Fractional order Riemann-Liouville integral inclusions ...

Opuscula Math. 33, no. 2 (2013), 209–222 http://dx.doi.org/10.7494/OpMath.2013.33.2.209

Opuscula Mathematica

To the memory of Professor B.G. Pachpatte

FRACTIONAL ORDER RIEMANN-LIOUVILLE INTEGRAL INCLUSIONS WITH TWO INDEPENDENT VARIABLES AND MULTIPLE DELAY Saïd Abbas and Mouffak Benchohra Communicated by P.A. Cojuhari Abstract. In the present paper we investigate the existence of solutions for a system of integral inclusions of fractional order with multiple delay. Our results are obtained upon suitable fixed point theorems, namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler for the nonconvex case. Keywords: integral inclusion, left-sided mixed Riemann-Liouville integral, time delay, solution, fixed point. Mathematics Subject Classification: 26A33.

1. INTRODUCTION Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. We can find numerous applications in rheology, control, porous media, viscoelasticity, electrochemistry, electromagnetism, etc. [18, 25]. There has been significant development in ordinary and partial fractional differential equations and inclusions in recent years; see the monographs of Abbas et al. [6], Kilbas et al. [21], the papers of Abbas et al. [1–5, 7], Benchohra et al. [9–11], Ibrahim and H.A. Jalab [20], Kilbas and Marzan [22], Pachpatte [26–29], Vityuk [30], Vityuk and Golushkov [31] and the references therein. Integral equations and inclusions occur in a natural way in the description of many physical phenomena; see the books by Corduneanu [14, 15]. c AGH University of Science and Technology Press, Krakow 2013

209

210

Saïd Abbas and Mouffak Benchohra

Motivated by the above literature, this paper is concerned with the existence of solutions for the following system of fractional integral inclusions u(x, y) −

m X

bi (x, y)u(x − ξi , y − µi ) ∈ Iθr F (x, y, u(x, y))

i=1

(1.1)

if (x, y) ∈ J := [0, a] × [0, b], u(x, y) = Φ(x, y) if (x, y) ∈ J˜ := [−ξ, a] × [−µ, b]\(0, a] × (0, b],

(1.2)

where a, b > 0, θ = (0, 0), ξi , µi ≥ 0, i = 1, . . . , m, ξ = maxi=1,...,m {ξi }, µ = maxi=1,...,m {µi }, F : J ×Rn → P(Rn ) is a set-valued function with nonempty values in Rn , P(Rn ) is the family of all nonempty subsets of Rn , Iθr F (x, y, u(x, y)) is the definite integral for the set-valued functions F of order r = (r1 , r2 ) ∈ (0, ∞) × (0, ∞), bi : J → R; i = 1, . . . m, and Φ : J˜ → Rn are given continuous functions such that Φ(x, 0) =

m X

bi (x, 0)Φ(x − ξi , −µi ),

x ∈ [0, a],

bi (0, y)Φ(−ξi , y − µi ),

y ∈ [0, b].

i=1

and Φ(0, y) =

m X i=1

We establish the existence results for the problem (1.1)–(1.2) when the right-hand side is convex as well as when it is non-convex valued. Our approach is based on appropriate fixed point theorems, namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler for the nonconvex case. This approach is now standard, however its utilization is new in the framework of the considered integral inclusions.

2. PRELIMINARIES In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By C(J) we denote the Banach space of all continuous functions from J into Rn with the norm kwk∞ = sup kw(x, y)k, (x,y)∈J n

where k.k denotes a norm on R . Also, C := C([−ξ, a] × [−µ, b]) is a Banach space endowed with the norm kwkC =

sup

kw(x, y)k.

