Tomus 49 (2013), 105–117
SEMILINEAR FRACTIONAL ORDER INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN BANACH SPACES
Khalida Aissani and Mouffak Benchohra
Abstract. This paper concerns the existence of mild solutions for fractional order integro-differential equations with infinite delay. Our analysis is based on the technique of Kuratowski’s measure of noncompactness and Mönch’s fixed point theorem. An example to illustrate the applications of main results is given.
1. Introduction
Fractional calculus is a generalization of the ordinary differentiation and integra- tion to arbitrary non-integer order. The subject is as old as the differential calculus since, starting from some speculations of G. W. Leibniz (1697) and L. Euler (1730), it has been progressing up to nowadays. Fractional differential and integral equa- tions have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering, radiative transfer, neutron transport and the kinetic theory of gases and others [5, 10, 11]. There has been a significant development in ordinary and partial fractional differential equations in recent years;
see the monographs of Abbas et al. [1], Baleanu et al. [6], Diethelm [13], Hilfer [18], Kilbas et al. [20], Miller and Ross [26], Podlubny [31], Samko et al. [32], and Tarasov [33], and the papers [2, 3, 8, 14, 23, 24, 28, 29, 30].
The theory of functional differential equations has emerged as an important branch of nonlinear analysis. Differential delay equations, or functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay; see for instance the books by Hale and Verduyn Lunel [16], Hino et al. [19], Kolmanovskii and Myshkis [21], Lakshmikantham et al. [22], and Wu [34], and the papers [12, 15].
2010Mathematics Subject Classification: primary 26A33; secondary 34G20, 34K30, 34K37.
Key words and phrases: semilinear differential equations, Caputo fractional derivative, mild solution, measure of noncompactness, fixed point, semigroup, Banach space.
Received February 13, 2013, revised May 2013. Editor R. Šimon Hilscher.
DOI: 10.5817/AM2013-2-105
In this work we discuss the existence of mild solutions for fractional order integro-differential equations with infinite delay of the form
(1)
Dtqx(t) =Ax(t) + Z t
0
a(t, s)f(s, xs, x(s))ds , t∈J = [0, T], x(t) =φ(t), t∈ (−∞,0],
whereDqt is the Caputo fractional derivative of order 0< q <1,Ais a generator of an analytic semigroup{S(t)}t≥0 of uniformly bounded linear operators onX, f:J× B ×X −→X,a:D→R(D={(t, s)∈[0, T]×[0, T] :t≥s}),φ∈ Bwhere B is called phase space to be defined in Section 2. For any functionxdefined on (−∞, T] and anyt∈J, we denote byxtthe element ofBdefined by
xt(θ) =x(t+θ), θ∈ (−∞,0].
Here xtrepresents the history of the state up to the present timet.
In the present paper we deal with an infinite time delay. Note that in this case, the phase spaceB plays a crucial role in the study of both qualitative and quantitative aspects of theory of functional equations (see [15]).
We present an existence result of mild solutions for the problem (1) by means of the application of Mönch’s fixed point theorem combined with the Kuratowski measure of noncompactness. An example illustrating the abstract theory will be presented.
2. Preliminaries Let (X,k · k) be a real Banach space.
C=C(J, X) be the space of allX-valued continuous functions onJ. L(X) be the Banach space of all linear and bounded operators onX.
L1(J, X) the space of X-valued Bochner integrable functions onJ with the norm kykL1 =
Z T 0
ky(t)kdt .
L∞(J,R) is the Banach space of essentially bounded functions, normed by kykL∞ = inf{d >0 :|y(t)| ≤d , a.e.t∈J}.
Definition 2.1. A functionf:J× B ×X −→X is said to be an Carathéodory function if it satisfies:
(i) for eacht∈J the function f(t,·,·) :B ×X −→X is continuous;
(ii) for each (v, w)∈ B ×X the functionf(·, v, w) :J→X is measurable.
Next we give the concept of a measure of noncompactness [7].
