Impulsive Hyperbolic System of Partial Differential Equations of Fractional Order with Delay

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Mouffak Benchohra, Zohra Boutefal^∗

Impulsive Hyperbolic System of Partial Differential Equations of Fractional Order with Delay

Abstract. This paper deals with the existence of solutions to impulsive partial hyperbolic differential equations with finite delay, involving the Caputo fractional derivative. Our results will be obtained using Krasnoselskii fixed point theorem.

2000 Mathematics Subject Classification: 26A33, 34A37, 34K37, 35R11.

Key words and phrases: Impulsive partial hyperbolic differential equations, frac- tional order, solution, left-sided mixed Riemann-Liouville integral, Caputo fractional- order derivative, finite delay, fixed point.

1. Introduction. Fractional order models are found to be more adequate than integer order models in some real world problems. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes; see the books by Baleanu et al. [3], Hilfer [7], Tarasov [12], and the references therein. Recent developments on fractional differential equations from theoretical point of view are given in the books by Abbas et al. [1, 2], and Lakshmikantham et al. [10], and the references therein.

The theory of impulsive differential equations have become important in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There has been a significant development in impulse theory in recent years, especially in the area of impulsive differential equations and inclusions with fixed moments; see the mono- graphs of Abbas et al. [1], Benchohra et al. [4], Lakshmikantham et al. [9, 10], Samoilenko and Perestyuk [11], and the references therein.

In this paper, we study the existence of solutions for the following impulsive partial hyperbolic differential equations:

(1) (^cD^r_z_ku)(t, x) = f (t, x, u(t,x)); if (t, x) ∈ J^k, k = 0, . . . , m,

∗Corresponding author

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(2) u(t⁺_k, x) = u(t⁻_k, x) + Ik(u(t⁻_k, x)), if x∈ [0, b], k = 1, . . . , m,

(3) u(t, x) = φ(t, x); if (t, x)∈ ˜J,

(4) u(t, 0) = ϕ(t), t∈ [0, a], u(0, x) = ψ(x); x ∈ [0, b],

where J0 = [0, t1] × [0, b], J^k := (tk, tk+1] × [0, b], k = 1, . . . , m, z^k = (tk, 0), k = 0, . . . , m, a, b, α, β > 0, J = [0, a] × [0, b], ˜J = [−α, a] × [−β, b]\(0, a] × (0, b], ^cD^r₀is the Caputo fractional derivative of order r = (r₁, r2) ∈ (0, 1]×(0, 1], ϕ : [0, a] → Rⁿ, ψ : [0, b]→ Rⁿare given absolutely continuous functions with ϕ(t) = φ(t, 0), ψ(x) = φ(0, x) for each (t, x) ∈ J, 0 = t0 < t1 < · · · < t^m < tm+1 = a, f : J × C → Rⁿ, Ik : Rⁿ → Rⁿ, k = 1, . . . , m, φ : ˜J → Rⁿ, are given functions and C := C([−α, 0] × [−β, 0], Rⁿ) is the space of piecewise continuous functions on [−α, 0] × [−β, 0].

If u : [−α, 0] × [−β, 0] −→ Rⁿ, then for any (t, x) ∈ J define u(t,x) by u(t,x)(s, τ) = u(t + s, x + τ).

This paper is organized as follows. In Section 2, we introduce some prelimi- nary results that will need later. Section 3 will be devoted to our main result. An illustrating example will be presented in Section 4.

2. Preliminaries. For further purpose, we collect in this section a few auxiliary results which will be needed in the sequel.

By C(J,Rⁿ) we denote the Banach space of continuous functions u : J → Rⁿ, with the usual supremum norm

kuk∞= sup{|u(t, x)|, (t, x) ∈ J}.

Let L¹(J,Rⁿ) be the Banach space of measurable functions u : J → Rⁿ which are Lebesgue integrable, equipped with the norm

kuk^L¹= Z a

0

Z b

0 |u(t, x)|dxdt.

C=C_(α,β)= {u : [−α, 0] × [−β, 0] → Rⁿ: continuous and there exist τk∈ (−α, 0) with u(τ_k⁻,x) and u(t˜ ⁺_k,x), k = 1, . . . , m, exist for any ˜˜ x∈ [−β, 0] with u(τ_k⁻,x) = u(τ˜ k,x)˜

.