(x,y)∈[−ξ,a]×[−µ,b]

Let L1 (J) be the space of Lebesgue-integrable functions w : J → Rn with the norm Za Zb kwkL1 =

kw(x, y)kdydx, 0

0

211

Fractional order Riemann-Liouville integral inclusions. . .

and L∞ (J) be the Banach space of measurable functions u : J → Rn which are essentially bounded, equipped with the norm kukL∞ = inf{c > 0 : ku(x, y)k ≤ c a.e. (x, y) ∈ J}. Definition 2.1 ([31]). Let r = (r1 , r2 ) ∈ (0, ∞) × (0, ∞), θ = (0, 0) and u ∈ L1 (J). The left-sided mixed Riemann-Liouville integral of order r of u is defined by (Iθr u)(x, y)

1 = Γ(r1 )Γ(r2 )

Zx Zy 0

(x − s)r1 −1 (y − t)r2 −1 u(s, t)dtds,

0

where Γ(·) is the (Euler’s) Gamma function defined by Γ(ξ) =

R∞

tξ−1 e−t dt, ξ > 0.

0

In particular, (Iθθ u)(x, y)

= u(x, y),

(Iθσ u)(x, y)

Zx Zy u(s, t)dtds for almost all (x, y) ∈ J,

= 0

0

where σ = (1, 1). For instance, Iθr u exists for all r1 , r2 ∈ (0, ∞), when u ∈ L1 (J). Note also that when u ∈ C(J), then (Iθr u) ∈ C(J), and moreover (Iθr u)(x, 0) = (Iθr u)(0, y) = 0,

x ∈ [0, a], y ∈ [0, b].

Example 2.2. Let λ, ω ∈ (−1, ∞) and r = (r1 , r2 ) ∈ (0, ∞) × (0, ∞), then Iθr xλ y ω =

Γ(1 + λ)Γ(1 + ω) xλ+r1 y ω+r2 for almost all (x, y) ∈ J. Γ(1 + λ + r1 )Γ(1 + ω + r2 )

Let (X, k · k) be a Banach space. Denote Pcl (X) = {Y ∈ P(X) : Y is closed}, Pbd (X) = {Y ∈ P(X) : Y is bounded}, Pcp (X) = {Y ∈ P(X) : Y is compact} and Pcp,cv (X) = {Y ∈ P(X) : Y is compact and convex}. Definition 2.3. A multivalued map T : X → P(X) is convex (closed) valued if T (x) is convex (closed) for all x ∈ X. T is bounded on bounded sets if T (B) = ∪x∈B T (x) is bounded in X for all B ∈ Pbd (X) (i.e. supx∈B supy∈T (x) kyk < ∞). T is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set T (x0 ) is a nonempty closed subset of X, and if for each open set N of X containing T (x0 ), there exists an open neighborhood N0 of x0 such that T (N0 ) ⊆ N . T is said to be completely continuous if T (B) is relatively compact for every B ∈ Pbd (X). T has a fixed point if there is x ∈ X such that x ∈ T (x). The fixed point set of the multivalued operator T will be denoted by F ixT . A multivalued map G : X → Pcl (Rn ) is said to be measurable if for every v ∈ Rn , the function x 7−→ d(v, G(x)) = inf{kv − zk : z ∈ G(x)} is measurable. For more details on multivalued maps see the books of Aubin and Cellina [8], Górniewicz [17], Hu and Papageorgiou [19], and Kisielewicz [23].