Definition 2.2. Let B be a bounded subset of a seminormed linear space Y. Kuratowski’s measure of noncompactness ofB is defined as
α(B) = inf{d >0 :B has a finite cover by sets of diameter ≤d}.
We note that this measure of noncompactness satisfies interesting regularity properties (for more information, we refer to [7]).
Lemma 2.3.
(1) If A⊆B thenα(A)≤α(B),
(2) α(A) =α(A), whereA denotes the closure ofA, (3) α(A) = 0⇔A is compact (Ais relatively compact), (4) α(λA) =|λ|A, with λ∈R,
(5) α(A∪B) = max{α(A), α(B)}, (6) α(A+B)≤α(A) +α(B), where
A+B={x+y:x∈A, y∈B}, (7) α(A+a) =α(A) for anya∈Y,
(8) α(convA) =α(A), whereconvA is the closed convex hull ofA.
ForH ⊂C(J, X), we define (2)
Z t 0
H(s)ds=nZ t 0
u(s)ds:u∈Ho
for t∈J , whereH(s) ={u(s)∈X :u∈H}.
Lemma 2.4 ([7]). If H⊂C(J, X)is a bounded, equicontinuous set, then
(3) α(H) = sup
t∈J
α(H(t)).
Lemma 2.5 ([17]). If{un}∞n=1 ⊂L1(J, X)and there exists m∈L1(J,R+) such that kun(t)k ≤m(t), a.e.t∈J, thenα({un(t)}∞n=1) is integrable and
(4) αnZ t
0
un(s)dso∞ n=1
≤2 Z t
0
α({un(s)}∞n=1)ds .
In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [15]. Specifically,Bwill be a linear space of functions mapping (−∞,0] intoX endowed with a seminormk.kB, and satisfies the following axioms:
(A1): Ifx: (−∞, T ]−→X is continuous onJ andx0∈ B, thenxt∈ B andxtis continuous int∈J and
(5) kx(t)k ≤CkxtkB, whereC≥0 is a constant.
(A2): There exist a continuous functionC1(t)>0 and a locally bounded functionC2(t)≥0 int≥0 such that
(6) kxtkB≤C1(t) sup
s∈[0,t]
kx(s)k+C2(t)kx0kB, fort∈[0, T] andxas in (A1).
(A3): The spaceB is complete.
Remark 2.6. Condition (5) in (A1) is equivalent to kφ(0)k ≤ CkφkB, for all φ∈ B.
For our purpose we will only need the following fixed point theorem.
Theorem 2.7([4, 27]). LetU be a bounded, closed and convex subset of a Banach space such that0∈U, and letN be a continuous mapping of U into itself. If the implication
V = convN(V)or V =N(V)∪ {0}=⇒α(V) = 0 holds for every subsetV ofU, thenN has a fixed point.
Let Ω be a set defined by Ω =
x: (−∞, T]→X such that x|(−∞,0]∈ B, x|J∈C(J, X) . 3. Existence of mild solutions
Following [25, 14] we will introduce now the definition of mild solution to (1).
Definition 3.1. A functionx∈Ω is said to be a mild solution of (1) ifxsatisfies
(7) x(t) =
φ(t), t∈ (−∞,0] ;
−Q(t)φ(0) + Z t
0
Z s 0
R(t−s)a(s, τ)f(τ, xτ, x(τ))dτ ds , t∈J , where
Q(t) = Z ∞
0
ξq(σ)S(tqσ)dσ , R(t) =q Z ∞
0
σtq−1ξq(σ)S(tqσ)dσ andξq is a probability density function defined on (0,∞) such that
ξq(σ) =1
qσ−1−1q$q(σ−1q)≥0, where
$q(σ) = 1 π
∞
X
n=1
(−1)n−1σ−qn−1Γ(nq+ 1)
n! sin(nπq), σ∈(0,∞). Remark 3.2. Note that{S(t)}t≥0 is a uniformly bounded semigroup, i.e,
there exists a constant M >0 such that kS(t)k ≤M for all t∈[0, T]. Remark 3.3. According to [25], a direct calculation gives that
(8) kR(t)k ≤Cq,Mtq−1, t >0, whereCq,M= qM
Γ(1 +q).
