Existence of solutions for fractional impulsive

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Research Article

Existence of solutions for fractional impulsive

neutral functional infinite delay integrodifferential equations with nonlocal conditions

A. Anguraj^a,∗, M. Latha Maheswari^b

aDepartment of Mathematics, P. S. G. College of Arts and Science, Coimbatore-641 014, Tamil Nadu, India

bDepartment of Mathematics with CA, P. S. G. College of Arts and Science, Coimbatore-641 014, Tamil Nadu, India

This paper is dedicated to Professor Ljubomir ´Ciri´c Communicated by Professor V. Berinde

Abstract

This paper is mainly concerned with the existence of solutions for fractional impulsive neutral functional integrodifferential equations with nonlocal initial conditions and infinite delay. The results are obtained by the fixed point theorem. c2012. All rights reserved.

Keywords: Existence of solution, Fractional, Integrodifferential equations, Impulsive conditions, Nonlocal conditions, Fixed point theorem.

2010 MSC: primary 34A37, 34K37.

1. Introduction

Fractional Calculus deals with the generalization of integrals and derivatives of noninteger order. Frac- tional calculus involves a wide area of applications by bringing into a broader paradigm concepts of physics, mathematics and engineering [11, 13]. Infact fractional differential equation is considered as an alternative model to nonlinear differential equations [8]. In [2, 12], the authors have proved the existence of solutions of abstract fractional differential equations by using fixed point tecniques. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details, see [14, 15, 16] and the

∗Corresponding author

Email addresses: [email protected](A. Anguraj),[email protected](M. Latha Maheswari) Received 2011-6-24

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references therein. Subsequently several authors have discussed the problem for different types of nonlinear differential and integro differential equations including functional differential equations in Banach spaces.

The theory of impulsive differential equations has undergone rapid development over the years and played a very important role in modern applied mathematical models of real processes arising in phenom- ena studied in physics, population dynamics, chemical technology and economics. In [1, 7], Benchohra et al. established sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative of order 0< q≤1 and 1< q≤2. In [10], Mophou proved the existence and uniqueness results of a mild solution to impulsive fractional semilinear differential equations. Anguraj and Karthikeyan [3] proved Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. Benchohra and Seba [6] studied the existence of fractional impulsive differential equations in Banach spaces while Balachandran and Kiruthika [5] discussed the existence of nonlocal cauchy problem for semilinear fractional evoluation equations. Balachandran and Trujillo [4] investigated the nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces.

In this paper, we consider the following fractional impulsive neutral integrodifferential systems with infinite delay











D^q_t(x(t)−u(t, xt)) =A(t, x)(x(t)−u(t, xt)) +f(t, xt,Rt

0h(t, s, xs)ds), t∈J = [0, b], t6=t_k

∆x|_t=t_k =I_k(x(t⁻_k)), t=t_k, k= 1,2, ..., n x(0) +g(x) =φ, φ∈B_ϑ

(1.1)

where 0< q <1 and the state x(.) belongs to Banach space X endowed with the norm k.k. Operator A generates a strongly continuous bounded linear operator on a Banach space X. D_t^q is the Caputo fractional derivative. u and f are two given continuous functions,I_k:X →X, ∆x(t_k) =x(t⁺_k)−x(t⁻_k) with x(t⁺_k) = lim_h→0⁺x(tk+h), x(t⁻_k) = lim_h→0⁻x(tk+h), k= 1,2,3, ...n, 0 =t0 < t1 < t2< ... < tn< tn+1 =b. Let xt(.) denote xt(θ) =x(t+θ),θ∈(−∞,0].

The rest of this papper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we study the existence and the uniqueness of solutions for the impulsive fractional system 1.1. In Section 4, an example.

2. Preliminaries

In this section, we shall introduce some basic definitions, notations, lemmas and proposition which are used throughout this paper.