212


Lemma 2.4 ([19]). Let G be a completely continuous multivalued map with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e. un → u, wn → w, wn ∈ G(un ) imply w ∈ G(u)). Definition 2.5. A multivalued map F : J ×Rn → P(Rn ) is said to be Carathéodory if (i) (x, y) 7−→ F (x, y, u) is measurable for each u ∈ Rn , (ii) u 7−→ F (x, y, u) is upper semicontinuous for almost all (x, y) ∈ J. For each u ∈ C(J), define the set of selections of F by SF,u = {w ∈ L1 (J) : w(x, y) ∈ F (x, y, u(x, y)) a.e. (x, y) ∈ J}. Lemma 2.6 ([24]). Let X be a Banach space. Let F : J × X −→ Pcp,cv (X) be a Carathéodory multivalued map and let Λ be a linear continuous mapping from L1 (J, X) to C(J, X), then the operator Λ ◦ SF : C(J, X) −→ Pcp,cv (C(J, X)), u 7−→ (Λ ◦ SF )(u) := Λ(SF,u ) is a closed graph operator in C(J, X) × C(J, X). Proposition 2.7 ([13]). Let X be a separable Banach space. Let F1 , F2 : J → Pcp (X) be measurable multivalued maps, then the multivalued map t → F1 (t) ∩ F2 (t) is measurable. Theorem 2.8 ([13]). Let X be a separable metric space, (T, L) a measurable space, F a multivalued map from T to complete non empty subsets of X. If for each open set U in X, F − (U ) = {t : F (t) ∩ U 6= ∅} belongs to L, then F admits a measurable selection. Let F : J × Rn → P(Rn ) be a set-valued function with nonempty values in Rn . are the definite integral for the set-valued functions F of order r = (r1 , r2 ) ∈ (0, ∞) × (0, ∞) which is defined as

Iθr F (x, y, u(x, y))

Iθr F (x, y, u(x, y)) =   Zx Zy   1 (x − s)r1 −1 (y − t)r2 −1 f (s, t)dtds : f (x, y) ∈ SF,u . =   Γ(r1 )Γ(r2 ) 0

0

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 d(a, B), sup d(A, b) , a∈A

b∈B

where d(A, b) = inf d(a, b), d(a, B) = inf d(a, b). Then (Pbd,cl (X), Hd ) is a metric a∈A

b∈B

space and (Pcl (X), Hd ) is a generalized metric space (see [23]).

213


Definition 2.9. A multivalued operator N : X → Pcl (X) is called (a) γ-Lipschitz if and only if there exists γ > 0 such that Hd (N (u), N (v)) ≤ γd(u, v) for each

u, v ∈ X,

(b) a contraction if and only if it is γ-Lipschitz with γ < 1. Lemma 2.10 ([12], Bohnenblust-Karlin). Let X be a Banach space and K ∈ Pcl,cv (X) and suppose that the operator G : K → Pcl,cv (K) is upper semicontinuous and the set G(K) is relatively compact in X. Then G has a fixed point in K. Lemma 2.11 ([16], Covitz-Nadler). Let (X, d) be a complete metric space. If N : X → Pcl (X) is a contraction, then N has fixed points.

3. EXISTENCE OF SOLUTIONS Let us start by defining what we mean by a solution of the problem (1.1)–(1.2). Definition 3.1. A function u ∈ C is said to be a solution of (1.1)–(1.2) if there exists an essentially bounded function f ∈ SF,u such that u satisfies the equation u(x, y) =

m X

bi (x, y)u(x − ξi , y − µi ) + (Iθr f )(x, y)

i=1

˜ on J and condition (1.2) on J. Set B = max { sup |bi (x, y)|}. i=1,...,m (x,y)∈J

Theorem 3.2 (Convex Case). Assume that the following conditions hold: (H1 ) The multifunction F is Carathéodory, (H2 ) There exists a non-negative function h ∈ L∞ (J) such that kF (x, y, u)kP ≤ h(x, y), a.e. (x, y) ∈ J for all u ∈ Rn . If mB < 1, then the problem (1.1)–(1.2) has at least one solution u on [−ξ, a]×[−µ, b]. Proof. Transform the problem (1.1)–(1.2) into a fixed point problem. Consider the multivalued operator N : C → P(C) defined by

N (u)(x, y) =

        

g

  y), (x, y) ∈ J˜  Φ(x,    m X  bi (x, y)u(x − ξi , y − µi )+ ∈ C : g(x, y) = .     i=1     +(Iθr f )(x, y), f ∈ SF,u , (x, y) ∈ J (3.1)

