FRACTIONAL ORDER IMPULSIVE PARTIAL HYPERBOLIC DIFFERENTIAL INCLUSIONS

WITH VARIABLE TIMES Sa¨ıd Abbas

Laboratoire de Math´ematiques, Universit´e de Sa¨ıda B.P. 138, 20000, Sa¨ıda, Alg´erie

e-mail: [email protected]

Mouffak Benchohra

Laboratoire de Math´ematiques, Universit´e de Sidi Bel-Abb`es B.P. 89, 22000, Sidi Bel-Abb`es, Alg´erie

e-mail: [email protected]

and

Lech G´orniewicz∗

Institute of Mathematics, Kazimierz Wielki University Weyssenhoffa 11, 85–072 Bydgoszcz, Poland

J. Schauder Center for Nonlinear Studies

University of Nicolaus Copernicus, 87–100 Toru´n, Poland e-mail: [email protected]

Abstract

This paper deals with the existence of solutions to some classes of partial impulsive hyperbolic differential inclusions with variable times involving the Caputo fractional derivative. Our works will be consid-ered by using the nonlinear alternative of Leray-Schauder type. Keywords and phrases: impulsive functional differential inclusions, fractional order, solution, left-sided mixed Riemann-Liouville integral, Caputo fractional-order derivative, variable times, fixed point. 2010 Mathematics Subject Classification: 26A33, 34A60.

∗

1. Introduction

The paper deals with the existence of solutions for the following impulsive partial fractional order initial value problem (IVP for short), for the system

(1)

(c_{D}r

0u)(x, y) ∈ F (x, y, u(x, y)), where (x, y) ∈ J,

x6= xk(u(x, y)), k = 1, . . . , m,

(2) u(x

+_{, y) = I}

k(u(x, y)), where (x, y) ∈ J,

x= xk(u(x, y)), k = 1, . . . , m,

(3) u(x, 0) = ϕ(x), u(0, y) = ψ(y), x ∈ [0, a], y ∈ [0, b],

where J = [0, a] × [0, b], a, b > 0, cD_{0}r is the fractional Caputo derivative
of order r = (r1, r2) ∈ (0, 1] × (0, 1], 0 = x0 < x1 <· · · < xm < xm+1 = a,

F : J × Rn _{→ P(R}n_{) is a compact valued multivalued map, P(R}n_{) is the}

family of all subsets of Rn, xk: Rn→ R, Ik : Rn → Rn, k= 1, 2, . . . , m are

given functions and ϕ : [0, a] → Rn_{, ψ} _{: [0, b] → R}n_{are absolutely continuous}

functions with ϕ(0) = ψ(0).

Next we consider the following initial value problem for impulsive partial neutral functional differential inclusions

(4)

c_{D}r

0[u(x, y) − g(x, y, u(x, y))] ∈ F (x, y, u(x, y)), where (x, y) ∈ J,

x6= xk(u(x, y)), k = 1, . . . , m,

(5) u(x

+_{, y) = I}

k(u(x, y)), where (x, y) ∈ J,

x= xk(u(x, y)), k = 1, . . . , m,

(6) u(x, 0) = ϕ(x), u(0, y) = ψ(y), x ∈ [0, a], y ∈ [0, b],

where F, ϕ, ψ, xk, Ik; k = 1, 2, . . . , m are as in problem (1)–(3) and g : J ×

Rn→ Rn_{is a given function.}

of fractional order (see [15, 17, 18, 21, 27]). In recent years, several quali-tative results for ordinary and partial fractional differential equations have been obtained; see the monographs of Kilbas et al. [25], Miller and Ross [28], Samko [30], the papers of Abbas and Benchohra [1, 2], Agarwal et al. [7], Belarbi et al. [8], Belmekki et al. [9], Benchohra et al. [10, 11, 13], Diethelm [15, 16], Kilbas and Marzan [23], Mainardi [27], Podlubny [29], Vityuk and Golushkov [34], and the references therein. The problem of existence of so-lutions to Cauchy-type problems for ordinary differential equations of frac-tional order in spaces of integrable functions was studied in numerous works (see [22, 32]), a similar problem in spaces of continuous functions was studied in [33].

The theory of impulsive integer order differential equations has 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 monographs of Benchohra et al. [12], Lakshmikantham et al. [26], and Samoilenko and Perestyuk [31], and the references therein. Recently in [3, 4, 6], we have considered some classes of hyperbolic differential equations involving the Caputo fractional derivative and impulses at fixed time. The theory of impulsive differential equations with variable time is relatively less developed due to the difficulties created by the state-dependent impulses [5].

