Existence results for fractional order functional differential equations with infinite delay

J. Math. Anal. Appl. 338 (2008) 1340–1350 www.elsevier.com/locate/jmaa

Existence results for fractional order functional differential equations with infinite delay M. Benchohra a , J. Henderson b , S.K. Ntouyas c,∗ , A. Ouahab a a Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algeria b Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA c Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 9 January 2006 Available online 22 June 2007 Submitted by I. Podlubny

Abstract The Banach fixed point theorem and the nonlinear alternative of Leray–Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay. © 2007 Elsevier Inc. All rights reserved. Keywords: Functional differential equations; Fractional derivative; Fractional integral; Existence; Fixed point

1. Introduction This paper is concerned with the existence of solutions for initial value problems (IVP for short) of fractional order functional differential equations with infinite delay. In Section 3 we will consider the IVP of the form D α y(t) = f (t, yt ), y(t) = φ(t),

for each t ∈ J = [0, b], 0 < α < 1,

t ∈ (−∞, 0],

(1) (2)

where D α is the standard Riemman–Liouville fractional derivative, f : J × B → R is a given function satisfying some assumptions that will be specified later, φ ∈ B, φ(0) = 0 and B is called a phase space that will be defined later (see Section 2). For any function y defined on (−∞, b] and any t ∈ J , we denote by yt the element of B defined by yt (θ ) = y(t + θ ),

θ ∈ (−∞, 0].

* Corresponding author.

E-mail addresses: [email protected] (M. Benchohra), [email protected] (J. Henderson), [email protected] (S.K. Ntouyas), [email protected] (A. Ouahab). 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.06.021

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Here yt (·) represents the history of the state from time −∞ up to the present time t. Section 4 is devoted to fractional neutral functional differential equations,   D α y(t) − g(t, yt ) = f (t, yt ), for each t ∈ J, (3) y(t) = φ(t),

t ∈ (−∞, 0],

(4)

where f and φ are as in problem (1)–(2), and g : J × B → R is a given function such that g(0, φ) = 0. The notion of the phase space B plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato [9] (see also Kappel and Schappacher [12] and Schumacher [22]). For a detailed discussion on this topic we refer the reader to the book by Hino et al. [11]. In fact, for the case where α = 1, in addition to [9,11], an extensive theory has been developed for the problems (1)–(2) and (3)–(4) by Corduneanu and Lakshmikantham [1], Lakshmikantham et al. [13] and Shin [23]. Fractional differential equations have been of great interest recently. In cause, in part to both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, engineering, etc. For details, see the monographs of Miller and Ross [16], Podlubny [19] and Samko et al. [21], and the papers of Delboso and Rodino [3], Diethelm et al. [2,4,5], Gaul et al. [6], Glockle and Nonnenmacher [7], Mainardi [14], Metzler et al. [15], Momani and Hadid [17], Momani et al. [18], Podlubny et al. [20], Yu and Gao [24] and the references therein. Our approach is based on the Banach fixed point theorem and on the nonlinear alternative of Leray–Schauder type [8]. These results can be considered as a contribution to this emerging field. 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. We consider R+ = {x ∈ R: x > 0}, C 0 (R+ ) the space of all continuous functions on R+ . Consider also the space 0 C (R+ 0 ) of all continuous real functions on R+ 0 = {x ∈ R: x  0}, which later identify by abuse of notation, with the class of all f ∈ C 0 (R+ ) such that limt→0+ f (t) = f (0+ ) ∈ R. By C(J, R) we denote the Banach space of all continuous functions from J into R with the norm    y∞ := sup y(t): t ∈ J , where | · | denotes a suitable complete norm on R. Definition 2.1. The fractional primitive of order α > 0 of a function h : R+ → R of order α ∈ R+ is defined by t I0α h(t) =

(t − s)α−1 h(s) ds, (α)