We make the following assumptions.
(H1) f:J× B ×X →X satisfies the Carathéodory conditions, and there exist two positive functionsµi(·)∈L1(J,R+) (i= 1,2) with
(9) kµ2kL1(J,R+)< q TqaCq,M , such that
(10) kf(t, v, w)k ≤µ1(t)kvkB+µ2(t)kwk, (t, v, w)∈J× B ×X .
(H2) For any bounded sets D1⊂ B, D2⊂X, and 0≤s≤t≤T, there exists an integrable positive functionη such that
α R(t−s)f(τ, D1, D2)
≤ηt(s, τ) α(D2) + sup
−∞<θ≤0
α(D1(θ)) ,
whereηt(s, τ) =η(t, s, τ) and sup
t∈J
Z t 0
Z s 0
ηt(s, τ)dτ ds=η∗<∞.
(H3) For eacht∈J,a(t, s) is measurable on [0, t] anda(t) = ess sup{|a(t, s)|,0≤ s≤t}is bounded onJ. The mapt→atis continuous fromJ toL∞(J,R), here,at(s) =a(t, s).
Seta= sup
t∈J
a(t).
Theorem 3.4. Suppose that the assumptions(H1)–(H3)hold with
(11) 16aη∗<1,
then the problem (1) has at least one mild solution on(−∞, T].
Proof. We transform the problem (1) into a fixed-point problem. Define a mapping Φ from Ω into itself by
Φ(x)(t) =
φ(t), t∈ (−∞,0] ;
−Q(t)φ(0) + Z t
0
Z s 0
R(t−s)a(s, τ)f τ, xτ, x(τ)
dτ ds, t∈J . Clearly, fixed points of the operator Φ are mild solutions of the problem (1).
Forφ∈ B, we will define the functiony(·) : (−∞, T]→X by y(t) =
(φ(t), if t∈ (−∞,0] ; 0, if t∈J .
Theny0 =φ. For eachz ∈C(J, X) withz(0) = 0, we denote byz the function defined by
z(t) =
(0, if t∈ (−∞,0] ; z(t), if t∈J .
If x(·) verifies (7), we can decompose it as x(t) =y(t) +z(t), for t ∈J, which impliesxt=yt+zt, for everyt∈J and the function z(t) satisfies
z(t) =−Q(t)φ(0) + Z t
0
Z s 0
R(t−s)a(s, τ)f τ, yτ+zτ, y(τ) +z(τ) dτ ds . Let
Z0={z∈Ω :z0= 0}. For anyz∈Z0, we have
kzkZ0 = sup
t∈J
kz(t)k+kz0kB= sup
t∈J
kz(t)k.
Thus (Z0,k · kZ0) is a Banach space. We define the operatorΦ :e Z0→Z0 by:
Φ(z)(t) =e −Q(t)φ(0) + Z t
0
Z s 0
R(t−s)a(s, τ)f τ, yτ+zτ, y(τ) +z(τ) dτ ds . Obviously, the operator Φ has a fixed point is equivalent toΦ has one, so it turnse to prove thatΦ has a fixed point. Lete r >0 and consider the set
Br={z∈Z0:kzkZ0 ≤r}.
We need the following lemma.
Lemma 3.5. Set
(12) C1∗= sup
t∈J
C1(t) ; C2∗= sup
η∈J
C2(η). Then for any z∈Br we have
kyt+ztkB≤C2∗kφkB+C1∗r:=r∗, and
(13) kf(t, yt+zt, y(t) +z(t)k ≤µ1(t)r∗+µ2(t)r . Proof. Using (6), (12) and (10), we obtain
kyt+ztkB≤ kytkB+kztkB
≤C1(t) sup
0≤τ≤t
ky(τ)k+C2(t)ky0kB+C1(t) sup
0≤τ≤t
kz(τ)k+C2(t)kz0kB
≤C2(t)kφkB+C1(t) sup
0≤τ≤t
kz(τ)k
≤C2∗kφkB+C1∗r:=r∗. Also, we get
kf(t, yt+zt, y(t) +z(t)k ≤µ1(t)kyt+ztkB+µ2(t)ky(t) +z(t)k
≤µ1(t)r∗+µ2(t)r.