Assume thatϑ: (−∞,0]→(0,+∞) is a continuous function satisfyi`=R0

−∞ϑ(t)dt <+∞. The Banach space (B_ϑ,||.||_B_ϑ) induced by the function ϑis defined as follows

B_ϑ =

(ϕ: (−∞,0]−→X : f or any c >0, ϕ(θ) is a bounded and measurable f unction on[−c,0]andR0

−∞ϑ(t)supt≤θ≤0|ϕ(θ)|dt <+∞

endowed with the normkϕk_B

ϑ =R0

−∞ϑ(s)sups≤θ≤0|ϕ(θ)|ds.

Let us define the space B_ϑ⁰ =

(ϕ: (−∞, b]→X :ϕ_k∈C(J_k, X), k= 0,1,2, ...n and there exist ϕ(t⁻_k)and ϕ(t⁺_k)withϕ(tk) =ϕ(t⁻_k), ϕ0 =ϕ(0) +g(ϕ) =φ∈Bϑ

where ϕ_k is the restriction of ϕ to J_k, J₀ = [0, t₁], J_k = (t_k, t_k+1], k = 1,2, ....n. Denote by ||.||_B

ϑ0, a seminorm in the spaceB_ϑ⁰, which is defined by

||ϕ||_B

ϑ0 =||ϕ||_B_ϑ +max||ϕ_k||_J_k, k= 1,2, ...nwhere ||ϕ_k||_J_k =sup_s∈J_k||ϕ_k(s)||.

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Definition 2.1. A functionx: (−∞, b]→Xis said to be a solution of system 1.1 ifx(0)+g(x) =φ∈B_ϑ⁰,the impulsive condition ∆x|_t=t_k = Ik(x(t⁻_k)), k = 1,2, ..., n is verified, the restriction of x(.) to the interval J_k(k= 0,1,2, ...n) is continuous and the following integral equation holds fort∈J,

x(t) =











[φ(0)−g(x)−u(0, φ)] +u(t, xt) +_Γ(q)¹ P

0<tk<t

R_t_k

tk−1(t_k−s)^q−1A(s, x)x(s)ds +_Γ(q)¹ Rt

tk(t−s)^q−1A(s, x)x(s)ds

−_Γ(q)¹ P

0<tk<t

Rtk

tk−1(t_k−s)^q−1A(s, x)u(s, x_s)ds

−_Γ(q)¹ Rt

tk(t−s)^q−1A(s, x)u(s, x_s)ds +_Γ(q)¹ P

0<tk<t

Rtk

tk−1(t_k−s)^q−1f(s, x_s,Rs

0 h(s, τ, x_τ)dτ)ds +_Γ(q)¹ Rt

tk(t−s)^q−1f(s, x_s,Rs

0 h(s, τ, x_τ)dτ)ds +P

0<tk<tI_k(x(t⁻_k))

(2.1)

Definition 2.2. The Riemann - Liouville fractional integral operator of orderq ≥0 of functionf ∈L¹(R⁺) is defined as

I₀^q+f(t) = 1 Γ(q)

Z t 0

(t−s)^q−1f(s)ds, t >0 where Γ(.) is the Euler gamma function.

Definition 2.3. The Caputo fractional derivative of order q≥0 ,n−1< q < n, is defined as D₀^q+f(t) = 1

Γ(n−q) Z t

0

(t−s)^(n−q−1)f⁽ⁿ⁾(s)ds, t >0

where the function f(t) has absolutely continuous derivatives up to order (n-1).

If 0< q <1, then D₀^q+f(t) = 1

Γ(1−q) Z t

0

(t−s)^(−q)f⁽¹⁾(s)ds

wheref⁽¹⁾(s) =Df(s) = ^df(s)_ds and f is an abstract function with values in X.

We shall state some properties of the operators I₀^α+ and D₀^α+. Lemma 2.4. . For α, β > o and f as a suitable function, we have

(i) I^αI^βf(t) =I^α+βf(t) (ii) I^αI^βf(t) =I^βI^αf(t)

(iii) I^α(f(t) +g(t)) =I^αf(t) +I^αg(t) (iv) I^{α c}D^αf(t) =f(t)−f(0),0< α <1

(v) ^cD^αI^αf(t) =f(t)

(vi) ^cD^αf(t) =I^(1−α)Df(t) =I^(1−α)f⁰(t),0< α <1, D= _dt^d (vii) ^cD^{α c}D^βf(t)6=^cD^(α+β)f(t)

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(viii) ^cD^{α c}D^βf(t)6=^cD^{β c}D^αf(t)

Zhang and Xiyue Huang [18] proved the existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay , in which A is a infinitesimal generator of strongly continuous semigroup. But in [4], Balachandran and Trujillo observed that both the R - L and the Caputo fractional differential operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives on integer order.