0

provided the right side exists pointwise on R+ .  is the gamma function. For instance, I α h exists for all α > 0, when + α 0 α h ∈ C 0 (R+ ) ∩ L1loc (R+ ); note also that when h ∈ C 0 (R+ 0 ) then I h ∈ C (R0 ) and moreover I h(0) = 0. Definition 2.2. The fractional derivative of order α > 0 of a continuous function h : R+ → R is given by d 1 d α h(t) = dt α (1 − α) dt

t

(t − s)−α h(s) ds =

d 1−α I h(t). dt a

a

In this paper, we assume that the state space (B,  · B ) is a seminormed linear space of functions mapping (−∞, 0] into R, and satisfying the following fundamental axioms which were introduced by Hale and Kato in [9].

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(A) If y : (−∞, b] → R, and y0 ∈ B, then for every t ∈ [0, b] the following conditions hold: (i) yt is in B, (ii) yt B  K(t) sup{|y(s)|: 0  s  t} + M(t)y0 B , (iii) |y(t)|  H yt B , where H  0 is a constant, K : [0, b] → [0, ∞) is continuous, M : [0, ∞) → [0, ∞) is locally bounded and H, K, M are independent of y(·). (A-1) For the function y(·) in (A), yt is a B-valued continuous function on [0, b]. (A-2) The space B is complete. 3. FDEs of fractional order Let us start by defining what we mean by a solution of problem (1)–(2). Let the space   Ω = y : (−∞, b] → R: y|(−∞,0] ∈ B and y|[0,b] is continuous . Definition 3.1. A function y ∈ Ω is said to be a solution of (1)–(2) if y satisfies the equation D α y(t) = f (t, yt ) on J , and the condition y(t) = φ(t) on (−∞, 0]. For the existence results on the problem (1)–(2) we need the following auxiliary lemma. Lemma 3.2. (See [3].) Let 0 < α < 1 and let h : (0, b] → R be continuous and limt→0+ h(t) = h(0+ ) ∈ R. Then y is a solution of the fractional integral equation y(t) =

1 (α)

t (t − s)α−1 h(s) ds, 0

if and only if y is a solution of the initial value problem for the fractional differential equation D α y(t) = h(t),

t ∈ (0, b],

y(0) = 0. Our first existence result for the IVP (1)–(2) is based on the Banach contraction principle. Theorem 3.3. Let f : J × B → R. Assume (H) there exists  > 0 such that   f (t, u) − f (t, v)  u − vB , If

for t ∈ J and every u, v ∈ B.

bα Kb  (α+1)

< 1, where    Kb = sup K(t): t ∈ [0, b] ,

then there exists a unique solution for the IVP (1)–(2) on the interval (−∞, b]. Proof. Transform the problem (1)–(2) into a fixed point problem. Consider the operator N : Ω → Ω defined by  φ(t), t ∈ (−∞, 0], t N (y)(t) = 1 α−1 f (s, ys ) ds, t ∈ [0, b]. (α) 0 (t − s) Let x(·) : (−∞, b] → R be the function defined by  0, if t ∈ [0, b], x(t) = φ(t), if t ∈ (−∞, 0].

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Then x0 = φ. For each z ∈ C([0, b], R) with z(0) = 0, we denote by z¯ the function defined by  z(t), if t ∈ [0, b], z¯ (t) = 0, if t ∈ (−∞, 0]. If y(·) satisfies the integral equation 1 y(t) = (α)

t (t − s)α−1 f (s, ys ) ds, 0

we can decompose y(·) as y(t) = z¯ (t) + x(t), 0  t  b, which implies yt = z¯ t + xt , for every 0  t  b, and the function z(·) satisfies 1 z(t) = (α)

t (t − s)α−1 f (s, z¯ s + xs ) ds. 0

Set



  C0 = z ∈ C [0, b], R : z0 = 0 ,

and let  · b be the seminorm in C0 defined by       zb = z0 B + sup z(t): 0  t  b = sup z(t): 0  t  b ,

z ∈ C0 .