This paper initiates the study of fractional order hyperbolic differential inclusions at variable times. Our existence results will be obtained using the nonlinear alternative of Leray-Schauder type.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts
from multivalued analysis which are used throughout this paper. By
L1(J, Rn_{) we denote the space of Lebesgue-integrable functions w : J → R}n

with the norm

kwk1 = Z a 0 Z b 0 kw(x, y)kdydx,

Let L∞

(J, Rn_{) be the Banach space of measurable functions w : J → R}n

which are bounded, equipped with the norm

kwkL∞ = inf{c > 0 : kw(x, y)k ≤ c, a.e. (x, y) ∈ J},

and AC(J, Rn) be the space of absolutely continuous functions from J into Rn.

Let a1 ∈ [0, a], z+ = (a+1,0) ∈ J, Jz = [a1, a] × [0, b], r1, r2 > 0 and

r= (r1, r2). For w ∈ L1(Jz, Rn), the expression

(I_{z}r+w)(x, y) =
1
Γ(r1)Γ(r2)
Z x
a+1
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{w(s, t)dtds}
is called the left-sided mixed Riemann-Liouville integral of order r, where
Γ(.) is the (Euler’s) Gamma function defined by Γ(ξ) =R∞

0 tξ−1e

−t_{dt, ξ >}_{0.}

Denote by D2

xy := ∂

2

∂x∂y the mixed second order partial derivative.

Definition 2.1 ([34]). For w ∈ L1_{(J}

z, Rn) where D2xyf is Lebesque

inte-grable on [xk, xk+1]×[0, b], k = 0, . . . , m, the Caputo fractional-order

deriva-tive of order r is defined by the expression (c_{D}r

z+w)(x, y) = (I

1−r

z+ D2_{xy}w)(x, y).
In the definition above by 1 − r we mean (1 − r1,1 − r2) ∈ (0, 1] × (0, 1].

Let (X, k · k) be a Banach space. Let Pcl(X) = {Y ∈ P(X) : Y closed},

Pb(X) = {Y ∈ P(X) : Y bounded}, Pcp(X) = {Y ∈ P(X) : Y compact}

and Pcp,c(X) = {Y ∈ P(X) : Y compact and convex}. A multi-valued map

G : X → 2X _{has convex (closed) values if G(x) is convex (closed) for all}

x ∈ X. G is bounded on bounded sets if G(B) is bounded in X for each
bounded set B of X, i.e., sup_{x∈B}{sup{kyk : y ∈ G(x)}} < ∞. G is called
upper semi-continuous (u.s.c.) on X if for each x0 ∈ X the set G(x0) is a

nonempty, closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood M of x0 such that G(M ) ⊆ N . G

is lower semi-continuous (l.s.c.) if the set {x ∈ X : G(x) ∩ A 6= ∅} is open for any open subset A ⊆ X. G is said to be completely continuous if G(B) is relatively compact for every bounded subset B ⊆ X. If the multi-valued G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., xn→ x∗, yn→ y∗, yn∈ G(xn) imply

y∗ ∈ G(x∗)). G has a fixed point if there is x ∈ X such that x ∈ G(x).

Definition 2.2. A multi-valued map F : J × Rn _{→ P(R}n_{) is said to be}

(i) (x, y) 7→ F (x, y, u) is measurable for each u ∈ Rn_{;}

(ii) u 7→ F (x, y, u) is upper semi-continuous for almost each (x, y) ∈ J; (iii) For each q > 0, there exists φq∈ L1(J, R+) such that

kF (x, y, u)kP = sup{kvk : v ∈ F (x, y, u)} ≤ φq(x, y)

for all kuk ≤ q and for almost each (x, y) ∈ J. F is said to be Carath´eodory if (i) and (ii) hold.

Let (X, d) be a metric space induced from the normed space (X, k · k). Consider Hd: P(X) × P(X) −→ R+∪ {∞} given by Hd(A, B) = max sup a∈A d(a, B), sup b∈B d(A, b) ,

where d(A, b) = infa∈Ad(a, b), d(a, B) = infb∈Bd(a, b). Then, (Pb,cl(X), Hd)

is a metric space and (Pcl(X), Hd) is a generalized metric space (see [24]).

For more details on multi-valued maps we refer the reader to the books of Deimling [14], G´orniewicz [20], Kisielewicz [24].

For a function u defined on J we define the set

SF,u= {f ∈ L1(J, Rn) : f (x, y) ∈ F (x, y, u(x, y)) for a.e. (x, y) ∈ J},

which is known as the set of selection functions.

Theorem 2.3 [19] (Nonlinear alternative of Leray-Schauder type). Let X be a Banach space and C a nonempty convex subset of X. Let U be a nonempty open subset of C with 0 ∈ U and T : U → Pcp,c(C) be an upper

semicontinuous and compact operator. Then, either

(a) T has fixed points, or

(b) there exist u ∈ ∂U and λ ∈ [0, 1] with u ∈ λT (u).