Theorem 2.5. ([17])

Let B be a convex, bounded and closed subset of a Banach space X andN :B →B be a condensing map.

Then N has a fixed point in B.

Lemma 2.6. ([9])

Assume that x∈B_ϑ⁰ then, fort∈J, x_t∈B_ϑ. Moreover

`||x(t)|| ≤ ||x_t||_B_ϑ ≤ ||φ||_B_ϑ+` sups∈[0,t]||x(s)||.

3. Main results

For φ∈B_ϑ, we define ˆφ by

φ(t) =ˆ







φ(t), t∈(−∞,0]

φ(0), t∈J then ˆφ∈B_ϑ⁰.

Let x(t) =y(t) + ˆφ(t), −∞< t < b.

It is evident that y satisfiesy0 = 0, t∈(−∞,0], and x(t) = [−g(y+ ˆφ)−u(0, φ)] +u(t, y_t+ ˆφ_t)

+ 1

Γ(q) X

0<tk<t

Z _t_k

tk−1

(t_k−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds

+ 1

Γ(q) Z _t

tk

(t−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds

− 1 Γ(q)

X

0<t_k<t

Z t_k tk−1

(t_k−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds

− 1 Γ(q)

Z t tk

(t−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds

+ 1

Γ(q) X

0<tk<t

Z tk

tk−1

(tk−s)^q−1f(s, ys+ ˆφs, Z s

0

h(s, τ, yτ+ ˆφτ)dτ)ds

+ 1

Γ(q) Z t

tk

(t−s)^q−1f(s, ys+ ˆφs, Z s

0

h(s, τ, yτ + ˆφτ)dτ)ds

+ X

0<t_k<t

I_k(y(t⁻_k) + ˆφ(t⁻_k)), t∈J

if and only if x satisfiesx(t) =φ(t), t∈(−∞,0] and x satisfies equation 2.1.

For brevity let us take H(x_s) =

Z s 0

h(s, τ, x_τ)dτ.

We assume the following conditions to prove the existence of solution of the equation 1.1.

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(H₁) A:J×B_ϑ→B(X) is a continuous bounded linear operator and there exists a constantM >0, such that||A(t, x)−A(t, y)|| ≤M||x−y||_B_ϑ, for all x, y∈Bϑ.

(H₂) The functionu:J×B_ϑ→X, and there exist two positive constantsλ₁ andλ₂ such that the function satisfies the Lipschitz condition

||u(s, x_t)−u(s, y_t)|| ≤λ₁(||x_t−y_t||_B

ϑ0), λ₂=supt∈J||u(t,0)||.

(H3) f :J×B_ϑ×X →X , and there exist two positive constantsK1, K2 such that

||f(t, φ₁, y₁)−f(t, φ₂, y₂)|| ≤K₁(||φ₁−φ₂||_B_ϑ+||y₁−y₂||), K2=supt∈J||f(t,0,0)||.

(H₄) h: ∆×B_ϑ→X , where ∆ =

(t, s) : 0≤s≤b , equipped with positive constantsP₁, P₂ satisfying

||h(t, s, φ₁)−h(t, s, φ2)|| ≤P1(||φ₁−φ2||_B

ϑ0), P2=sup_(t,s)||h(t, s,0)||.

(H5) Ik:X →X are continuous, and there exists a constantµ >0 such that

||I_k(x)−I_k(y)|| ≤µ||x−y||, k= 1,2,3, ...n.