C0 is a Banach space with norm  · b . Let the operator P : C0 → C0 be defined by 1 (P z)(t) = (α)

t (t − s)α−1 f (s, z¯ s + xs ) ds,

t ∈ [0, b].

(5)

0

That the operator N has a fixed point is equivalent to P has a fixed point, and so we turn to proving that P has a fixed point. We shall show that P : C0 → C0 is a contraction map. Indeed, consider z, z∗ ∈ C0 . Then we have for each t ∈ [0, b]   P (z)(t) − P (z∗ )(t) 

1 (α)

t

 

(t − s)α−1 f (s, z¯ s + xs ) − f s, z¯ s∗ + xs  ds

0

1  (α)

t

(t − s)α−1  z¯ s − z¯ s∗ B ds

0



1 (α)

t 0

Kb  (α)

t

(t − s)α−1 Kb sup z(s) − z∗ (s) ds s∈[0,t]

(t − s)α−1  dsz − z∗ b .

0

Therefore α P (z) − P (z∗ )  b Kb  z − z∗ b , b (α + 1) and hence P is a contraction. Therefore, P has a unique fixed point by Banach’s contraction principle.

2

Now we give an existence result based on the nonlinear alternative of Leray–Schauder type. For this, we state the following generalization of Gronwall’s lemma for singular kernels, whose proof can be found in [10, Lemma 7.1.1], which will be essential for the main result of this section.

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Lemma 3.4. Let v : [0, b] → [0, ∞) be a real function and w(·) is a nonnegative, locally integrable function on [0, b] and there are constants a > 0 and 0 < α < 1 such that t v(s) ds. v(t)  w(t) + a (t − s)α 0

Then there exists a constant K = K(α) such that t w(s) v(t)  w(t) + Ka ds, (t − s)α 0

for every t ∈ [0, b]. Theorem 3.5. Assume that the following hypotheses hold: (H1) f is a continuous function; (H2) there exist p, q ∈ C(J, R+ ) such that   f (t, u)  p(t) + q(t)uB for t ∈ J and each u ∈ B, and I α p∞ < +∞. Then the IVP (1)–(2) has at least one solution on (−∞, b]. Proof. Let P : C0 → C0 be defined as in (5). We shall show that the operator P is continuous and completely continuous. Step 1. P is continuous. Let {zn } be a sequence such that zn → z in C0 . Then   (P zn )(t) − (P z)(t) 

1 (α)

b

  (t − s)α−1 f (s, z¯ ns + xs ) − f (s, z¯ s + xs ) ds.

0

Since f is a continuous function, we have bα P (zn ) − P (z)  f (·, z¯ n + x(·) ) − f (·, z¯ (·) + x(·) ) → 0 as n → ∞. (·) b ∞ (α + 1) Step 2. P maps bounded sets into bounded sets in C0 . Indeed, it is enough to show that for any η > 0, there exists a positive constant  such that for each z ∈ B η = {z ∈ C0 : zb  η} one has P (z)∞  . Let z ∈ Bη . Since f is a continuous function, we have for each t ∈ [0, b],   (P z)(t) 

1 (α)

b (t − s)α−1 f (s, z¯ s + xs ) ds 0

1  (α)

b

  (t − s)α−1 p(s) + q(s)¯zs + xs B ds

0

bα q∞ bα p∞ + η∗ =: ,  (α + 1) (α + 1) where ¯zs + xs B  ¯zs B + xs B  Kb η + Mb φB := η∗ ,

M. Benchohra et al. / J. Math. Anal. Appl. 338 (2008) 1340–1350

and

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   Mb = sup M(t): t ∈ [0, b] .