The lemma is proved.
Now we prove thatΦ has a fixed point. The proof will be given in three steps.e Step 1:Φ is continuous.e
Let{zk}k∈Nbe a sequence such thatzk →zinBrask→ ∞. Then for eacht∈J, we have
kΦ(ze k)(t)−Φ(z)(t)k ≤e Z t
0
Z s 0
kR(t−s)a(s, τ)
f τ, yτ+zkτ, y(τ) +zk(τ)
−f τ, yτ+zτ, y(τ) +z(τ) kdτ ds
≤a Cq,M
Z t 0
Z s 0
(t−s)q−1kf τ, yτ+zkτ, y(τ) +zk(τ)
−f τ, yτ+zτ, y(τ) +z(τ) kdτ ds.
Sincef is of Carathéodory type, we have by the Lebesgue Dominated Convergence Theorem that
kΦ(ze k)(t)−Φ(z)(t)k →e 0 when k→ ∞. Consequently,
k→∞lim kΦ(ze k)−Φ(z)ke Z0= 0. ThusΦ is continuous.e
Step 2:Φ mapse Br into itself. Let r≥MkφkB+T
qaCq,Mr∗kµ1kL1 (J,R+ ) q
1−TqaCq,Mkµq2kL1 (J,R+ ) . Then for eachz∈Br andt∈J we have
kΦ(z)(t)k ≤ kQ(t)φ(0)ke +
Z t 0
Z s 0
kR(t−s)a(s, τ)f τ, yτ+zτ, y(τ) +z(τ) kdτ ds
≤MkφkB+a Cq,M Z t
0
Z s 0
(t−s)q−1[µ1(τ)r∗+µ2(τ)r] dτ ds
≤MkφkB+a Cq,Mr∗ Z t
0
Z s 0
(t−s)q−1µ1(τ)dτ ds +a Cq,Mr
Z t 0
Z s 0
(t−s)q−1µ2(τ)dτ ds
≤MkφkB+Tqa Cq,M
q
r∗kµ1kL1(J,R+)+rkµ2kL1(J,R+)
≤r .
Step 3:Φ(Be r) is bounded and equicontinuous.
By Step 2, it is obvious that Φ(Be r)⊂Br is bounded. For the equicontinuity of Φ(Be r). Letτ1, τ2∈J withτ1> τ2, and letz∈Br. Then
eΦ(z)(τ1)−Φ(z)(τe 2)
≤ kQ(τ1)−Q(τ2)kkφ(0)k +k
Z τ1
0
Z s 0
R(τ1−s)a(s, τ)f τ, yτ+zτ, y(τ) +z(τ) dτ ds
− Z τ2
0
Z s 0
R(τ2−s)a s, τ)f(τ, yτ+zτ, y(τ) +z(τ) dτ dsk. Set
G ·, y·+z·, y(·) +z(·)
= Z ·
0
a(·, τ)f τ, yτ+zτ, y(τ) +z(τ) dτ , then
eΦ(z)(τ1)−Φ(z)(τe 2)
≤ kQ(τ1)−Q(τ2)kkφ(0)k +k
Z τ1 0
R(τ1−s)G s, ys+zs, y(s) +z(s) ds
− Z τ2
0
R(τ2−s)G s, ys+zs, y(s) +z(s) dsk
≤ kQ(τ1)−Q(τ2)kkφ(0)k +k
Z τ2 0
R(τ1−s)G s, ys+zs, y(s) +z(s) ds +
Z τ1 τ2
R(τ1−s)G s, ys+zs, y(s) +z(s) ds
− Z τ2
0
R(τ2−s)G s, ys+zs, y(s) +z(s) dsk
≤ kQ(τ1)−Q(τ2)kkφ(0)k +
Z τ2
0
[R(τ1−s)−R(τ2−s)]G s, ys+zs, y(s) +z(s) ds
+
Z τ1
τ2
kR(τ1−s)kkG s, ys+zs, y(s) +z(s) kds
≤I1+I2+I3, where
I1=kQ(τ1)−Q(τ2)kkφ(0)k I2=
Z τ2
0
[R(τ1−s)−R(τ2−s)]G s, ys+zs, y(s) +z(s) ds
I3= Z τ1
τ2
kR(τ1−s)kkG s, ys+zs, y(s) +z(s) kds.