214


Since the selection function f is essentially bounded, the operator N is well defined, that is, maps C into P(C). Clearly, the fixed points of N are solutions to (1.1)–(1.2). Let h∗ = khkL∞ , ar1 br2 h∗ R = max kΦk, (1 − mB)Γ(1 + r1 )Γ(1 + r2 )

,

and BR = {u ∈ C : kukC ≤ R}. Clearly, BR is a closed, bounded and convex subset of C. We shall show that N satisfies the assumptions of Lemma 2.10. The proof will be given by several steps. Step 1. N (BR ) ⊂ BR . Let u ∈ BR . We must show that N (u) ⊂ BR . For each g ∈ N (u), there exists f ∈ SF,u such that, for each (x, y) ∈ J, we have

kg(x, y)k ≤

m X

|bi (x, y)|ku(x − ξi , y − µi )k+

i=1

1 + Γ(r1 )Γ(r2 )

Zx Zy 0

(x − s)r1 −1 (y − t)r2 −1 kf (s, t)kdtds ≤

0

1 ≤ mBkuk + Γ(r1 )Γ(r2 )

Zx Zy 0

(x − s)r1 −1 (y − t)r2 −1 h(s, t)dtds ≤

0

ar1 br2 h∗ ≤ mBR + = R, Γ(1 + r1 )Γ(1 + r2 ) ˜ we have and, for all (x, y) ∈ J, kg(x, y)k ≤ kΦk ≤ R. Thus, for all (x, y) ∈ [−ξ, a] × [−µ, b] and g(x, y) ∈ N (u), we get kg(x, y)k ≤ R. Step 2. N (BR ) is a relatively compact set. We must show that N is a compact operator. Since BR is a bounded closed and convex set and N (BR ) ⊂ BR , it follows that N (BR ) is a bounded closed and convex

215


set. Moreover, for 0 ≤ x1 ≤ x2 ≤ a and 0 ≤ y1 ≤ y2 ≤ b and u ∈ BR , then for each g ∈ N (u), there exists f ∈ SF,u such that, for each (x, y) ∈ J, we have kg(x1 , y1 ) − g(x2 , y2 )k ≤ m X ≤ kbi (x1 , y1 )u(x1 − ξi , y1 − µi ) − bi (x2 , y2 )u(x2 − ξi , y2 − µi )k + i=1

1 + Γ(r1 )Γ(r2 )

Zx1 Zy1 [(x2 − s)r1 −1 (y2 − t)r2 −1 − 0

0

− (x1 − s)r1 −1 (y1 − t)r2 −1 ]kf (s, t)kdtds+ Zx2 Zy2 1 (x2 − s)r1 −1 (y2 − t)r2 −1 kf (s, t)kdtds+ + Γ(r1 )Γ(r2 ) +

1 Γ(r1 )Γ(r2 )

x1 y1 Zx1 Zy2

(x2 − s)r1 −1 (y2 − t)r2 −1 kf (s, t)kdtds+

0 y1

+

1 Γ(r1 )Γ(r2 )

Zx2 Zy1 (x2 − s)r1 −1 (y2 − t)r2 −1 kf (s, t)kdtds ≤ x1 0

≤B

m X

ku(x1 − ξi , y1 − µi ) − u(x2 − ξi , y2 − µi )k +

i=1

h∗ [2y r2 (x2 − x1 )r1 + 2xr21 (y2 − y1 )r2 + Γ(r1 )Γ(r2 ) 2 + xr11 y1r2 − xr21 y2r2 − 2(x2 − x1 )r1 (y2 − y1 )r2 ]. +

As x1 → x2 and y1 → y2 , the right-hand side of the above inequality tends to zero. The equicontinuity for the cases x1 < x2 < 0, y1 < y2 < 0 and x1 ≤ 0 ≤ x2 , y1 ≤ 0 ≤ y2 is obvious. An application of the Arzelà-Ascoli Theorem yields that N maps BR into C a compact set in C, that is N : BR → P(C) is a compact operator. Thus N (BR ) is relatively compact. Step 3. N is upper semi-continuous on BR . 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  hn (x, y) = Φ(x, y),    m  X   hn (x, y) = bi (x, y)un (x − ξi , y − µi )+ i=1    Rx Ry  1  + (x − s)r1 −1 (y − t)r2 −1 fn (s, t)dtds, Γ(r1 )Γ(r2 )

0 0

˜ (x, y) ∈ J,

(x, y) ∈ J.