This set is a Banach space with the norm

kuk^C = sup

(t,x)∈[−α,0]×[−β,0]|u(t, x)|.

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Definition 2.1 ([8]) Let r1, r2 > 0 and r = (r1, r2). For u ∈ L¹(J,Rⁿ), the expression

(I_z^r_ku)(t, x) = 1 Γ(r1)Γ(r2)

Z t tk

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹u(s, τ )dτ ds,

where Γ(.) is the gamma function, is called the left-sided mixed Riemann-Liouville integral of order r.

Definition 2.2 ([8]) For u ∈ L¹(J,Rⁿ), the Caputo fractional-order derivative of order r is defined by the expression

(^cD_z^r_ku)(t, x) = (I_z^1−r_k ∂²

∂t∂xu)(t, x).

In the sequel we will make use of the following generalization of Gronwall’s lemma for two independent variables and singular kernel.

Lemma 2.3 ([6]) Let υ : J → [0, ∞) be a real function and ω(., .) be a nonnegative, locally integrable function on J. If there are constants c > 0 and 0 < r1, r2< 1such that

υ(t, x)¬ ω(t, x) + c Z t

0

Z x 0

υ(s, τ )

(t − s)^r¹(x − τ)^r²dτ ds, then there exists a constant δ = δ(r1, r2)such that

υ(t, x)¬ ω(t, x) + δc Z t

0

Z x 0

ω(s, τ )

(t − s)^r¹(x − τ)^r²dτ ds, for every (t, x) ∈ J.

Theorem 2.4 (Burton-Kirk, [5]) Let X be a Banach space, and A, B : X → X two operators satisfying:

(i) Ais completely continuous, and (ii) B is a contraction.

Then either

(a) the operator equation u = A(u) + B(u) has a solution, or

(b) the set E = {u ∈ X : u = λA(u) + λB(^uλ)} is unbounded for λ ∈ (0, 1).

3. Main Results. In this section, we give our main existence result for problem (1)-(4).

Set Jk = (tk, tk+1] × (0, b]. Consider the Banach space P C := P C(J,Rⁿ)

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=

u : J → Rⁿ: u ∈ C(J^k,Rⁿ), k = 1, . . . , m, and there exist u(t⁻_k, x) and u(t⁺_k, x), k = 1, . . . , m, with u(t⁻_k, x) = u(tk, x)

, with the norm

kuk^{P C}= sup

(t,x)∈J|u(t, x)|.

Set

Ω = {u : [−α, a] × [−β, b] −→ Rⁿ), u|[−α,0]×[−β,0]∈ C and u|[0,a]×[0,b]∈ P C}.

Definition 3.1 A function u ∈ Ω such that its mixed derivative D²tx exists on Jk; k = 0, 1, . . . , m is said to be a solution of (1)-(4) if u satisfies the condition (3) on ˜J, the equation (1) on Jk and conditions (2), (4) are satisfied on J.

Lemma 3.2 ([1]) Let 0 < r1, r2¬ 1 and let h : J → Rⁿ be continuous. A function uis a solution of the fractional integral equation

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u(t, x) =











φ(t, x) if (t, x) ∈ ˜J,

µ(t, x) + X

0<tk<t

Ik(u(t⁻_k, x)) − Ik(u(t⁻_k,0))

if (t, x) ∈ J, +_Γ(r₁_)Γ(r¹ ₂₎ X

0<tk<t

Z tk t_k−1

Z x 0

(tk− s)^r¹⁻¹(x − y)^r²⁻¹h(s, y)dyds k= 1, . . . , m, +_Γ(r₁_)Γ(r¹ ₂₎

Z t t_k

Z x 0

(t − s)^r¹⁻¹(x − y)^r²⁻¹h(s, y)dyds,

if and only if u is a solution of the fractional initial value problem (6) ^cD^ru(t, x) = h(t, x), (t, x) ∈ J^k, k = 0, . . . , m,

(7) u(t⁺_k, x) = u(t⁻_k, x) + Ik(u(t⁻_k, x)), k = 1, . . . , m.

Our result is based upon the fixed point theorem due to Burton and Kirk. Let us introduce the following hypotheses which are assumed to hold in the sequel.

(H1) The functions Ik:Rⁿ→ Rⁿ and f : J × C → Rⁿ are continuous.