To define the solutions to problem (1)–(3), we shall consider the space
Ω =u : J → Rn_{: there exist 0 = x}

0 < x1< x2 < ... < xm< xm+1 = a

such that xk= xk(u(xk, .)), and u(x−k, .), u(x+k, .) exist with

where Jk := (xk, xk+1] × [0, b]. This set is a Banach space with the norm

kukΩ = max{kukk, k = 0, . . . , m},

where uk is the restriction of u to Jk, k = 0, . . . , m.

3. Existence of solutions

In what follows, we will assume that F is an L1_{-Carath´eodory function. Let}

us start by defining what we mean by a solution to problem (1)–(3). Set

J′ := J\{(x1, y), . . . , (xm, y), y ∈ [0, b]}.

Definition 3.1. A function u ∈ Ω ∩ ∪m_{k=1}AC(Jk, Rn) whose r-derivative

exists on J′

is said to be a solution to (1)–(3) if there exists a function
f(x, y) ∈ F (x, y, u(x, y)) such that u satisfies (c_{D}r

0u)(x, y) = f (x, y) on J ′

and conditions (2), (3) are satisfied.

Let h, w ∈ C([xk, xk+1] × [0, b], Rn), zk = (xk,0), and

µk(x, y) = u(x, 0) + u(x+_{k}, y) − u(x+_{k},0), k = 0, . . . , m.

For the existence of solutions to problem (1)–(3), we need the following lemma:

Lemma 3.2 [3]. A function u ∈ AC([xk, xk+1] × [0, b], Rn); k = 0, . . . , m is

a solution of the differential equation

(cD_{z}r_{k}u)(x, y) = h(x, y); (x, y) ∈ [xk, xk+1] × [0, b],

if and only if u(x, y) satisfies

(7) u(x, y) = µk(x, y) + (Izrkh)(x, y); (x, y) ∈ [xk, xk+1] × [0, b].

Corollary 3.3. A function u∈ AC([xk, xk+1] × [0, b], Rn); k = 0, . . . , m is

a solution of the differential equation

c_{D}r

if and only if u(x, y) satisfies

(8) u(x, y) = µk(x, y)+w(x, y)−w(x, 0)−w(xk, y)+w(xk,0)+(Izrkh)(x, y), for all (x, y) ∈ [xk, xk+1] × [0, b].

In what follows, set µ(x, y) := µ0(x, y); (x, y) ∈ J.

Lemma 3.4. Let 0 < r1, r2 ≤ 1 and let h : J → Rn be continuous. A

function u is a solution of the fractional integral equation

(9) u(x, y) =
µ(x, y) + _{Γ(r} 1
1)Γ(r2)
Rx
0
Ry
0(x − s)r1
−1_{(y − t)}r2−1_{h(s, t)dtds;}
if (x, y) ∈ [0, x1] × [0, b],

ϕ(x) + Ik(u(xk, y)) − Ik(u(xk,0))

+_{Γ(r} 1
1)Γ(r2)
Rx
xk
Ry
0(x − s)r1
−_{1}
(y − t)r2−1_{h(s, t)dtds;}
if (x, y) ∈ (xk, xk+1] × [0, b], k = 1, . . . , m,

if and only if u is a solution of the fractional IVP

c_{D}r_{u(x, y) = h(x, y),} _{(x, y) ∈ J}′

, (10)

u(x+_{k}, y) = Ik(u(xk, y)), k= 1, . . . , m.

(11)

P roof. Assume u satisfies (10)–(11). If (x, y) ∈ [0, x1] × [0, b], then
c_{D}r_{u(x, y) = h(x, y).}
Lemma 3.2 implies
u(x, y) = µ(x, y) + 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds.}

If (x, y) ∈ (x1, x2] × [0, b], then Lemma 3.2 implies

+ 1
Γ(r1)Γ(r2)
Z x
x1
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds}

= ϕ(x) + I1(u(x1, y)) − I1(u(x1,0))

+ 1
Γ(r1)Γ(r2)
Z x
x1
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds.}

If (x, y) ∈ (x2, x3] × [0, b], then from Lemma 3.2 we get

u(x, y) = µ2(x, y) +
1
Γ(r1)Γ(r2)
Z x
x2
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds}
= ϕ(x) + u(x+_{2}, y) − u(x+_{2},0)
+ 1
Γ(r1)Γ(r2)
Z x
x2
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds}
= ϕ(x) + I2(u(x2, y)) − I2(u(x2,0))

+ 1
Γ(r1)Γ(r2)
Z x
x2
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{h(s, t)dtds.}

If (x, y) ∈ (xk, xk+1] × [0, b], then again from Lemma 3.2 we get (9).

Conversely, assume that u satisfies the impulsive fractional integral equation (9). If (x, y) ∈ [0, x1] × [0, b] and using the fact thatcDr is the left

inverse of Ir_{, we get}

c_{D}r_{u(x, y) = h(x, y),} _{for each (x, y) ∈ [0, x}

1] × [0, b].

