0) k ds : t, s ∈ Ω.   

(H5) : Ii : X → X is continuous and there exist constants $i, such that

k Ii(x) − Ii(y) k≤ $ik x − y k, i = 1, 2, ..., m, x, y ∈ X.

(H6) : The function g : [P C([−h, 0], X)]p → X is continuous and there exist two

constants %, ˆ% > 0, such that

k [g(x_t1, ..., xtp](s)−[g(yt1, ..., ytp](s) k≤ % k x−y kP C; x, y ∈ P C([−h, b], X), s ∈ [−h, 0]

ˆ

% = max{k [g(xt1, ..., xtp](s) k: x, y ∈ P C([−h, b], X), s ∈ [−h, 0]}.

kθ(0)k + ˆ% + (m + 1) λ [(M + β + bω) r + (κ + ß + bˆω)] + m ($r + ˆ$) ≤ r, κ = C [kx1k + kθ (0)k + ˆ% + (m + 1) λ [(M + β + bω) r + ß + bˆω] + m ($r + ˆ$), λ = b α Γ (α + 1). 3. Main Results

Using the well-known Banach fixed point theorem, we give the following theorem. Theorem 1. If the hypotheses (Hj)j=1,7 and

Λ := [% + (m + 1) λ [γ + (M + β + bω)] + m$] < 1 (5) are satisfied, then the problem (1) is controllable on J .

Proof. Using (H2), we define the control u as follows:

u (t) =                                                                    w−1[x1− θ (0) + [g(xt1, ..., xtp)](0) − 1 Γ(α) X 0<ti<b ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ − 1 Γ(α) b Z ti (b − τ )α−1A (τ ) x (τ ) dτ − 1 Γ(α) X 0<ti<b ti Z ti (ti− τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ − 1 Γ(α) b Z ti (b − τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ + X 0<ti<t Ii x t−i

)] (t) . (6)

Now, we shall prove that the operator Φ : P C([−h, b] , Br) → P C([−h, b] , Br) defined by Φx (t) =                                                                        θ(0) − [g(xt1, ..., xtp)](0) + 1 Γ(α) t Z ti (t − τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− µ)α−1φ (µ, x) dµ + _Γ(α)1 t Z ti (t − µ)α−1φ (µ, x) dµ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ +_Γ(α)1 t Z ti (t − τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ + X 0<ti<t Ii x t−i

(7) has a fixed point. So, let us take

φ (µ, x) =                                                  Bw−1[x1− θ (0) + [g(xt1, ..., xtp)](0) − 1 Γ(α) X 0<ti<b ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ − 1 Γ(α) b Z ti (b − τ )α−1A (τ ) x (τ ) dτ − 1 Γ(α) X 0<ti<b ti Z ti (ti− τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ − 1 Γ(α) b Z ti (b − τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ + X 0<ti<t Ii x t−i

)] (t) . (8) Then, we have

kφ (µ, x)k ≤ C [kx1k + kθ (0)k + ˆ%+

+ (m + 1) λ [(M + β + bω) r + ß + bˆω] + m ($r + ˆ$) = κ (9) and

kφ (µ, x) − φ (µ, y)k ≤ C [(m + 1) λ [(M + β + bω)] + m ($)] kx − yk ≤ γ kx − yk . 1* We have to prove that ΦBr ⊂ Br. Let x ∈ Br, then we can write:

kΦx ((t))k ≤ kθ(0)k + [g(xt1, ..., xtp)](0) +_Γ(α)1 t Z ti (t − τ )α−1kA (τ )k kx (τ )k dτ + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1_{kA (τ )k kx (τ )k dτ +} 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1_{kφ (µ, x)k dµ} + 1 Γ(α) Rt ti(t − µ) α−1_{kφ (µ, x)k dµ +} 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
dτ + 1 Γ(α) Rt ti(t − τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
dτ + P 0<ti<t Ii x t−i