(H2) There exist p, q ∈ C(J, R⁺) such that

kf(t, x, u)k ¬ p(t, x) + q(t, x)kuk^C, for (t, x)∈ J and each u ∈ C.

(H3) There exists l > 0 such that

kI^k(u) − I^k(v)k ¬ lku − vk for each u, v ∈ Rⁿ, k = 1, . . . , m.

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Theorem 3.3 Assume that hypotheses (H1)-(H3) hold. If

(8) 2ml < 1,

then the IV P (1)-(4) has at least one solution on J.

Proof. We shall reduce the existence of solutions of (1)-(4) to a fixed point problem.

Consider the operator N : Ω −→ Ω defined by

N (u)(t, x) =











φ(t, x) if (t, x) ∈ ˜J,

µ(t, x) + X

0<tk<t

Ik(u(t⁻_k, x))− I^k(u(t⁻_k, 0))

+_Γ(r₁_)Γ(r¹ ₂₎ X

0<tk<t

Z tk

tk−1

Z x 0

(tk− s)^r¹⁻¹(x − y)^r²⁻¹

×f(s, y, u(s,y))dyds +_Γ(r₁_)Γ(r¹ ₂₎

Z t tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹

×f(s, y, u(s,y))dyds if (t, x) ∈ J,

k = 1, . . . , m, Consider the operators F, G : Ω → Ω defined by,

G(u)(t, x) =











φ(t, x), (t, x) ∈ ˜J,

+_Γ(r₁_)Γ(r¹ ₂₎ X

0<tk<t

Z tk

tk−1

Z x

0 (tk− s)^r¹⁻¹(x − y)^r²⁻¹ k = 1, . . . , m

×f(s, y, u^(s,y))dyds +_Γ(r ¹

1)Γ(r₂)

Z t tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹

×f(s, y, u(s,y))dyds, (t, x) ∈ J.

and

F (u)(t, x) =





0, (t, x) ∈ ˜J,

µ(t, x) + X

0<tk<t

(Ik(u(t⁻_k, x))− I^k(u(t⁻_k, 0))), (t, x)∈ J.

Then the problem of finding the solution of the IV P (1)–(3) is reduced to finding the solutions of the operator equation F (u) + G(u) = u. We shall show that the operators F and G satisfy the conditions of Theorem 2.4. The proof will be given by several steps.

Step 1: G is continuous.

Let {uⁿ} be a sequence such that uⁿ→ u in C, then for each (t, x) ∈ J kG(un)(t, x) − G(u)(t, x)k

¬ 1

Γ(r1)Γ(r2) Xm k=1

Z tk t_k−1

Z x 0

(tk− s)^r¹⁻¹(x − y)^r²⁻¹kf(s, y, un(s,y)) − f(s, y, u(s,y))kdyds

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+ 1 Γ(r1)Γ(r2)

Z t tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹kf(s, y, un(s,y)) − f(s, y, u(s,y))kdyds.

¬kf(., ., un(.,.)) − f(., ., u(.,.))k Γ(r1)Γ(r2)

Xm k=1

Z tk tk−1

Z x 0

(tk− s)^r¹⁻¹(x − y)^r²⁻¹dyds

+ kf(., ., un(.,.)) − f(., , u(.,.))k Γ(r1)Γ(r2)

Z t tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹dyds.

Since f is a continuous function, we have

kG(uⁿ) − G(u)k∞¬2a^r¹b^r²kf(., ., un(.,.)) − f(., ., u(.,.))k∞

Γ(r1+ 1)Γ(r2+ 1) → 0 as n → ∞ Thus G is continuous.

Step 2: G maps bounded sets into bounded sets in C.

Indeed, it is enough show that for any η^∗, there exists a positive constant l such that, for each u ∈ B^η^∗= {u ∈ C : kuk∞¬ η^∗} we have kG(u)k^C¬ l.

By (H2) we have for each (t, x) ∈ J,

kG(u)(t, x)k ¬ 1 Γ(r1)Γ(r2)

Xm k=1

Z tk

t_k−1

Z x 0

(tk− s)^r¹⁻¹(x − y)^r²⁻¹kf(s, y, u^(s,y))kdyds

+ 1

Γ(r1)Γ(r2) Z t

tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹kf(s, y, u(s,y))kdyds

¬kpk∞+ kqk∞η^∗ Γ(r1)Γ(r2)

Xm k=1

Z tk

tk−1

Z x

0 (tk− s)^r¹⁻¹(x − y)^r²⁻¹dyds + kpk∞+ kqk∞η^∗

Γ(r1)Γ(r2) Z t

tk

Z x

0 (t − s)^r¹⁻¹(x − y)^r²⁻¹dyds.