(H6) g:B_ϑ⁰ →X is continuous and there exist some positive constantδ1, δ2 such that

||g(x)−g(y)|| ≤δ₁||x−y||_B⁰

ϑ and||g(x)|| ≤δ₁||x||_B⁰

ϑ+δ₂. (H7)

δ1L+δ2+λ1(L1+||φ||_B_ϑ) + 2λ2+T+µn(r+`⁻¹||φ||_B_ϑ+|φ(0)|)

≤r whereT = ^b_Γ(q+1)^q⁽ⁿ⁺¹⁾[(M L+K)(L+λ₁L₁+λ₂) +K₁L₁+K₁b(P₁L₁+P₂) +K₂] and L=r+||φ||_B_ϑ+|φ(0)|,L1 =`(r+|φ(0)|) +||φ||_B_ϑ.

Theorem 3.1. Suppose that conditions (H₁)−(H₇) are satisfied with δ₁+λ₁` <1 then system 1.1 has a solution.

Proof 1. Define Θ :B_ϑ⁰ →B_ϑ⁰ by

Θy(t) = 0, t∈(−∞,0]

Θy(t) = [−g(y+ ˆφ)−u(0, φ)] +u(t, yt+ ˆφt) + 1

Γ(q) X

0<tk<t

Z tk

tk−1

(tk−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds

+ 1 Γ(q)

Z t tk

− 1 Γ(q)

X

0<tk<t

Z tk

tk−1

(tk−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds

− 1 Γ(q)

Z t tk

(t−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds + 1

Γ(q) X

0<tk<t

Z tk

tk−1

(t_k−s)^q−1f(s, y_s+ ˆφ_s, , H(y_s+ ˆφ_s))ds

+ 1 Γ(q)

Z t t_k

(t−s)^q−1f(s, ys+ ˆφs, , H(ys+ ˆφs))ds

+ X

0<tk<t

Ik(y(t⁻_k) + ˆφ(t⁻_k)), t∈J.

Clearly, y is a fixed point of Θ theny+ ˆφis a solution of the system 1.1. We shall show that Θ satisfies the hypotheses of Theorem 2.5.

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Define the Banach space (B_ϑ⁰⁰,||.||_B

ϑ0) induced by B_ϑ⁰ , Bϑ⁰⁰ =

y ∈Bϑ⁰ :y0 = 0∈Bϑ with norm||y||_B

ϑ0 =sup

|y(s)|:s∈[0, b] . SetBr=

y∈B_ϑ⁰⁰:||y||_B

ϑ0 ≤r for somer >0 thenBr, for each r, is bounded, closed convex subset of X.

For anyy∈B_r, by Lemma 2.6, we have

||y_t+ ˆφt||_B_ϑ ≤ ||φ||_B_ϑ+`[r+|φ(0)|],

||y+ ˆφ||_B

ϑ0 ≤ r+||φ||_B_ϑ+|φ(0)|, supt∈J

y(t) + ˆφ(t)

≤ r+`⁻¹||φ||_B_ϑ+|φ(0)|. Now we proceed in Two steps.

Step I:We claim that there exists a positive integer r∈N such that Θ(Br)⊂Br.

There exists a positive number r such that B_r is clearly a closed bounded convex set in B_ϑ⁰. For each positive integer r, there exist yr∈Br and t(r)∈(−∞, b] such that

||Θ(y^r)(t(r))|| ≤ ||[−g(y^r+ ˆφ)−u(0, φ)]||+||u(t(r), y_t(r)+ ˆφ_t(r))||

+ 1 Γ(q)

X

0<tk<t(r)

Z tk

tk−1

(t_k−s)^q−1||A(s, y^r+ ˆφ)(y^r+ ˆφ)(s)||ds

+ 1 Γ(q)

Z t(r) tk

(t(r)−s)^q−1||A(s, y^r+ ˆφ)(y^r+ ˆφ)(s)||ds

− 1 Γ(q)

X

0<tk<t(r)

Z _t_k

tk−1

(t_k−s)^q−1||A(s, y^r+ ˆφ)u(s, y^r_s+ ˆφ_s)||ds

− 1 Γ(q)

Z t(r) tk

(t(r)−s)^q−1||A(s, y^r+ ˆφ)u(s, y_s^r+ ˆφs)||ds + 1

Γ(q) X

0<t_k<t(r)