If (x, y) ∈ [xk, xk+1) × [0, b], k = 1, . . . , m and using the fact thatcDrC= 0,

where C is a constant, we get

c_{D}r_{u(x, y) = h(x, y), for each (x, y) ∈ [x}

k, xk+1) × [0, b].

Also, we can easily show that

Corollary 3.5. Let 0 < r1, r2 ≤ 1 and let h, w : J → Rn be continuous. A

function u is a solution of the fractional integral equation

(12) u(x, y) =
µ(x, y) + w(x, y) − w(x, 0) − w(0, y) + w(0, 0)
+_{Γ(r} 1
1)Γ(r2)
Rx
0
Ry
0(x − s)r1
−_{1}
(y − t)r2−1_{h(s, t)dtds;}
if (x, y) ∈ [0, x1] × [0, b],
ϕ(x) + w(x, y) − w(x, 0) − w(xk, y) + w(xk,0)

+Ik(u(xk, y)) − Ik(u(xk,0))

+_{Γ(r} 1
1)Γ(r2)
Rx
xk
Ry
0(x − s)r1
−1_{(y − t)}r2−1_{h(s, t)dtds;}
if (x, y) ∈ (xk, xk+1] × [0, b], k = 1, . . . , m,

if and only if u is a solution of the fractional IVP

c_{D}r_{(u(x, y) − w(x, y)) = h(x, y),} _{(x, y) ∈ J}′

, (13)

u(x+_{k}, y) = Ik(u(xk, y)), k= 1, . . . , m.

(14)

Now, we are concerned with the existence result for problem (1)–(3). The following hypotheses will be assumed hereafter.

(H1) The function xk∈ C1(Rn, R) for k = 1, . . . , m. Moreover,

0 = x0(u) < x1(u) < . . . < xm(u) < xm+1(u) = a, for all u ∈ Rn,

(H2) there exist constants ck, dk>0 such that

kIk(u)k ≤ ckkuk + dk, for each u ∈ Rn, k= 1, . . . , m,

(H3) there exist a continuous nondecreasing function δ : [0, ∞) → (0, ∞), and p ∈ L∞

(J, R+) such that

kF (x, y, u)kP ≤ p(x, y)δ(kuk) a.e. (x, y) ∈ J, and each u ∈ Rn,

(H4) there exists l ∈ L∞

(J, R+); k = 1, . . . , m, such that

and

d(0, F (x, y, 0)) ≤ l(x, y), a.e. (x, y) ∈ Jk, k= 0, . . . , m,

(H5) for all (s, t, u) ∈ J × Rn_{,} _{there exists f ∈ S}

F,u such that

x′_{k}(u)[ϕ′(s) + r1− 1
Γ(r1)Γ(r2)
Z s
xk
Z t
0
(s − θ)r1−2_{(t − η)}r2−1_{f}_{(θ, η)dηdθ] 6= 1;}
k= 1, . . . , m,
(H6) for all u ∈ Rn_{, x}

k(Ik(u)) ≤ xk(u) < xk+1(Ik(u)) for k = 1, . . . , m,

(H7) there exists M > 0 such that

min
(
M
kµk∞+
p∗ar1_{b}r2_{δ(M )}
Γ(r1+ 1)Γ(r2+ 1)
,
(15) M
kϕk∞+ 2ckM+ 2dk+
p∗_{a}r1_{b}r2_{δ(M )}
Γ(r1+ 1)Γ(r2+ 1)
; k = 1, . . . , m
)
>1,
where p∗
= kpkL∞.

Theorem 3.6. Assume that hypotheses (H1)–(H7) hold. Then the initial-value problem (1)–(3) has at least one solution on J.

The proof of this theorem will be given in several steps. Step 1. Consider the problem

(cD_{0}ru)(x, y) ∈ F (x, y, u(x, y)), where (x, y) ∈ J,
(16)

u(x, 0) = ϕ(x), u(0, y) = ψ(y), x ∈ [0, a], y ∈ [0, b]. (17)

Clearly, the fixed points of N are solutions to (16)–(17). We shall show that the operator N is completely continuous. The proof will be given in several Claims.

Claim 1. N_{(u) is convex for each u ∈ C(J, R}n_{)).}

Indeed, if h1, h2 belong to N (u), then there exist f1, f2 ∈ SF,u such that

for each (x, y) ∈ J we have
hi(x, y) = µ(x, y)+
1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x−s)r1−1_{(y−t)}r2−1_{f}
i(s, t)dtds, i = 1, 2.

Let 0 ≤ d ≤ 1. Then, for each (x, y) ∈ J we have
(dh1+ (1 − d)h2)(x, y) = µ(x, y) + 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1
× [f1(s, t) + (1 − d)f2(s, t)]dtds.