I1 tends to zero asτ2→τ1, sinceQ(t) is a strongly continuous operator.
ForI2, using (8) and (13), we have I2≤
Z τ2
0
[q Z ∞
0
σ(τ1−s)q−1ξq(σ)S (τ1−s)qσ dσ
−q Z ∞
0
σ(τ2−s)q−1ξq(σ)S (τ2−s)qσ
dσ]G s, ys+zs, y(s) +z(s) ds
≤q Z τ2
0
Z ∞ 0
σk[(τ1−s)q−1−(τ2−s)q−1]ξq(σ)S (τ1−s)qσ
×G s, ys+zs, y(s) +z(s) kdσ ds +q
Z τ2
0
Z ∞ 0
σ(τ2−s)q−1ξq(σ)kS (τ1−s)qσ
−S (τ2−s)qσ k
× kG(s, ys+zs, y(s) +z(s))kdσ ds
≤Cq,M Z τ2
0
(τ1−s)q−1−(τ2−s)q−1
kG s, ys+zs, y(s) +z(s) kds +q
Z τ2
0
Z ∞ 0
σ(τ2−s)q−1ξq(σ)kS (τ1−s)qσ
−S (τ2−s)qσ k
× kG s, ys+zs, y(s) +z(s) kdσ ds
≤a
r∗kµ1kL1(J,R+)+rkµ2kL1(J,R+)
×[Cq,M Z τ2
0
(τ1−s)q−1−(τ2−s)q−1 ds +q
Z τ2
0
Z ∞ 0
σ(τ2−s)q−1ξq(σ)kS (τ1−s)qσ
−S (τ2−s)qσ
kdσ ds]. Clearly, the first term on the right-hand side of the above inequality tends to zero asτ2→τ1. From the continuity ofS(t) in the uniform operator topology fort >0, the second term on the right-hand side of the above inequality tends to zero as τ2→τ1. In view of (13), we have
I3≤Cq,M
Z τ1 τ2
(τ1−s)q−1kG(s, ys+zs, y(s) +z(s))kds
≤a Cq,M
r∗kµ1kL1(J,R+)+rkµ2kL1(J,R+)
Z τ1
τ2
(τ1−s)q−1ds . Asτ2→τ1, I3 tends to zero.
SoΦ(Be r) is equicontinuous.
Now let V be a subset of Br such thatV ⊂conv(Φ(Ve )∪ {0}). Moreover, for any ε >0 and bounded set D,we can take a sequence {vn}∞n=1 ⊂D such that α(D)≤2α({vn}) +ε ([9, p.125]). Thus, for{vn}∞n=1 ⊂V, and using Lemmas
2.3–2.5 and (H2), we get α ΦVe
≤2α
Φve n +ε= 2 sup
t∈J
α
Φve n(t) +ε
= 2 sup
t∈J
αnZ t 0
R(t−s) Z s
0
a(s, τ)f τ, yτ+vnτ, y(τ) +vn(τ)
dτ dso +ε
≤4 sup
t∈J
Z t 0
αn
R(t−s) Z s
0
a(s, τ)f τ, yτ+vnτ, y(τ) +vn(τ)
dτ dso +ε
≤8 sup
t∈J
Z t 0
Z s 0
α({R(t−s)a(s, τ)f τ, yτ+vnτ, y(τ) +vn(τ)
dτ ds}) +ε
≤8 asup
t∈J
Z t 0
Z s 0
α {R(t−s)f τ, yτ+vnτ, y(τ) +vn(τ) dτ ds}
+ε
≤8 a sup
t∈J
Z t 0
Z s 0
ηt(s, τ)
α(vn(τ)) + sup
−∞<θ≤0
α(vn(θ+τ))
dτ ds+ε
≤8 a sup
t∈J
Z t 0
Z s 0
ηt(s, τ)
α(vn) + sup
0<µ≤τ
α(vn(µ))
dτ ds+ε
≤16a α(vn) sup
t∈J
Z t 0
Z s 0
ηt(s, τ)dτ ds+ε
≤16a η∗α(V) +ε .