Z tk

tk−1

(tk−s)^q−1||f(s, y_s^r+ ˆφs, H(y_s^r+ ˆφs))||ds

+ 1 Γ(q)

Z t(r) t_k

(t(r)−s)^q−1||f(s, y_s^r+ ˆφ_s, H(y^r_s+ ˆφ_s))||ds

+ X

0<tk<t(r)

||I_k(y^r(t⁻_k) + ˆφ(t⁻_k))||

≤ δ₁L+δ₂+λ₁(L₁+||φ||_B_ϑ) + 2λ₂ +b^q(n+ 1)

Γ(q+ 1)[(M L+K)(L+λ₁L₁+λ₂) +K₁L₁ +K1b(P1L1+P2) +K2] +µn(r+`⁻¹||φ||_B_ϑ+|φ(0)|)

≤ r

whereL=r+||φ||_B_ϑ+|φ(0)|, L₁ =`(r+|φ(0)|) +||φ||_B_ϑ. Using (H7) , for some positive integer r , Θ(Br)⊂Br.

Step II:Now we claim that the operator Θ = Θ1+ Θ2 is condensing , that is Θ1 is a contraction and Θ2 is compact.

The operators Θ1 and Θ2 are defined onBr respectively by,

(Θ1y)(t) =







0, t∈(−∞,0]

[−g(y+ ˆφ)−u(0, φ)] +u(t, yt+ ˆφt), t∈J.

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(Θ2y)(t) =











0 , t∈(−∞,0]

1 Γ(q)

P

0<tk<t

Rtk

tk−1(tk−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds + _Γ(q)¹ Rt

tk(t−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds

− _Γ(q)¹ P

0<tk<t

Rt_k

tk−1(tk−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds

− _Γ(q)¹ R_t

tk(t−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds + _Γ(q)¹ P

0<tk<t

Rt_k

tk−1(t_k−s)^q−1f(s, ys+ ˆφs, H(ys+ ˆφs))ds + _Γ(q)¹ Rt

tk(t−s)^q−1f(s, y_s+ ˆφ_s, , H(y_s+ ˆφ_s))ds +P

0<tk<tI_k(y(t⁻_k) + ˆφ(t⁻_k)), t∈J.

We takey1, y2∈Br arbitrarily.

By (H2) and (H6), we have

||(Θ₁y1)(t)−(Θ1y2)(t)|| ≤ δ1||y₁−y2||_B

ϑ0 +λ1`||y₁−y2||_B

ϑ0

since||y_t||_B_ϑ ≤`||y||_B

ϑ0

||(Θ₁y₁)(t)−(Θ₁y₂)(t)|| ≤ (δ₁+λ₁`)||y₁−y₂||_B

ϑ0

≤ ||y₁−y₂||_B

ϑ0

sinceδ1+λ1` <1. Therefore Θ1 is a contraction.

Next, we prove that Θ₂ is continuous onB_r. Let

y_m ^∞_k=0 ⊆B_r, withy_m →y inB_r . By (H1), (H3), (H4) and (H5) we have

||(Θ₂y)(t)−(Θ2ym)(t)|| ≤ 1 Γ(q)

X

0<tk<t

Z tk

tk−1

(tk−s)^q−1||A(s, y+ ˆφ)(y+ ˆφ)(s)

−A(s, y_m+ ˆφ)(ym+ ˆφ)(s)||ds + 1

Γ(q) Z t

t_k

(t−s)^q−1||A(s, y+ ˆφ)(y+ ˆφ)(s)

−A(s, y_m+ ˆφ)(y_m+ ˆφ)(s)||ds + 1

Γ(q) X

0<tk<t

Z tk

tk−1

(t_k−s)^q−1||A(s, y+ ˆφ)u(s, y_s+ ˆφ_s)

−A(s, y_m+ ˆφ)u(s, yms+ ˆφs)||ds + 1

Γ(q) Z t

tk

(t−s)^q−1||A(s, y+ ˆφ)u(s, ys+ ˆφs)