Thus

kG(u)k^C¬ 2a^r¹b^r²(kpk∞+ kqk∞η^∗) Γ(r1+ 1)Γ(r2+ 1) := l^∗. Hence kG(u)kf^{P C}¬ l^∗.

Step 3: G maps bounded sets into equicontinuous sets in C.

Let (µ₁, x1), (µ₂, x2) ∈ (0, a] × (0, b], µ1 < µ2, x1< x2, Bη be a bounded set as in Step 2,

let u ∈ B^η^∗ be a bounded set of C as in Step 2. Then

kG(u)(µ2, x2) − G(u)(µ1, x1)k

¬ 1

Γ(r1)Γ(r2)

Xm k=1

Z tk t_k−1

Z x₁ 0

(tk− s)^r¹⁻¹[(x2− τ)^r²⁻¹− (x1− τ)^r²⁻¹] × kf(s, τ, u(s,τ))kdτds

+ 1

Γ(r1)Γ(r2)

Xm k=1

Z tk tk−1

Z x2 x₁

(tk− s)^r¹⁻¹(x2− τ)^r²⁻¹kf(s, τ, u(s,τ))kdτds

+ 1

Γ(r1)Γ(r2)

Z µ₁ 0

Z x₁ 0

[(µ2− s)^r¹⁻¹(x2− τ)^r²⁻¹− (µ1− s)^r¹⁻¹(x1− τ)^r²⁻¹] × kf(s, τ, u(s,τ))kdτds

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+ 1 Γ(r1)Γ(r2)

Z µ2 µ1

Z x2 x1

(µ₂− s)^r¹⁻¹(x₂− τ)^r²⁻¹|f(s, τ, u(s,τ))|dτds

+ 1

Γ(r1)Γ(r2)

Z µ₁ 0

Z x₂ x1

(µ2− s)^r¹⁻¹(x2− τ)^r²⁻¹kf(s, τ, u(s,τ))kdτds

+ 1

Γ(r1)Γ(r2)

Z µ₂ µ1

Z x₁ 0

(t2− s)^r¹⁻¹(x2− τ)^r²⁻¹kf(s, τ, u(s,τ))kdτds

¬ kpk∞+ kqk∞η^∗ Γ(r₁+ 1)Γ(r₂+ 1)

Xm k=1

Z tk tk−1

Z x1 0

(t_k− s)^r¹⁻¹[(x₂− τ)^r²⁻¹− (x1− τ)^r²⁻¹]dτds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Xm k=1

Z t_k tk−1

Z x₂ x1

(tk− s)^r¹⁻¹(x2− τ)^r²⁻¹dτ ds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Z µ₁ 0

Z x₁ 0

[(µ2− s)^r¹⁻¹(x2− τ)^r²⁻¹− (µ1− s)^r¹⁻¹(x1− τ)^r²⁻¹]dτds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Z µ2 µ1

Z x2 x1

(µ₂− s)^r¹⁻¹(x₂− τ)^r²⁻¹dτ ds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Z µ₁ 0

Z x₂ x1

(µ2− s)^r¹⁻¹(x2− τ)^r²⁻¹dτ ds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Z µ₂ µ1

Z x₁ 0

(µ2− s)^r¹⁻¹(x2− τ)^r²⁻¹dτ ds

¬ kpk∞+ kqk∞η^∗ Γ(r₁+ 1)Γ(r₂+ 1)

Xm k=1

Z tk tk−1

Z x1 0

(t_k− s)^r¹⁻¹[(x₂− τ)^r²⁻¹− (x1− τ)^r²⁻¹]dτds

+ kpk∞+ kqk∞η^∗ Γ(r1+ 1)Γ(r2+ 1)

Xm k=1

Z t_k tk−1

Z x₂ x1

(tk− s)^r¹⁻¹(x2− τ)^r²⁻¹dτ ds

+ kpk∞+ kqk∞η^∗

Γ(r1+ 1)Γ(r2+ 1)[2x^r₂²(µ2− µ1)^r¹+ 2µ^r₂¹(x2− x1)^r² + µ^r₁¹x^r₁²− µ^r₂¹x^r₂²− 2(µ2− µ1)^r¹(x2− x1)^r²].