Hence (P z)∞  . Step 3. P maps bounded sets into equicontinuous sets of C0 . Let t1 , t2 ∈ [0, b], t1 < t2 , and let Bη be a bounded set of C0 as in Step 2. Let z ∈ Bη . Then for each t ∈ [0, b], we have   (P z)(t2 ) − (P z)(t1 )  t1  t2   1 

1  = (t2 − s)α−1 − (t1 − s)α−1 f (s, z¯ s + xs ) ds + (t2 − s)α−1 f (s, z¯ s + xs ) ds    (α)  (α) t1

0

p∞ + q∞ η∗  (α)

t1

  p∞ + q∞ η∗ (t1 − s)α−1 − (t2 − s)α−1 ds + (α)

t2 (t2 − s)α−1 ds t1

0

 p∞ + q∞ η∗ p∞ + q∞ η∗   (t2 − t1 )α + t1α − t2α + (t2 − t1 )α (α + 1) (α + 1) 2[p∞ + q∞ η∗ ] (t2 − t1 )α .  (α + 1) As t1 → t2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t1 < t2  0 and t1  0  t2 is obvious. As a consequence of Steps 1–3, together with the Arzela–Ascoli theorem, we can conclude that P : C0 → C0 is continuous and completely continuous. Step 4 (A priori bounds). We now show there exists an open set U ⊆ C0 with z = λP (z) for λ ∈ (0, 1) and z ∈ ∂U . Let z ∈ C0 and z = λP (z) for some 0 < λ < 1. Then for each t ∈ [0, b] we have 

t 1 α−1 (t − s) f (s, z¯ s + xs ) ds . z(t) = λ (α) 0

This implies by (H2)   z(t) 

1 (α)

t (t − s)α−1 q(s)¯zs + xs B ds + 0

But

bα p∞ , (α + 1)

t ∈ [0, b].

   ¯zs + xs B  ¯zs B + xs B  K(t) sup z(s): 0  s  t    + M(t)z0 B + K(t) sup x(s): 0  s  t + M(t)x0 B     Kb sup z(s): 0  s  t + Mb φB .

If we name w(t) the right-hand side of (6), then we have ¯zs + xs B  w(t), and therefore   z(t) 

1 (α)

t (t − s)α−1 q(s)w(s) ds + 0

bα p∞ , (α + 1)

t ∈ [0, b].

(6)

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Using the above inequality and the definition of w we have that Kb bα p∞ Kb q∞ + w(t)  Mb φB + (α + 1) (α)

t (t − s)α−1 w(s) ds,

t ∈ [0, b].

0

Then from Lemma 3.4, there exists K = K(α) such that we have α   w(t)  Mb φB + Kb b p∞ + K(α) Kb q∞ (α + 1) (α)

t (t − s)α−1 R ds, 0

where R = Mb φB +

Kb bα p∞ . (α + 1)

Hence w∞  R +

RK(α)bα Kb  := M. (α + 1)

Then α  I α q + b p∞ := M ∗ . z∞  M ∞ (α + 1)

Set

  U = z ∈ C0 : zb < M ∗ + 1 .

P : U → C0 is continuous and completely continuous. From the choice of U , there is no z ∈ ∂U such that z = λP (z), for λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray–Schauder type [8], we deduce that P has a fixed point z in U . 2 4. NFDEs of fractional order In this section we give existence results for the IVP (3)–(4). Definition 4.1. A function y ∈ Ω is said to be a solution of (3)–(4) if y satisfies the equation D α [y(t) − g(t, yt )] = f (t, yt ) on J , and y(t) = φ(t) on (−∞, 0]. Our first existence result for the IVP (3)–(4) is also based on the Banach contraction principle. Theorem 4.2. Assume that (H) holds and moreover (A) there exists a nonnegative constant c1 such that   g(t, u) − g(t, v)  c1 u − vB , for every u, v ∈ B. If Kb [c1 +

bα  (α+1) ] < 1,

then there exists a unique solution for the IVP (3)–(4) on the interval (−∞, b].

Proof. Consider the operator N1 : Ω → Ω defined by  φ(t), if t ∈ (−∞, 0], t N1 (y)(t) = 1 α−1 g(t, yt ) + (α) 0 (t − s) f (s, ys ) ds, if t ∈ [0, b]. In analogy to Theorem 3.5, we consider the operator P1 : C0 → C0 defined by  0, t  0, t (P1 z)(t) = 1 α−1 g(t, z¯ t + xt ) + (α) 0 (t − s) f (s, z¯ s + xs ) ds, t ∈ (0, b].