. Consequently, kΦx ((t))k ≤ kθ(0)k + [g(xt1, ..., xtp)](0) +_Γ(α)1 Rt ti(t − τ ) α−1_{kA (τ )k kx (τ )k dτ} + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1_{kA (τ )k kx (τ )k dτ +} 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1_{kφ (µ, x)k dµ} + 1 Γ(α) Rt ti(t − µ) α−1_{kφ (µ, x)k dµ} + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
− f (τ, 0, 0) + f (τ, 0, 0) dτ + 1 Γ(α) Rt ti(t − τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
− f (τ, 0, 0) + f (τ, 0, 0) dτ +P 0<ti<t Ii x t−i

− Ii(0) + Ii(0) .

Therefore, kΦx ((t))k ≤ kθ(0)k + [g(xt1, ..., xtp)](0) + 1 Γ(α) Rt ti(t − τ ) α−1_{kA (τ )k kx (τ )k dτ} +_Γ(α)1 P 0<ti<t Rti ti−1(ti− τ ) α−1_{kA (τ )k kx (τ )k dτ +} 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1_{kφ (µ, x)k dµ} + 1 Γ(α) Rt ti(t − µ) α−1_{kφ (µ, x)k dµ} +_Γ(α)1 P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ,R₀τa (τ, s, xs) ds
− f (τ, 0, 0) dτ + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1_{kf (τ, 0, 0)k dτ} + 1 Γ(α) Rt ti(t − τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
− f (τ, 0, 0) dτ + 1 Γ(α) Rt ti(t − τ ) α−1 f (τ, 0, 0) dτ +P 0<ti<t  Ii x t−i

− Ii(0) + kIi(0)k .

This implies that,

kΦx ((t))k ≤ kθ(0)k + ˆ% + (m + 1) λ [(M + β + bω) r + (κ + ß + bˆω)] + m ($r + ˆ$) (10) and then,

kΦx ((t))k ≤ r. (11)

Hence ΦBr ⊂ Br, which means that the operator Φ maps Br into itself.

2* Now we prove that Φ is a contraction mapping on Br. Let x, y ∈ Br, then we

can write: kΦx (t) − Φy (t)k ≤ [g(xt1, ..., xtp) − g(yt1, ..., ytp)](0) + + 1 Γ(α) Rt ti(t − τ ) α−1_{kA (τ )k kx (τ ) − y (τ )k dτ} +_Γ(α)1 P 0<ti<t Rti ti−1(ti− τ ) α−1_{kA (τ )k kx (τ ) − y (τ )k dτ} + 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1_{kφ (µ, x) − φ (µ, y)k dµ} + 1 Γ(α) Rt ti(t − µ) α−1_{kφ (µ, x) − φ (µ, y)k dµ} + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
− f τ, yτ, Rτ 0 a (τ, s, ys) ds
dτ + 1 Γ(α) Rt ti(t − τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
− f τ, yτ, Rτ 0 a (τ, s, ys) ds
dτ +P 0<ti<t Ii x t−_i

− Ii y t−_i

. Consequently,

kΦx (t) − Φy (t)k ≤ [% + (m + 1) λ [γ + (M + β + bω)] + m$] kx − yk . (12) Then,

kΦx (t) − Φy (t)k ≤ Λ kx − yk . (13) Hence, the operator Φ has a unique fixed point which is a solution of (1) . Conse-quently, the system (1) is controllable on J. Theorem 1 is thus proved.

Our second result is based on Krasnoselskii’s fixed point theorem [11]. Let us consider the following conditions:

(H₃0) : i) For each t ∈ J , the function f (t,·,·) : X × X → X is continuous. ii) There exists a function βf (· ) ∈ L1(J, R+) such that

kf (t, x1, y1) − f (t, x2, y2)k ≤ βf(t)(kx1− x2k + ky1− y2k),

ß = max

t∈J kf (t, 0, 0)k

for any t ∈ J, xl, yi∈ X for l = 1, 2.

iii) The function f : J × X × X → X is compact.