As µ1→ µ², x1→ x² the right-hand side of the above inequality tends to zero.

As a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we can conclude that G : gP C→ gP C is continuous and completely continuous.

Step 4: F is a contraction.

Let u, v ∈ C, then we have for each (t, x) ∈ J kF (u)(t, x) − F (v)(t, x)k

¬ Xm k=1

(kIk(u(t⁻_k, x))− I^k(v(t⁻_k, x))k + kI^k(u(t⁻_k, 0))− I^k(v(t⁻_k, 0))k)

¬ Xm k=1

l(ku − vk^C+ ku − vk^C)

¬ 2mlku − vk^C.

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Thus

kF (u) − F (v)k^C¬ 2mlku − vk^C. Hence by (8), F is a contraction.

Step 5: (A priori bounds)

Now it remains to show that the set E = {u ∈ C : u = λF (u

λ) + λG(u) for some λ ∈ (0, 1)}

is bounded. Let u ∈ E, then u = λF (^uλ) + λG(u). Thus, for each (t, x) ∈ J we have

u(t, x) = λµ(t, x) + Xm k=1

λ

Ik

u(t⁻_k, x) λ

+ Ik

u(t⁻_k, 0) λ

+ λ

Γ(r1)Γ(r2) Xm k=1

Z tk

t_k−1

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹f (s, τ, u(s,τ))dτds

+ λ

Γ(r₁)Γ(r₂) Z t

tk

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹f (s, τ, u(s,τ))dτds.

This implies by (H2) and (H3) that, for each (t, x) ∈ J, we have

ku(t, x)k ¬ kµ(t, x)k +

Xm k=1

(kIku(t⁻_k, x)k − kIk(0)k + kIku(t⁻_k,0)k − kIk(0)k)

+ 2

Xm k=1

kIk(0)k + kpk∞

Γ(r1)Γ(r2)

Xm k=1

Z tk t_k−1

Z x 0

(t − s)^r¹⁻¹(x − τ)^r²⁻¹ku(s,τ)kCdτ ds

+ kqk∞

Γ(r1)Γ(r2)

Xm k=1

Z tk tk−1

Z x 0

(t − s)^r¹⁻¹(x − τ)^r²⁻¹dτ ds

+ kpk∞

Γ(r₁)Γ(r₂)

Z t 0

Z x 0

+ kqk∞

Γ(r1)Γ(r2)

Z t 0

Z x 0

(t − s)^r¹⁻¹(x − τ)^r²⁻¹dτ ds

¬ kµ(t, x)k + l

Xm k=1

(ku(t⁻_k, x)k + ku(t⁻_k,0)k) + 2I^∗

+ kpk∞

Γ(r₁)Γ(r₂)

Xm k=1

Z tk tk−1

Z x 0

+ kqk∞

Γ(r1)Γ(r2)

Xm k=1

Z tk tk−1

Z x 0

(t − s)^r¹⁻¹(x − τ)^r²⁻¹dτ ds

+ kpk∞

Γ(r1)Γ(r2)

Z t 0

Z x 0

+ kqk∞

Γ(r1)Γ(r2)

Z t 0

Z x 0

(t − s)^r¹⁻¹(x − τ)^r²⁻¹dτ ds,

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where

I^∗= Xm k=1

kI^k(0)k.

We consider the function γ defined by

γ(t, x) = sup{|u(s, τ)| : −α ¬ s ¬ t, −β ¬ τ ¬ x, 0 ¬ t ¬ a, 0 ¬ x ¬ b}.