Therefore, in view of Lemma 2.3, we have
α(V)≤α(eΦV)≤16a η∗α(V) +ε , sinceεis arbitrary we obtain that
α(V)≤16a η∗α(V). This means that
α(V)(1−16a η∗)≤0.
By (11) it follows that α(V) = 0. In view of the Ascoli-Arzelà theorem, V is relatively compact inBr. Applying now Theorem 2.7, we conclude thatΦ has ae fixed point which is a solution of the problem (1).
4. An example
In this section we give an example to illustrate the above results. Consider the following fractional integrodifferential equations
(14)
∂q
∂tqv(t, ζ) = ∂2
∂ζ2v(t, ζ) + Z t
0
(t−s) Z 0
−∞
γ1(θ) sin(s|v(s+θ, ζ)|)dθ ds +
Z t 0
(t−s)s2
2 sin|v(s, ζ)|
Z s 0
cosv(ι, ζ)dι ds
v(θ, ζ) =v0(θ, ζ), −∞< θ≤0,
where t, ζ ∈ [0,1], γ1: (−∞,0] → R, v0: (−∞,0]×[0,1] → R are continuous functions, and
Z 0
−∞
|γ1(θ)|dθ <∞.
SetX =L2([0,1],R) and defineAby
D(A) =H2((0,1))∩H01((0,1)), Au=u00.
Then,Agenerates a compact, analytic semigroup S(t) of uniformly bounded, linear operators such thatkS(t)k ≤1.
Let the phase spaceB=C((−∞,0], X), the space of bounded uniformly continuous functions endowed with the following norm:
kϕkB= sup
−∞<θ≤0
|ϕ(θ)|, ∀ϕ∈ B, then we can see thatC1(t) = 0 in (6).
Fort∈[0,1],ζ∈[0,1] and ϕ∈C((−∞,0], X),we set x(t)(ζ) =v(t, ζ),
φ(θ)(ζ) =v0(θ, ζ), θ∈(−∞,0], a(t, s) =t−s ,
f t, ϕ, x(t) (ζ) =
Z 0
−∞
γ1(θ) sin s|ϕ(θ)(ζ)|
dθ+s2
2 sin|x(t)(ζ)|
Z s 0
cosx(ι)(ζ)dι . Thus, problem (14) can be rewritten as the abstract problem (1).
Moreover, fort∈[0,1], we can see kf(t, ϕ, x(t))(ζ)k ≤tkϕkB
Z 0
−∞
|γ1(θ)|dθ+t3 2kx(t)k
=µ1(t)kϕkB+µ2(t)kx(t)k, where
µ1(t) =t Z 0
−∞
|γ1(θ)|dθ , µ2(t) =t3/2. Then (14) has a mild solution by Theorem 3.4.
For example, if we put
γ1(θ) =eθ, q= 1 2, then
Cq,M = 1
Γ 12 = 1/√
π , kµ2kL1(J,R+)= 1/8. Thus, we see
aTqCq,Mkµ2kL1(J,R+)
q = 1
4√ π <1.
Acknowledgement. The authors are grateful to the referee for the careful reading of the paper.
References
[1] Abbas, S., Benchohra, M., N’Guérékata, G.M.,Topics in Fractional Differential Equations, Springer, New York, 2012.
[2] Agarwal, R. P., Belmekki, M., Benchohra, M.,A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differential Equations 2009(2009), 1–47, Article ID 981728.