216


We must show that there exists f∗ ∈ SF,u∗ such that  h∗ (x, y) = Φ(x, y),    m  X   h∗ (x, y) = bi (x, y)u∗ (x − ξi , y − µi )+ i=1    Rx Ry  1  + (x − s)r1 −1 (y − t)r2 −1 f∗ (s, t)dtds, Γ(r1 )Γ(r2 )

˜ (x, y) ∈ J,

(x, y) ∈ J.

0 0

Now, we consider the linear continuous operator Λ : L∞ ([−ξ, a] × [−µ, b]) −→ C, f 7−→ Λ(f )(x, y) such that  Λ(f )(x, y) = Φ(x, y),    m  X   Λ(f )(x, y) = bi (x, y)u∗ (x − ξi , y − µi )+ i=1    Rx Ry  1  + (x − s)r1 −1 (y − t)r2 −1 f (s, t)dtds, Γ(r1 )Γ(r2 )

˜ (x, y) ∈ J,

(x, y) ∈ J.

0 0

From Lemma 2.6, it follows that Λ ◦ SF is a closed graph operator. Clearly, for each (x, y) ∈ J, we have

" # m

X

bi (x, y)un (x − ξi , y − µi ) −

hn (x, y) −

i=1 # " m

X

bi (x, y)u∗ (x − ξi , y − µi ) → 0 as n → ∞. − h∗ (x, y) −

i=1

Moreover, from the definition of Λ, we have hn (x, y) −

m X

bi (x, y)un (x − ξi , y − µi ) ∈ Λ(SF,un ).

i=1

Since un → u∗ , it follows from Lemma 2.6 that, for some f∗ ∈ Λ(SF,u∗ ), we have h∗ (x, y) −

m X

bi (x, y)u∗ (x − ξi , y − µi ) =

i=1

1 = Γ(r1 )Γ(r2 )

Zx Zy 0

(x − s)r1 −1 (y − t)r2 −1 f∗ (s, t)dtds.

0

From Lemma 2.4, we can conclude that N is u.s.c.

217


Step 4. N has convex values. Let u ∈ C and g1 , g2 ∈ N (u), then there exists f1 , f2 ∈ SF,u such that gk (x, y) =

m X

bi (x, y)u(x − ξi , y − µi ) + Iθr fk (x, y),

k = 1, 2.

i=1

Let 0 ≤ ζ ≤ 1, then for each (x, y) ∈ J, we have [ζg1 + (1 − ζ)g2 ](x, y) =

m X

bi (x, y)[ζu(x − ξi , y − µi ) + (1 − ζ)u(x − ξi , y − µi )]+

i=1

+ Iθr [ζf1 + (1 − ζ)f2 ](x, y) = m X = bi (x, y)u(x − ξi , y − µi ) + Iθr [ζf1 + (1 − ζ)f2 ](x, y), i=1

˜ we have [ζg1 + (1 − ζ)g2 ](x, y) = Φ(x, y). and for each (x, y) ∈ J, The convexity of SF,u and F (x, y, u) implies that [ζg1 + (1 − ζ)g2 ] ∈ N (u). Hence N (u) is convex for each u ∈ C. As a consequence of Lemma 2.10, we deduce that N has a fixed point which is a solution for the problem (1.1)–(1.2). Theorem 3.3 (Non-convex Case). Assume that the following conditions hold: (H3 ) The multifunction F : J × Rn → Pcp (Rn ) has the property that F (·, ·, u) : J → Pcp (Rn ) is measurable for each u ∈ Rn , (H4 ) There exists a non-negative function m ∈ L∞ (J) such that Hd (F (x, y, u), F (x, y, v)) ≤ m(x, y)ku − vk for every u, v ∈ Rn , and d(0, F (x, y, 0)) ≤ m(x, y) a.e. (x, y) ∈ J. If mB +

m∗ ar1 br2 < 1, Γ(1 + r1 )Γ(1 + r2 )