Let (t^∗, x^∗) ∈ [−α, t] × [−β, x] be such that γ(t, x) = |u(t^∗, x^∗)|. If (t^∗, x^∗) ∈ J, then by the previous inequality, we have for (t, x) ∈ J

γ(t, x) ¬ kµ(t, x)k + 2mlγ(t, x) + 2I^∗

+ kpk∞

Γ(r₁)Γ(r₂)

X^m

k=1

Z tk

tk−1

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹γ(s, τ )dτ ds +

Z t 0

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹γ(s, τ )dτ ds

+ 2a^r¹b^r²kqk∞

Γ(r1+ 1)Γ(r2+ 1). If (t^∗, x^∗) ∈ ˜J, then γ(t, x) =kφk^C and the previous inequality holds. If (t, x) ∈ J, by Lemma 2.3 implies that there exists ˜k = ˜k(r2, r2) such that

γ(t, x) ¬ 1

1 − 2ml

kµ(t, x)k + 2I^∗+ 2a^r¹b^r²kqk∞

Γ(r1+ 1)Γ(r2+ 1)

×

1 + ˜k kpk∞

Γ(r2)Γ(r2) Z t

0

Z x

0 (t − s)^r¹⁻¹(x − τ)^r²⁻¹dτ ds

¬ 1

1 − 2ml

kµk∞+ 2I^∗+ 2a^r¹b^r²kqk∞

Γ(r1+ 1)Γ(r2+ 1)

×

1 + ˜k kpk∞a^r¹b^r² Γ(r2+ 1)Γ(r2+ 1)

:= ˜R.

Since for every (t, x) ∈ J, ku(t, x)k^C¬ γ(t, x). This shows that the set E is bounded.

As a consequence of Theorem 2.4 we deduce that F + G has a fixed point u which is a solution of problem (1)-(4).

4. An Example. As an application of our results we consider the following impulsive partial hyperbolic differential equations of the form

(9) (^cD₀^ru)(t, x) = 1

(10e^t+x+2)(1 + |u(t − 1, x − 2)|); if (t, x) ∈ (tk, t_k+1] × [0, 1], k = 1, . . . , m,

(10) u(t⁺_k, x) = u(t⁻_k, x) + 1

(6e^t+x+4)(1 + |u(t⁻k, x)|); x ∈ [0, 1], k = 1, . . . , m,

(11) u(t, x) = t + x², (t, x)∈ [−1, 1] × [−2, 1]\(0, 1] × (0, 1],

(12) u(t, 0) = t, t∈ [0, 1], u(0, x) = x², x∈ [0, 1].

(10)

Set

f (t, x, u) = 1

(10e^t+x+2)(1 + |u|); if (t, x) ∈ [0, 1] × [0, 1]

Ik(u(t⁻_k, x)) = 1

(6e^t+x+4)(1 + |u(t⁻k, x)|); x ∈ [0, 1].

For each u, ¯u ∈ Rⁿ and (t, x) ∈ [0, 1] × [0, 1] we have kI^k(u) − I^k(v)k ¬ 1

6e⁴ku − vk f^{P C}.

Hence (H3) is satisfied with l = _6e¹4. We shall show that condition (8) holds with a = b = 1 and m = 1. Indeed,

2ml = 1 3e⁴ < 1,

which is satisfied for each (r1, r2) ∈ (0, 1]×(0, 1]. Consequently Theorem 2.4 implies that problem (9)-(12) has a solution defined on [−1, 1] × [−2, 1].

References

[1] S. Abbas, M. Benchohra and G.M. N’Gu´er´ekata, Topics in Fractional Differential Equations, Springer, New York, 2012.

[2] S. Abbas, M. Benchohra and G.M. N’Gu´er´ekata, Advanced Fractional Differential and Inte- gral Equations, Nova Science Publishers, New York, 2014.

[3] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012.

[4] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclu- sions, Hindawi Publishing Corporation, Vol 2, New York, 2006.

[5] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr.

189 (1998), 23-31.

[6] D. Henry, Geometric theory of Semilinear Parabolic Partial Differential Equations, Springer- Verlag, Berlin-New York, 1989.

[7] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

[8] A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Frac- tional Differential Equations. North-Holland Mathematics Studies20. Elsevier Science B.V., Amsterdam, 2006.

[9] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[10] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.

[11] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

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[12] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.

Mouffak Benchohra

Laboratory of Mathematics, University of Sidi Bel-Abb`es P.O. Box 89, 22000 Sidi Bel-Abb`es, Algeria

and Department of Mathematics, Faculty of Science, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail: benchohra@univ-sba.dz Zohra Boutefal

Laboratory of Mathematics, University of Sidi Bel-Abb`es P.O. Box 89, 22000 Sidi Bel-Abb`es, Algeria

(Received: 21.12.2013)