[3] Agarwal, R. P., Benchohra, M., Hamani, S.,A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math.109 (2010), 973–1033.
[4] Agarwal, R. P., Meehan, M., O’Regan, D.,Fixed Point Theory and Applications, Cambridge University Press, 2001.
[5] Appell, J. M., Kalitvin, A. S., Zabrejko, P. P.,Partial Integral Operators and Integrodiffe- rential Equations, vol. 230, Marcel Dekker, Inc., New York, 2000.
[6] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.,Fractional Calculus Models and Nume- rical Methods, World Scientific Publishing, New York, 2012.
[7] Banaś, J., Goebel, K.,Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1980.
[8] Benchohra, M., Henderson, J., Ntouyas, S., Ouahab, A.,Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl.338(2008), 1340–1350.
[9] Bothe, D.,Multivalued perturbations ofm–accretive differential inclusions, Israel J. Math.
108(1998), 109–138.
[10] Case, K. M., Zweifel, P. F.,Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.
[11] Chandrasekhe, S.,Radiative Transfer, Dover Publications, New York, 1960.
[12] Corduneanu, C., Lakshmikantham, V.,Equations with unbounded delay, Nonlinear Anal.4 (1980), 831–877.
[13] Diethelm, K.,The Analysis of Fractional Differential Equations, Springer, Berlin, 2010.
[14] El-Borai, M.,Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons & Fractals14(2002), 433–440.
[15] Hale, J. K., Kato, J.,Phase space for retarded equations with infinite delay, Funkcial. Ekvac.
21(1) (1978), 11–41.
[16] Hale, J. K., Lunel, S. Verduyn,Introduction to Functional–Differential Equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993.
[17] Heinz, H.-P.,On the behaviour of measures of noncompactness with respect to differentiation and integration of vector–valued functions, Nonlinear Anal.7(12) (1983), 1351–1371.
[18] Hilfer, R.,Applications of fractional calculus in physics, Singapore, World Scientific, 2000.
[19] Hino, Y., Murakami, S., Naito, T.,Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991.
[20] Kilbas, A. A., Srivastava, Hari M., Trujillo, Juan J.,Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
[21] Kolmanovskii, V., Myshkis, A., Introduction to the Theory and Applications of Functional–Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[22] Lakshmikantham, V., Wen, L., Zhang, B.,Theory of Differential Equations with Unbounded Delay, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1994.
[23] Li, F., Zhang, J.,Existence of mild solutions to fractional integrodifferential equations of neutral type with infinite delay, Adv. Differential Equations2011(2011), 1–15, Article ID 963463.
[24] Liang, J., Xiao, T.-J., Casteren, J. van,A note on semilinear abstract functional differential and integrodifferential equations with infinite delay, Appl. Math. Lett.17(4) (2004), 473–477.
[25] Mainardi, F., Paradisi, P., Gorenflo, R.,Probability distributions generated by fractional diffusion equations, Econophysics: An Emerging Science (Kertesz, J., Kondor, I., eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
[26] Miller, K. S., Ross, B.,An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
[27] Mönch, H.,Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal.4(1980), 985–999.
[28] Mophou, G. M., Nakoulima, O., N’Guérékata, G. M.,Existence results for some fractional differential equations with nonlocal conditions, Nonlinear Stud.17(2010), 15–22.
[29] Mophou, G. M., N’Guérékata, G. M.,Existence of mild solution for some fractional diffe- rential equations with nonlocal conditions, Semigroup Forum79(2009), 315–322.
[30] Obukhovskii, V., Yao, J.-C.,Some existence results for fractional functional differential equations, Fixed Point Theory11(2010), 85–96.
[31] Podlubny, I.,Fractional Differential Equations, Academic Press, San Diego, 1999.
[32] Samko, S. G., Kilbas, A. A., Marichev, O. I.,Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[33] Tarasov, V. E.,Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.
[34] Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer Verlag, New York, 1996.
Corresponding author:
Mouffak Benchohra,
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie
E-mail:[email protected]