(3.2)

where m∗ = kmkL∞ , then the problem (1.1)–(1.2) has at least one solution on [−ξ, a] × [−µ, b]. Proof. For each u ∈ C the set SF,u is nonempty since by (H3 ), F has a nonempty measurable selection (Theorem 2.8). We shall show that N defined in Theorem 3.2 satisfies the assumptions of Lemma 2.11. The proof will be given in two steps.

218


Step 1. N (u) ∈ Pcl (C) for each u ∈ C. Indeed, let (gn )n≥0 ∈ N (u) such that gn → g. Then, g ∈ C and there exists fn ∈ SF,u such that, for each (x, y) ∈ J,  ˜ gn (x, y) = Φ(x, y), (x, y) ∈ J,    m  X   gn (x, y) = bi (x, y)u(x − ξi , y − µi )+ i=1    Rx Ry  1  + Γ(r1 )Γ(r (x − s)r1 −1 (y − t)r2 −1 fn (s, t)dtds, (x, y) ∈ J. 2) 0 0

Using the fact that F has compact values and from (H4 ), we may pass to a subsequence if necessary to get that fn (·, ·) converges to f almost everywhere on J. Using (H4 ) we have for a.e. (x, y) ∈ J |fn (x, y)| ≤ m(x, y)kuk∞ + m(x, y),

n ∈ N.

The Lebesgue dominated convergence theorem implies that kfn − f kL1 → 0 as n → ∞. Hence f ∈ SF,u . Then, for each (x, y) ∈ [−ξ, a] × [−µ, b], gn (x, y) → g(x, y), where  ˜ g(x, y) = Φ(x, y), (x, y) ∈ J,    m  X   g(x, y) = bi (x, y)u(x − ξi , y − µi )+ i=1    Rx Ry  1  + (x − s)r1 −1 (y − t)r2 −1 f (s, t)dtds, (x, y) ∈ J. Γ(r1 )Γ(r2 )

0 0

So, g ∈ N (u). Step 2. There exists γ < 1 such that Hd (N (u), N (v)) ≤ γku − vkC for each u, v ∈ C. Let u, v ∈ C and g ∈ N (u). Then, there exists f ∈ SF,u such that  ˜ g(x, y) = Φ(x, y), (x, y) ∈ J,    m  X   g(x, y) = bi (x, y)u(x − ξi , y − µi )+ i=1    Rx Ry  1 +  (x − s)r1 −1 (y − t)r2 −1 f (s, t)dtds, (x, y) ∈ J. Γ(r1 )Γ(r2 )

0 0

From (H4 ) it follows that Hd (F (x, y, u(x, y)), F (x, y, v(x, y))) ≤ m(x, y)ku(x, y) − v(x, y)k. Hence, there exists w ∈ SF,v such that kf (x, y) − w(x, y)k ≤ m(x, y)ku(x, y) − v(x, y)k,

(x, y) ∈ J.


219

Consider U : J → P(Rn ) given by U (x, y) = {w(x, y) | w : J → Rn is Lebesgue integrable and kf (x, y) − w(x, y)k ≤ m(x, y)ku(x, y) − v(x, y)k}. Since the multivalued operator U (x, y) ∩ F (x, y, v(x, y)) is measurable (Proposition 2.7), there exists a function f (x, y) which is a measurable selection for U. So, f (x, y) ∈ SF,v , and for each (x, y) ∈ J, kf (x, y) − f (x, y)k ≤ m(x, y)ku(x, y) − v(x, y)k. Let us define  ˜ g(x, y) = Φ(x, y), (x, y) ∈ J,    m  X   g(x, y) = bi (x, y)v(x − ξi , y − µi )+ i=1    Rx Ry  1  + Γ(r1 )Γ(r (x − s)r1 −1 (y − t)r2 −1 f (s, t)dtds, (x, y) ∈ J. 2) 0 0