(H₄0) : i) For each t, s ∈ J , the function a(t, s,·) : X → X is continuous . ii) There exists an integrable function ωa : J × J → [0, ∞) such that

ka(t, s, x) − a(t, s, y)k ≤ ωa(t, s) kx − yk , t, s ∈ J, x, y ∈ X,

ˆ ω = max Z t 0 k a(t, s, 0) k ds : t, s ∈ Ω.  .

(H₅0): i) Ii: X → X is continuous and there exist constants $i such that

k Ii(x) − Ii(y) k≤ $ik x − y k, i = 1, 2, ..., m, x, y ∈ X.

ii) Ii : X → X is compact.

kθ(0)k + ˆ% + (m + 1) λ h M + kβfk_L1 + b kωak L1  r +κ + kβfk_L1 + b kωak L1  + m ($r + ˆ$) ≤ r, κ = C h kx1k + kθ (0)k + ˆ% + (m + 1) λ h M + kβfk_L1 + b kωak L1  r + ß + bˆω i + m ($r + ˆ$) i , and λ = b α Γ (α + 1). (H8): There exists a constant Υ such that

Υ = % + (m + 1) λhM + Ch(m + 1) λhM + kβfk_L1 + b kωak

L1

i

+ m ($)ii. Theorem 2.(Krasnoselskii Theorem) Let K be a closed convex and nonempty subset of a Banach space X. Let P, Q be the operators such that

(i) P x + Qy ∈ K, x, y ∈ K. (ii) P is compact and continuous. (iii) Q is a contraction mapping. Then there exists x∗, such that

x∗= P x∗+ Qx∗.

Theorem 3. If the hypotheses (H1), (H6), (H8), (H

0

j)j=3,5 and (H

0

7) are

satis-fied and if Υ < 1, then the system (1) is controllable on J. Proof. On Br, we define the operators R and S as:

                                                       Rx (t) = θ(0) − [g(xt1, ..., xtp)](0) + 1 Γ(α) t Z ti (t − τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ + 1 Γ(α) X 0<ti<t ti Z ti−1 (ti− µ)α−1φ (µ, x) dµ +_Γ(α)1 t Z ti (t − µ)α−1φ (µ, x) dµ (14) and                      Sx (t) = _Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ +_Γ(α)1 t Z ti (t − τ )α−1f  τ, xτ, τ Z 0 a (τ, s, xs) ds  dτ + X 0<ti<t Ii x t−_i

. (15) For x, y ∈ Br, we have kRx (t) + Sy (t)k ≤ kRx (t)k + kSx (t)k . (16)

Then we can write kRx (t) + Sy (t)k ≤ kθ(0)k + [g(xt1, ..., xtp)](0) + 1 Γ (α) t Z ti (t − τ )α−1A (τ ) x (τ ) dτ + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1_{A (τ ) x (τ ) dτ} + 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1_{φ (µ, x) dµ} + 1 Γ(α) Rt ti(t − µ) α−1 φ (µ, x) dµ + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
dτ + 1 Γ(α) Rt ti(t − τ ) α−1_{f τ, x} τ, Rτ 0 a (τ, s, xs) ds
dτ + P 0<ti<tIi x t − i

. It follows then that

kRx (t) + Sy (t)k ≤ kθ(0)k + ˆ% + (m + 1) λM + kβfk_L1 + b kωak L1  r + (m + 1) λκ + kβfk_L1 + b kωak L1  + m ($r + ˆ$) . Consequently, kRx (t) + Sy (t)k ≤ r. Hence, Rx + Sy ∈ Br.