−A(s, y_m+ ˆφ)u(s, yms+ ˆφs)||ds + 1

Γ(q) X

0<tk<t

Z tk

tk−1

(t_k−s)^q−1||f(s, y_s+ ˆφ_s, H(y_s+ ˆφ_s))

−f(s, y_ms+ ˆφ_s, H(y_ms+ ˆφ_s))||ds + 1

Γ(q) Z t

tk

(t−s)^q−1||f(s, y_s+ ˆφ_s, H(y_s+ ˆφ_s))

−f(s, yms+ ˆφs, , H(yms+ ˆφs))||ds

+ X

0<tk<t

||I_k(y(t⁻_k) + ˆφ(t⁻_k))−I_k(y_m(t⁻_k) + ˆφ(t⁻_k))||

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≤ (n+ 1)b^q

Γ(q+ 1)[(M[r+||φ||_B_ϑ+|φ(0)|] +K)(1 +`λ₁) +M(r+|φ(0)|+||φ||_B_ϑ+λ1[`(r+|φ(0)|) +||φ||_B_ϑ] +λ2) +K1`(1 + P1b

(q+ 1))]||y−ym||_B

ϑ0

+ X

0<tk<t

||I_k(y(t⁻_k) + ˆφ(t⁻_k))−I_k(y_m(t⁻_k) + ˆφ(t⁻_k))||.

||(Θ₂y)(t)−(Θ2ym)(t)|| →0 as m→ ∞ Thus , Θ2 is continuous.

Next, we prove that

Θ₂y :y∈B_r is a family of equicontinuous functions.

Let 0< t1 < t2≤b. Then

||(Θ₂y)(t₂)−(Θ₂y)(t₁)|| ≤ I₁+I₂+I₃+I₄+I₅+I₆+I₇ where

I₁ = 1

Γ(q)[|| X

0<tk<t2

Z _t_k

tk−1

− X

0<tk<t1

Z _t_k

tk−1

(t_k−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds||],

I₂ = 1 Γ(q)

h||

Z t2

tk

(t₂−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds− Z t1

tk

(t₁−s)^q−1A(s, y+ ˆφ)(y+ ˆφ)(s)ds||i ,

I3 = 1

Γ(q)[|| X

0<tk<t2

Z tk

tk−1

(tk−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds

− X

0<tk<t1

Z t_k tk−1

(t_k−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds||],

I4 = 1 Γ(q)

h

||

Z t2

tk

(t2−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds− Z t1

tk

(t1−s)^q−1A(s, y+ ˆφ)u(s, ys+ ˆφs)ds||i ,

I5 = 1 Γ(q)

h

|| X

0<tk<t2

Z tk

tk−1

(tk−s)^q−1f(s, ys+ ˆφs, H(ys+ ˆφs))ds

− X

0<tk<t1

Z tk

tk−1

(tk−s)^q−1f(s, ys+ ˆφs, H(ys+ ˆφs))ds||i ,

I6 = 1 Γ(q)

h

||

Z t2

t_k

(t2−s)^q−1f(s, ys+ ˆφs, H(ys+ ˆφs))ds− Z t1

t_k

(t1−s)^q−1f(s, ys+ ˆφs, H(ys+ ˆφs))ds||i ,

I₇ = || X

0<tk<t2

I_k(y(t⁻_k) + ˆφ(t⁻_k))− X

0<tk<t1

I_k(y(t⁻_k) + ˆφ(t⁻_k))||.

Clearly I1, I2, I3, I4, I5, I6, I7 tends to 0 whent1→t2.

Therefore, we conclude thatlim_t₁→t₂I_i= 0, i= 1,2,3,4,5,6,7. Hence Θ₂B_r is equicontinuous.

Next , we claim that Θ2Br is precompact.