Then, for each (x, y) ∈ [−ξ, a] × [−µ, b], we get kg(x, y) − g(x, y)k ≤

m X

|bi (x, y)|ku(x − ξi , y − µi ) − v(x − ξi , y − µi )k+

i=1

1 + Γ(r1 )Γ(r2 )

Zx Zy 0

(x − s)r1 −1 (y − t)r2 −1 ×

0

× kf (s, t) − f (s, t)kdtds ≤ ≤ mBku − vkC + Zx Zy 1 + (x − s)r1 −1 (y − t)r2 −1 × Γ(r1 )Γ(r2 ) 0

0

× m(s, t)ku − vkC dtds ≤ ar1 br2 ku − vkC = ≤ mBku − vkC + m∗ Γ(1 + r1 )Γ(1 + r2 ) m∗ ar1 br2 = mB + ku − vkC . Γ(1 + r1 )Γ(1 + r2 ) Thus, for each (x, y) ∈ [−ξ, a] × [−µ, b], we get m∗ ar1 br2 kg − gkC ≤ mB + ku − vkC . Γ(1 + r1 )Γ(1 + r2 ) By an analogous relation, obtained by interchanging the roles of u and v, it follows that m∗ ar1 br2 Hd (N (u), N (v)) ≤ mB + ku − vkC . Γ(1 + r1 )Γ(1 + r2 )

220


So by (3.2), N is a contraction and thus, by Lemma 2.11, N has a fixed point u which is a solution to (1.1)–(1.2) on [−ξ, a] × [−µ, b].

4. AN EXAMPLE As an application of our results we consider the following system of fractional integral inclusions of the form x3 y x4 y 2 1 + u(x, y) − u(x − 1, y − 3) + u x − 2, y − 8 12 4 1 3 + u x − , y − 2 ∈ Iθr F (x, y, u) if (x, y) ∈ J := [0, 1] × [0, 1], 14 2 u(x, y) = Φ(x, y) if (x, y) ∈ J˜ := [−2, 1] × [−3, 1]\(0, 1] × (0, 1], where m = 3, r = 12 , 15 , F (x, y, u) =

ex+y ex+y (1 + |u|) , 2 + |u| 2 + |u|

(4.1)

(4.2)

a.a. (x, y) ∈ J and for all u ∈ R,

and Φ : J˜ → R is a continuous function satisfying Φ(x, 0) =

1 3 Φ x − , −2 , 14 2

Φ(0, y) =

1 3 Φ − ,y − 2 x, y ∈ [0, 1]. 14 2

(4.3)

Notice that condition (4.3) is satisfied by Φ ≡ 0. Set x4 y 2 1 x3 y , b2 (x, y) = , b3 (x, y) = . b1 (x, y) = 8 12 14 Then, B =

1 8

and kF (x, y, u)k ≤ ex+y for a.e. (x, y) ∈ J and all u ∈ R.

It is clear that F is Carathéodory, hence condition (H1 ) is satisfied. Also, condition (H2 ) holds with h∗ = e2 , and we have mB = 38 < 1. In view of Theorem 3.2, the problem (4.1)–(4.2) has a solution defined on [−2, 1] × [−3, 1]. Acknowledgments The authors are grateful to the referees for the careful reading of the paper and for their helpful remarks.

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Saïd Abbas [email protected] Université de Saïda Laboratoire de Mathématiques B.P. 138, 20000, Saïda, Algeria

Mouffak Benchohra [email protected] Université de Sidi Bel-Abbès Laboratoire de Mathématiques B.P. 89, 22000, Sidi Bel-Abbès, Algeria

Received: January 18, 2012. Revised: July 12, 2012. Accepted: August 28, 2012.