Since SF,u is convex (because F has convex values), then for each (x, y) ∈ J,

dh1+ (1 − d)h2∈ N (u).

Claim 2. N maps bounded sets into bounded sets in C(J, Rn_{).}

Indeed, it is sufficient to show that for any q > 0 there exists a positive constant ℓ such that for each u ∈ Bq = {u ∈ C(J, Rn) : kuk∞≤ q} we have

kN (u)k ≤ ℓ. Let u ∈ Bq and h ∈ N (u), then there exists f ∈ SF,u such that

then we obtain

khk ≤ kµk∞+

ar1_{b}r2_{p}∗_{δ(q)}
Γ(r1+ 1)Γ(r2+ 1)

:= ℓ.

Claim 3. N maps bounded sets into equicontinuous sets of C(J, Rn_{).}

Let (τ1, y1), (τ2, y2) ∈ J, τ1 < τ2 and y1 < y2, Bq be the bounded set

of C(J, Rn_{) as in Claim 2. Let u ∈ B}

q and h ∈ N (u). Then, there exists

f ∈ SF,u such that for each (x, y) ∈ J we have

h(x, y) = µ(x, y) + 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{f}_{(s, t)dtds.}
Then, for each (x, y) ∈ J we have

≤ kµ(τ1, y1) − µ(τ2, y2)k + p ∗ δ(q) Γ(r1+ 1)Γ(r2+ 1) [2yr2 2 (τ2− τ1)r1+ 2τ2r1(y2− y1)r2 + τr1 1 y r2 1 − τ r1 2 y r2 2 − 2(τ2− τ1)r1(y2− y1)r2].

As τ1 −→ τ2and y1 −→ y2, the right-hand side of the above inequality tends

to zero. As a consequence of Claims 1 to 3 together and the Arzela-Ascoli
theorem, we can conclude that N : C(J, Rn_{) → P(C(J, R}n_{)) is a completely}

continuous multi-valued operator. Claim 4. N has a closed graph.

Let un → u∗, hn ∈ N (un) and hn → h∗. We need to show that h∗ ∈

N(u∗). hn ∈ N (un) means that there exists fn ∈ SF,un such that for each
(x, y) ∈ J,
hn(x, y) = µ(x, y) +
1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{f}
n(s, t)dtds.

We must show that there exists f∗∈ SF,u∗ such that for each (x, y) ∈ J,
h∗(x, y) = µ(x, y) +
1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{f}
∗(s, t)dtds.

Since F (x, y, ·) is upper semicontinuous, then for every ε > 0, there exist n0(ǫ) ≥ 0 such that for every n ≥ n0 we have

fn(x, y) ∈ F (x, y, un(x,y)) ⊂ F (x, y, u∗(x,y)) + εB(0, 1), a.e. (x, y) ∈ J.

Since F (., ., .) has compact values, then there exists a subsequence fnm such that

fnm(·, ·) → f∗(·, ·) as m → ∞ and

f∗(x, y) ∈ F (x, y, u∗(x,y)), a.e. (x, y) ∈ J.

For every w ∈ F (x, y, u∗(x,y)) we have

Then,

kfnm(x, y) − f∗(x, y)k ≤ d(fnm(x, y), F (x, y, u∗(x,y))).

By an analogous relation, obtained by interchanging the roles of fnm and f∗, it follows that

kfnm(x, y) − u∗(x, y)k ≤ Hd(F (x, y, un(x,y)), F (x, y, u∗(x,y))) ≤ l(x, y)kun− u∗k∞.

Let l∗

:= klkL∞,then by (H4) we obtain for each (x, y) ∈ J,
khn(x,y)− h∗(x,y)k
≤ 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{kf}
m(s, t) − f∗(s, t)kdtds
≤ kunm− u∗k∞
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{l(s, t)dtds}
≤ a
r1_{b}r2_{l}∗_{ku}
nm− u∗k∞
Γ(r1+ 1)Γ(r2+ 1)
.
Hence,
khnm− h∗k∞ ≤
ar1_{b}r2_{l}∗_{ku}
nm− u∗k∞
Γ(r1+ 1)Γ(r2+ 1) → 0 as m → ∞.

Claim 5. A priori bounds on solutions.

Let u be a possible solution to problem (1)–(3). Then, there exists f ∈ SF,u such that for each (x, y) ∈ J,

This implies that for each (x, y) ∈ J we have
kuk∞
kµk∞+
p∗_{a}r1_{b}r2_{δ(kuk}
∞)
Γ(r1+ 1)Γ(r2+ 1)
<1.

Then, by condition (15), there exists M such that kuk∞6= M. Let

U = {u ∈ C(J, Rn) : kuk∞< M}.