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We shall show that the operator P1 is a contraction. Let z, z∗ ∈ Ω. Then following the steps of Theorem 3.3, we have     P1 (z)(t) − P1 (z∗ )(t)  g(t, z¯ t + xt ) − g(t, z¯ ∗t + xt ) 1 + (α)

t

   f (s, z¯ s + xs ) − f (s, z¯ ∗s + xs )(t − s)α−1  ds

0

t 1  c1 ¯zt − z¯ ∗t B + (t − s)α−1 ¯zs − z¯ ∗s B ds (α) 0      c1 Kb sup z(s) − z∗ (s): s ∈ [0, t] 1 + (α)

t

   (t − s)α−1 Kb sup z(s) − z∗ (s): s ∈ [0, t] ds.

0

Consequently

 P1 (z) − P1 (z∗ )  Kb c1 + b

 bα z − z∗ b , (α + 1)

which implies that P1 is a contraction. Hence P1 has a unique fixed point by Banach’s contraction principle.

2

Our second existence result for the IVP (3)–(4) is based on the nonlinear alternative of Leray–Schauder. Theorem 4.3. Assume (H1)–(H2) and the following condition: (H3) the function g is continuous and completely continuous, and for any bounded set B in Ω, the set {t → g(t, yt ): y ∈ B} is equicontinuous in C([0, b], R), and there exist constants 0  Kb d1 < 1, d2  0 such that   g(t, u)  d1 uB + d2 , t ∈ [0, b], u ∈ B. Then the IVP (3)–(4) has at least one solution on (−∞, b]. Proof. Let P1 : C0 → C0 be defined as in Theorem 4.2. We shall show that the operator P1 is continuous and completely continuous. Using (H3) it suffices to show that the operator P2 : C0 → C0 defined by 1 P2 (z)(t) = g(t, z¯ s + xs ) + (α)

t (t − s)α−1 f (s, z¯ s + xs ) ds,

t ∈ [0, b],

0

is continuous and completely continuous. This was proved in Theorem 3.5. We now show there exists an open set U ⊆ C0 with z = λP1 (z) for λ ∈ (0, 1) and z ∈ ∂U . Let z ∈ C0 and z = λN1 (z) for some 0 < λ < 1. Then 

t 1 α−1 (t − s) f (s, z¯ s + xs ) ds , t ∈ [0, b], z(t) = λ g(t, z¯ t + xt ) + (α) 0

and α   z(t)  d1 ¯zt + xt B + d2 + b p∞ + 1 (α + 1) (α)

t (t − s)α−1 q(s)¯zs + xs B ds, 0

t ∈ (0, b].

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Thus 

t 1 Kb bα p∞ Kb q ∗ ∞ w(t)  + (t − s)α−1 w(s) ds , 2Kb d2 + 1 − Kb d1 (α + 1) (α)

t ∈ (0, b],

0

and consequently w∞  R1 +

bα R1 Kb q ∗ ∞ := L, (1 − Kb d1 )(α + 1)

where q ∗ ∞ =

q∞ , 1 − Kb d1

and R1 =

  1 Kb bα p∞ 2Kb d2 + . 1 − Kb d1 (α + 1)

Then z∞  d1 φB + 2d2 + Ld1 + Set

bα p∞ + L I α q ∞ := L∗ . (α + 1)

  U1 = y ∈ C0 : yb < L∗ + 1 .