kRx (t) − Ry (t)k =k [θ(0) − [g(xt1, ..., xtp)](0) + 1 Γ(α) t Z ti (t − τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1A (τ ) x (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− µ)α−1φ (µ, x) dµ + 1 Γ(α) t Z ti (t − µ)α−1φ (µ, x) dµ] − [θ(0) − [g(yt1, ..., ytp)](0) + 1 Γ(α) t Z ti (t − τ )α−1A (τ ) y (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− τ )α−1A (τ ) y (τ ) dτ +_Γ(α)1 X 0<ti<t ti Z ti−1 (ti− µ)α−1φ (µ, y) dµ +_Γ(α)1 t Z ti (t − µ)α−1φ (µ, x) dµ]k. (17) Therefore, kRx (t) − Ry (t)k ≤ [g(yt1, ..., ytp)](0) − [g(xt1, ..., xtp)](0) + 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 A (τ ) (x (τ ) − y (τ )) dτ + 1 Γ(α) Rt ti(t − τ ) α−1 A (τ ) (x (τ ) − y (τ )) dτ + 1 Γ(α) P 0<ti<t Rti ti−1(ti− µ) α−1 [φ (µ, x) − φ (µ, y)] dµ + 1 Γ(α) Rt ti(t − µ) α−1 [φ (µ, x) − φ (µ, y)] dµ ≤h% + (m + 1) λhM + Ch(m + 1) λhM + kβfk_L1+ b kωak L1 i + m ($)iiikx − yk ≤ Υ kx − yk .

Since Υ < 1, then the operator R is a contraction.

Now, we shall prove that the operator S is completely continuous on Br: First, we

show that S is continuous on Br. So, let {xn(t)}∞₀ ⊂ Br, with xn→ x in Br. Then,

there exists r > 0, such that kxn(t)k ≤ r, for all n. Hence xn∈ Br and x ∈ Br.

i) Ii is continuous for each i = 1, 2, ..., m. ii) ft, xn_t,Rt 0a (t, s, x n s) ds  → ft, xt, Rt 0a (t, s, xs) ds 

for each t ∈ J and f  t, xn_t, Z t 0 a (t, s, xn_s) ds  → f  t, xt, Z t 0 a (t, s, xs) ds  ≤ 2r kβ_fk_L1.

Using Lebesgue dominated convergence theorem, yields

kSxn (t) − Sx (t)k = k 1 Γ(α) X 0<ti<t ti Z ti−1 (ti− τ )α−1  f  τ, xnτ, τ Z 0 a (τ, s, xns) ds  − f  τ, xτ, τ Z 0 a (τ, s, xs) ds    dτ +_Γ(α)1 t Z ti (t − τ )α−1  f  τ, xnτ, τ Z 0 a (τ, s, xns) ds  − f  τ, xτ, τ Z 0 a (τ, s, xs) ds    dτ + X 0<ti<t Ii xn t−i

− Ii x t−i


k . Furthermore, kSxn(t) − Sx (t)k ≤ P + Q + Z, where, P = 1 Γ (α) X 0<ti<t Z ti ti−1 (ti− τ )α−1 f  τ, xnτ, Z τ 0 a (τ, s, xns) ds  − f  τ, xτ, Z τ 0 a (τ, s, xs) ds  dτ, Q = + 1 Γ (α) Z t ti (t − τ )α−1 f  τ, xnτ, Z τ 0 a (τ, s, xns) ds  − f  τ, xτ, Z τ 0 a (τ, s, xs) ds  dτ and Z = X 0<ti<t Ii xn t−_i

− Ii x t−_i

. Consequently, kSxn(t) − Sx (t)k → 0, n → ∞. Hence, S is continuous on Br.