The operator N : U → P(C(J, Rn_{)) is upper semicontinuous and completely}

continuous. From the choice of U , there is no u ∈ ∂U such that u ∈ λN (u) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type (Theorem 2.3), we deduce that N has a fixed point which is a solution to (16)–(17). Denote this solution by u1. Define the function

rk,1(x, y) = xk(u1(x, y)) − x, for x ≥ 0, y ≥ 0.

Hypothesis (H1) implies that rk,1(0, 0) 6= 0 for k = 1, . . . , m.

If rk,1(x, y) 6= 0 on J for k = 1, . . . , m; i.e.,

x6= xk(u1(x, y)), on J for k = 1, . . . , m,

then u1 is a solution to problem (1)–(3).

It remains to consider the case when r1,1(x, y) = 0 for some (x, y) ∈ J.

Now, since r1,1(0, 0) 6= 0 and r1,1 is continuous, there exists x1 >0, y1 >0

such that r1,1(x1, y1) = 0, and r1,1(x, y) 6= 0, for all (x, y) ∈ [0, x1) × [0, y1).

Thus, by (H1) we have

r1,1(x1, y1) = 0 and r1,1(x, y) 6= 0, for all (x, y) ∈ [0, x1) × [0, y1] ∪ (y1, b].

Suppose that there exist (¯x,y) ∈ [0, x¯ 1) × [0, y1] ∪ (y1, b] such that

r1,1(¯x,y) = 0. The function r¯ 1,1 attains a maximum at some point (s, t) ∈

[0, x1) × [0, b]. Since

(cD_{0}ru1)(x, y) ∈ F (x, y, u1(x, y)), for (x, y) ∈ J,

Hence
∂u_{1}(x, y)
∂x exists, and
∂r_{1,1}(s, t)
∂x = x
′
1(u1(s, t))
∂u_{1}(s, t)
∂x − 1 = 0.
Since
∂u1(x, y)
∂x = ϕ
′
(x) + r1− 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−2_{(y − t)}r2−1_{f}_{(s, t)dtds,}
then
x′_{1}(u1(s, t))[ϕ′(s) +
r1− 1
Γ(r1)Γ(r2)
Z s
0
Z t
0
(s − θ)r1−2_{(t − η)}r2−1_{f}_{(θ, η)dηdθ] = 1,}

witch contradicts (H5). From (H1) we have

rk,1(x, y) 6= 0 for all (x, y) ∈ [0, x1) × [0, b] and k = 1, . . . , m.

Step 2. In what follows, set

Xk := [xk, a] × [0, b]; k = 1, . . . , m.

Consider now the problem

(cD_{0}ru)(x, y) ∈ F (x, y, u(x, y)), where (x, y) ∈ X1,

(18)

u(x+_{1}, y) = I1(u1(x1, y)).

(19)

Consider the operator N1: C(X1, Rn) → P(C(X1, Rn)) defined as

N1(u) =
h∈ C(X1, Rn) :
h(x, y) = ϕ(x)+I1(u1(x1, y))−I1(u1(x1,0))
+_{Γ(r} 1
1)Γ(r2)
Rx
x1
Ry
0(x − s)r1
−_{1}
(y − t)r2−1
×f (s, t)dtds; (x, y) ∈ X1, f ∈ SF,u.
As in Step 1 we can show that N1 is upper semicontinuous and completely

there exists f ∈ SF,u such that for each (x, y) ∈ X1,

ku(x, y)k ≤ kϕ(x)k + kI1(u1(x1, y))k + kI1(u1(x1,0))k

+ 1
Γ(r1)Γ(r2)
Z x
x1
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{kf (s, t)kdtds}
≤ kϕk∞+ 2c1kuk + 2d1
+ 1
Γ(r1)Γ(r2)
Z x
0
Z y
0
(x − s)r1−1_{(y − t)}r2−1_{p(s, t)δ(kuk)dtds.}
Then,
kuk∞
kϕk∞+ 2c1kuk∞+ 2d1+
p∗ar1_{b}r2_{δ(kuk}
∞)
Γ(r1+ 1)Γ(r2+ 1)
<1.

Then by condition (15), there exists M′

such that kuk∞6= M′.Let

U′ = {u ∈ C(X1, Rn) : kuk∞< M′}.

The operator N1 : U ′

→ P(C(X1, Rn)) is upper semicontinuous and

com-pletely continuous. From the choice of U′

, there is no u ∈ ∂U′

such that u∈ λN1(u) for some λ ∈ (0, 1). As a consequence of the nonlinear

alterna-tive of Leray-Schauder type (Theorem 2.3), we deduce that N1 has a fixed

point which is a solution to (18)–(19). Denote this solution by u2. Define

rk,2(x, y) = xk(u2(x, y)) − x, for (x, y) ∈ X1.