From the choice of U there is no y ∈ ∂U1 such that y = λP2 (y) for λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray–Schauder type [8], we deduce that P2 has a fixed point z in U1 . Then N1 has a fixed point, which is a solution of the IVP (3)–(4). 2 5. An example In this section we give an example to illustrate the usefulness of our main results. Let us consider the fractional functional differential equation, ce−γ t+t yt   D α y(t) = t , e + e−t (1 + yt ) y(t) = φ(t),

b

t ∈ J := [0, b], α ∈ (0, 1),

(7)

t ∈ (−∞, 0],

(8)

s α−1 e−s

ds, and c0 > 1 fixed. Let γ be a positive real constant and where c(α) = c0 0  

 Bγ = y ∈ C (−∞, 0], R : lim eγ θ y(θ ), exists in R . θ→−∞

The norm of Bγ is given by   yγ = sup eγ θ y(θ ). −∞<θ0

Let y : (−∞, b] → R be such that y0 ∈ Bγ . Then lim eγ θ yt (θ ) = lim eγ θ y(t + θ ) = lim eγ (θ−t) y(θ ) = e−γ t lim eγ θ y0 (θ ) < ∞.

θ→−∞

θ→−∞

θ→−∞

Hence yt ∈ Bγ . Finally we prove that    yt γ  K(t) sup y(s): 0  s  t + M(t)y0 γ , where K = M = 1 and H = 1. We have |yt (θ )| = |y(t + θ )|. If θ + t  0, we get      yt (θ )  sup y(s): −∞ < s  0 . For t + θ  0, then we have      yt (θ )  sup y(s): 0 < s  t .

θ→−∞

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Thus for all t + θ ∈ [0, b], we get         yt (θ )  sup y(s): −∞ < s  0 + sup y(s): 0  s  t . Then

   yt γ  y0 γ + sup y(s): 0  s  t .

It is clear that (Bγ ,  · γ ) is a Banach space. We can conclude that Bγ is a phase space. Set e−γ t+t x , c(et + e−t )(1 + x) Let x, y ∈ Bγ . Then we have f (t, x) =

(t, x) ∈ [0, b] × Bγ .

  e−γ t+t  x et−γ t |x − y| y  = − c(et + e−t )  1 + x 1 + y  c(et + e−t )(1 + x)(1 + y) et |x − y|Bγ 1 et e−γ t |x − y|   x − yBγ .  t −t t −t c(e + e )(1 + x)(1 + y) c(e + e ) c

  f (t, x) − f (t, y) =

Hence the condition (H) holds. Assume that

bα c(α+1)

< 1. Since K = 1, then

bα < 1. c(α + 1) Then by Theorem 3.3 the problem (7)–(8) has a unique solution on (−∞, b]. Acknowledgment The authors are grateful to the referee for her/his comments and remarks.

References [1] C. Corduneanu, V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal. 4 (1980) 831–877. [2] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II–Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217–224. [3] D. Delboso, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996) 609–625. [4] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [5] K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997) 231–253. [6] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991) 81–88. [7] W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995) 46–53. [8] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. [9] J. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978) 11–41. [10] D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin, 1989. [11] Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, 1991. [12] F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (1980) 141–183. [13] V. Lakshmikantham, L. Wen, B. Zhang, Theory of Differential Equations with Unbounded Delay, Math. Appl., Kluwer Academic Publishers, Dordrecht, 1994. [14] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291–348. [15] F. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995) 7180–7186. [16] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [17] S.M. Momani, S.B. Hadid, Some comparison results for integro-fractional differential inequalities, J. Fract. Calc. 24 (2003) 37–44. [18] S.M. Momani, S.B. Hadid, Z.M. Alawenh, Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. Math. Sci. 2004 (2004) 697–701. [19] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [20] I. Podlubny, I. Petraš, B.M. Vinagre, P. O’Leary, L. Dorˇck, Analogue realizations of fractional-order controllers, in: Fractional Order Calculus and Its Applications, Nonlinear Dynam. 29 (2002) 281–296.

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M. Benchohra et al. / J. Math. Anal. Appl. 338 (2008) 1340–1350

[21] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. [22] K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay, Arch. Ration. Mech. Anal. 64 (1978) 315–335. [23] J.S. Shin, An existence theorem of functional differential equations with infinite delay in a Banach space, Funkcial. Ekvac. 30 (1987) 19–29. [24] C. Yu, G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 310 (2005) 26–29.