(9)

Let 0< t≤b be fixed and letbe a real number satisfying 0< < b. Fory∈B_r , we define (Θ₂y)(t) = 1

Γ(q) X

0<tk<t

Z tk

tk−1

+ 1 Γ(q)

Z t−

tk

− 1 Γ(q)

X

0<tk<t

Z tk

tk−1

(t_k−s)^q−1A(s, y+ ˆφ)u(s, y_s+ ˆφ_s)ds

− 1 Γ(q)

Z t−

tk

(t−s)^q−1A(s, y+ ˆφ)u(s, y_s+ ˆφ_s)ds + 1

Γ(q) X

0<tk<t

Z _t_k

tk−1

(t_k−s)^q−1f(s, y_s+ ˆφ_s, H(y_s+ ˆφ_s))ds

+ 1 Γ(q)

Z t−

tk

(t−s)^q−1f(s, y_s+ ˆφ_s, , H(y_s+ ˆφ_s))ds

+ X

0<tk<t

I_k(y(t⁻_k) + ˆφ(t⁻_k))

since the operator A(t,x) is compact fort >0, for everysufficiently small, 0< < b,

||(Θ₂y)(t)−(Θ₂y)(t)|| = || 1 Γ(q)

Z t t−

− 1 Γ(q)

Z t t−

(t−s)^q−1A(s, y+ ˆφ)u(s, y_s+ ˆφ_s)ds + 1

Γ(q) Z t

t−

(t−s)^q−1f(s, y_s+ ˆφ_s, , H(y_s+ ˆφ_s))ds||

≤ 1

Γ(q+ 1)[(M L+K)(`⁻¹L₁+λ₁L₁+λ₂) +(K1L1+K1b(P1L1+P2) +K2)]^q.

Therefore, letting→0, we see that there are relatively compact sets arbitrarily close to the set

(Θ₂y)(t) : y∈Br . Hence

(Θ2y)(t) :y∈Br is relatively compact in B_ϑ⁰.

As a consequence of the above steps and the Arzela-Ascoli Theorem, we can conclude that Θ₂ is a compact operator. These arguments enable us to conclude that Θ = Θ₁+ Θ₂ is a condensing map on B_r and Theorem 2.5 gives the conclusion that the system 1.1 has a solution.

4. An Example

Consider the following fractional integrodifferential equation with impulsive condition of the form











D^q_t(x(t)−_(9+e^e_t^−t_)(1+x)^x ) = ¹₄sinx(t)[x(t)− _(9+e^e^−t_t_)(1+x)^x ] +_(t+2)¹ 2

|x|

(1+|x|)+¹₄Rs

0 e⁻³^xsds

∆x|_t=1 2 = ^|x(

1 2)⁻| 3+|x(¹₂)⁻|

x(0) + _5+x^x =x₀

(4.1)

where 0< q≤1 TakeJ = [0,1], b = 1 Let A(t, x) = ¹₄sinx(t) ,H(x_s) =Rs

0 e⁻³^xsds,

(10)

f(t, x, H(x_s)) = _(t+2)¹ 2

|x|

(1+|x|)+¹₄Rs

0 e⁻³^xsds,

u(t, x) = _(9+e^et^−t)(1+x)^x , wheret∈J,x∈X=R.

Ifx, y∈X andt∈J then we have

||A(t, x)−A(t, y)|| ≤ ¹₄||x−y||,||H(x_s)−H(ys)|| ≤ ¹₃||x−y||,

||f(t, x, H(xs))−f(t, y, H(ys))|| ≤ ¹₄[||x−y||+||H(x_s)−H(ys)||],

||I_k(x)−I_k(y)|| ≤ ¹₃||x−y||,||u(t, x_t)−u(t, y_t)|| ≤ ₁₀¹||x_t−y_t||,||g(x)−g(y)|| ≤ ¹₅||x−y||.

hereP1= ¹₃, K1= ¹₄, µ= ¹₃, λ1 = ₁₀¹, δ1= ¹₅. Letϑ(t) =e^t, therefore `=R0

−∞ϑ(t)dt=R0

−∞e^tdt= 1<+∞

henceδ₁+λ₁`= ₁₀³ <1.

For someq∈[0,1], all the hypotheses of the Theorem 2.5 are satisfied. Hence the problem 4.1 has a solution.

Acknowledgements:

The authors thank the referee for the valuable comments and suggestions.

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