Now, we prove that S is relatively compact as well as equi-continuous on X. To prove the compactness of S, we shall prove that S(Br) ⊆ P C([−h, b] , X) is

equi-continuous and S(Br)(t) is pre-compact for any r > 0, t ∈ J . Let x ∈ Br and

kSx (t + k) − Sx (t)k ≤k 1 Γ (α) X 0<ti<t+k Zti ti−1 (ti− τ )α−1f  τ, xτ, Zτ 0 a (τ, s, xs) ds  dτ +_Γ(α)1 Rt+k ti ((t + k) − τ ) α−1 f τ, xτ,R₀τa (τ, s, xs) ds
dτ +P 0<ti<t+kIi x t − i

− 1 Γ(α) P 0<ti<t Rti ti−1(ti− τ ) α−1 f τ, xτ,R₀τa (τ, s, xs) ds
dτ + 1 Γ(α) Rt ti(t − τ ) α−1 f τ, xτ, Rτ 0 a (τ, s, xs) ds
dτ + P0<ti<tIi x t − i

k. Therefore, kSx (t + k) − Sx (t)k ≤k 1 Γ (α) X t<ti<t+k Z ti ti−1 (ti− τ )α−1f  τ, xτ, Z τ 0 a (τ, s, xs) ds  dτ +_Γ(α)1 [Rt+k ti ((t + k) − τ ) α−1 f τ, xτ,R₀τa (τ, s, xs) ds
−Rt ti(t − τ ) α−1 f τ, xτ,R₀τa (τ, s, xs) ds
dτ ] +P t<ti<t+hIi x t − i

k.

The inequality (27) is independent of x. Thus S is equi-continuous, and as k → 0, the right hand side of this inequality tends to zero. Then S (Br) is relatively

compact, and S is compact. Finally by Krasnoselskii theorem, there exists a fixed point x in Br, such that (Φx) (t) = x (t) ; this point is a solution of (1) . It is clear

that (Φx) (b) = x (b) = x1, which implies that the system (1) is controllable on J.

References

[1] K. Balachandran and J.H. Kim, Remarks on the paper of Controllability of second order differential inclusion in Banach spaces, J. Math. Anal. Appl., 285, (2006), 537-550. [2] K. Balachandran and J.Y. Park, Controllability of fractional integro-differential systems in Banach spaces, Nonlinear Analysis: Hybrid Systems, 3(4), (2009), 363-367.

[3] K. Balachandran, S.Kiruthika and J.J. Trujillo, Existence results for frac-tional impulsive integro-differential equations in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 16(4), (2010), 1970-1977.

[4] M. Bragdi, M. Hazi, Existence and Controllability Result or an Evolution Fractional Integro-differential Systems, Int. J. Contemp. Math. Sciences, 5(19), (2010), 901-910.

[5] M. Benchohra and B.A. Slimani, Existence and uniqueness of solutions to im-pulsive fractional differential equations, Electronic Journal of Differential Equations, 2009(10), (2009), 1-11.

[6] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338, (2008), 1340-1350.

[7] B. Bonila, M. Rivero, L. Rodriquez-Germa and J.J. Trujilio, Fractional dif-ferential equations as alternative models to nonlinear difdif-ferential equations, Appl. Math. Comput., 187, (2007), 79-88.

[8] Y.K. Chang, J.J. Nieto and W.S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142, (2009), 267-273.

[9] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equa-tions of fractional order, Springer Verlag, Wien, (1997) 223-276.

[10] J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Techno., 15(2), (1999), 86-90.

[11] A.A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Appli-cations of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, (2006).

[12] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Analysis, 70, (2009), 2521-2529. [13] Y.F. Luchko, M. Rivero, J.J. Trujillo and M.P. Velasco, Fractional models, nonlocality and complex systems, Comp. Math. Appl., 59, (2010), 1048-1056.

[14] M. Mallika Arjunan, V. Kavitha, Controllability of impulsive fractional evo-lution integro-differential equations in Banach spaces, J. KSIAM, 15(3), (2011), 177-190.

[15] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, (1999).

[16] B. Radhakrishnan, K. Balachandran, Controllability of nonlocal impulsive functional integro-differential evolution systems, J. Nonlinear Sci. Appl. 4(4), (2011), 281-291

Soumia BELARBI, Faculty of Mathematics, USTHB of Algiers, Algeria

Zoubir DAHMANI, Laboratory LPAM, Faculty SEI, UMAB, University of Mostaganem, Algeria