If rk,2(x, y) 6= 0 on (x1, a] × [0, b] and for all k = 1, . . . , m, then

u(x, y) =(u1(x, y), if (x, y) ∈ [0, x1) × [0, b], u2(x, y), if (x, y) ∈ [x1, a] × [0, b],

is a solution to problem (1)–(3). It remains to consider the case where r2,2(x, y) = 0, for some (x, y) ∈ (x1, a] × [0, b]. By (H6), we have

r2,2(x+1, y1) = x2(u2(x+1, y1) − x1

= x2(I1(u1(x1, y1))) − x1

> x1(u1(x1, y1)) − x1

Since r2,2 is continuous, there exist x2 > x1, y2 > y1 such that r2,2(x2, y2)

= 0, and r2,2(x, y) 6= 0 for all (x, y) ∈ (x1, x2) × [0, b].

It is clear by (H1) that

rk,2(x, y) 6= 0 for all (x, y) ∈ (x1, x2)] × [0, b], k = 2, . . . , m.

Now suppose that there are (s, t) ∈ (x1, x2) × [0, b] such that r1,2(s, t) = 0.

From (H6) it follows that

r1,2(x+1, y1) = x1(u2(x+1, y1) − x1

= x1(I1(u1(x1, y1))) − x1

≤ x1(u1(x1, y1)) − x1

= r1,1(x1, y1) = 0.

Thus, r1,2 attains a nonnegative maximum at some point (s1, t1) ∈ (x1, a) ×

[0, x2) ∪ (x2, b]. Since

(cD_{0}ru2)(x, y) ∈ F (x, y, u2(x, y)), for (x, y) ∈ X1,

Therefore,
x′_{1}(u2(s1, t1))
ϕ′(s1) +
r1− 1
Γ(r1)Γ(r2)
Z s1
x1
Z t1
0
(s1− θ)r1−2(t1− η)r2−1f(θ, η)dηdθ
= 1,
which contradicts (H5).

Step 3. We continue this process and take into account that um+1 := u

_{X}

m is a solution to the problem

(cD_{0}ru)(x, y) ∈ F (x, y, u(x, y)), a.e. (x, y) ∈ (xm, a] × [0, b],

u(x+_{m}, y) = Im(um−1(xm, y)).

The solution u to problem (1)–(3) is then defined by

u(x, y) = u1(x, y), if (x, y) ∈ [0, x1] × [0, b], u2(x, y), if (x, y) ∈ (x1, x2] × [0, b], . . . um+1(x, y), if (x, y) ∈ (xm, a] × [0, b].

Now we present (without proof), by using Corollaries 3.3 and 3.5, an ex-istence result as an extension of the result presented in Theorem 3.6 to problem (4)–(6).

Definition 3.7. A function u ∈ Ω ∩ ∪m

k=1AC(Jk, Rn) whose r-derivative

exists on J′

is said to be a solution to (4)–(6) if there exists a function
f(x, y) ∈ F (x, y, u(x, y)) such that u satisfies (cDr_{0}u)(x, y) = f (x, y) on J′

and conditions (5), (6) are satisfied.

Theorem 3.8. Assume (H1)–(H4), (H6) and the following conditions (H8) The function g is nonnegative and completely continuous and there

exist constants 0 ≤ l1<1, l2 ≥ 0 such that

kg(x, y, u)k ≤ l1kuk + l2; (x, y) ∈ J, u ∈ Rn.

(H9) For all (s, t, u) ∈ J × Rn _{there exists f}_{∈ S}

x′_{k}(u)[ϕ′(s) +∂g(s, t, u(s, t))
∂x −
∂g(s, 0, u(s, 0))
∂x
+ r1− 1
Γ(r1)Γ(r2)
Z s
xk
Z t
0
(s − θ)r1−2_{(t − η)}r2−1_{f}_{(θ, η)dηdθ] 6= 1; k = 1, . . . , m.}

(H10) There exists a number M′ _{>}

0 such that
min
(
M′
kµk∞+ 4l1M′+ 4l2+
p∗_{a}r1_{b}r2_{δ(M}′_{)}
Γ(r1+ 1)Γ(r2+ 1)
,
(20)
M′
kϕk∞+ (2ck+ 4l1)M′+ 2dk+ 4l2+
p∗ar1_{b}r2_{δ(M}′_{)}
Γ(r1+ 1)Γ(r2+ 1)
; k = 1, . . . , m
)
>1,

hold. Then, the initial-value problem (4)–(6) has at least one solution on J.

4. An example

As an application of our results we consider the following impulsive partial hyperbolic functional differential inclusions of the form

(21)

(cD_{0}ru)(x, y) ∈ F (x, y, u(x, y)),

where (x, y) ∈ J, x 6= xk(u(x, y)); k = 1, . . . , m,

(22) u(x+_{k}, y) = Ik(u(xk, y)), y ∈ [0, 1], k = 1, . . . , m,

(23) u(x, 0) = x, u(0, y) = y2, x∈ [0, 1], y ∈ [0, 1],

where J = [0, 1] × [0, 1], r = (r1, r2), 0 < r1, r2 ≤ 1, xk(u) = 1 − _{2}k 1

(1+u2_{)};
k= 1, . . . , m, and for each u ∈ R we have xk+1(Ik(u)) > xk(u); k = 1, . . . , m,

and also there exist constants ck, dk>0 such that

Let u ∈ R, then we have
x_{k+1}(u) − xk(u) =

1

2k+1_{(1 + u}2_{)} >0; k = 1, ..., m.

Hence, 0 < x1(u) < x2(u) < . . . < xm(u) < 1 for each u ∈ R. Set

F_{(x, y, u(x, y)) = {u ∈ R : f}_{1}(x, y, u(x, y)) ≤ u ≤ f2(x, y, u(x, y))},

where f1, f2 : [0, 1] × [0, 1] × R → R. We assume that for each (x, y) ∈

J, f1(x, y, .) is lower semi-continuous (i.e., the set {z ∈ R : f1(x, y, z) > ν}

is open for each ν ∈ R), and assume that for each (x, y) ∈ J, f2(x, y, .) is

upper semi-continuous (i.e., the set {z ∈ R : f2(x, y, z) < ν} is open for

each ν ∈ R). Assume that there are p ∈ C(J, R+_{) and δ : [0, ∞) → (0, ∞)}

continuous and nondecreasing such that

max(|f1(x, y, z)|, |f2(x, y, z)|) ≤ p(x, y)δ(|z|), for a.e. (x, y) ∈ J and z ∈ R.

It is clear that F is compact and convex valued, and it is upper semi-continuous (see [14]). Moreover, assume that conditions (H5) and (H7) are satisfied. Since all conditions of Theorem 3.6 are satisfied, problem (21)–(23) has at least one solution on [0, 1] × [0, 1].

References

[1] S. Abbas and M. Benchohra, Partial hyperbolic differential equations with fi-nite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62–72.

[2] S. Abbas and M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal. Hybrid Syst. 3 (2009), 597–604.

[3] S. Abbas and M. Benchohra, Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), 406–413.

[4] S. Abbas and M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses, Discuss. Math. Differ. Incl. Control Optim. 30 (1) (2010), 141–161.

[6] S. Abbas, M. Benchohra and L. G´orniewicz, Existence theory for impulsive par-tial hyperbolic functional differenpar-tial equations involving the Caputo fractional derivative, Sci. Math. Jpn. online e- 2010, 271–282.

[7] R.P Agarwal, M. Benchohra and S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (3) (2010), 973–1033.

[8] A. Belarbi, M. Benchohra and A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fr´echet spaces, Appl. Anal. 85 (2006), 1459–1470.

[9] M. Belmekki, M. Benchohra and L. G´orniewicz, Functional differential equa-tions with fractional order and infinite delay, Fixed Point Theory 9 (2008), 423–439.

[10] M. Benchohra, J.R. Graef and S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal. 87 (7) (2008), 851–863.

[11] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1–12. [12] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, NY, USA, 2006. [13] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results

for functional differential equations of fractional order, J. Math. Anal. Appl. 338(2008), 1340–1350.

[14] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.

[15] K. Diethelm and A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: ”Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Re-action Engineering and Molecular Properties” (F. Keil, W. Mackens, H. Voss and J. Werther, Eds), pp. 217–224, Springer-Verlag, Heidelberg, 1999. [16] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J.

Math. Anal. Appl. 265 (2002), 229–248.

[17] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81–88.

[18] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46–53.

[20] L. G´orniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dor-drecht, 1999.

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

[22] A.A. Kilbas, B. Bonilla and J. Trujillo, Nonlinear differential equations of fractional order in a space of integrable functions, Dokl. Ross. Akad. Nauk 374(4) (2000), 445–449.

[23] A.A. Kilbas and S.A. Marzan, Nonlinear differential equations with the Ca-puto fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84–89.

[24] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.

[25] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

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

[27] F. Mainardi, Fractional calculus: Some basic problems in continuum and sta-tistical mechanics, in: ”Fractals and Fractional Calculus in Continuum Me-chanics” (A. Carpinteri and F. Mainardi, Eds), pp. 291–348, Springer-Verlag, Wien, 1997.

[28] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Dif-ferential Equations, John Wiley, New York, 1993.

[29] I. Podlubny, I. Petraˇs, B.M. Vinagre, P. O’Leary and L. Dorˇcak, Analogue realizations of fractional-order controllers, fractional order calculus and its ap-plications, Nonlinear Dynam. 29 (2002), 281–296.

[30] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Deriva-tives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

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

[32] N.P. Semenchuk, On one class of differential equations of noninteger order, Differents. Uravn. 10 (1982), 1831–1833.

[34] A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of par-tial differenpar-tial equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318–325.