Electronic Journal of Differential Equations, Vol. 2008(2008), No. 69, pp. 1–19.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
PERTURBED FUNCTIONAL AND NEUTRAL FUNCTIONAL
EVOLUTION EQUATIONS WITH INFINITE DELAY IN
FRÉCHET SPACES
SELMA BAGHLI, MOUFFAK BENCHOHRA
Abstract. This article shows sufficient conditions for the existence of mild
solutions, on the positive half-line, for two classes of first-order functional
and neutral functional perturbed differential evolution equations with infinite
delay. Our main tools are: the nonlinear alternative proved by Avramescu for
the sum of contractions and completely continuous maps in Fréchet spaces,
and the semigroup theory.
1. Introduction
In this paper, we study the existence of mild solutions, defined on the positive
semi-infinite real interval J := [0, +∞), for two classes of first-order perturbed
functional and neutral functional differential evolution equations with infinite delay
in Fréchet spaces. Firstly, in Section 3, we study the following partial perturbed
evolution equation with infinite delay
y 0 (t) = A(t)y(t) + f (t, yt ) + g(t, yt ),
a.e. t ∈ J,
y0 = φ ∈ B,
(1.1)
(1.2)
where f, g : J × B → E and φ ∈ B are given functions and {A(t)}0≤t<+∞ is a
family of linear closed (not necessarily bounded) operators from a real Banach space
(E, | · |) into E that generates an evolution system of operators {U (t, s)}(t,s)∈J×J
for 0 ≤ s ≤ t < +∞.
For any continuous function y defined on (−∞, +∞) and any t ≥ 0, we denote
by yt the element of B defined by yt (θ) = y(t + θ) for θ ∈ (−∞, 0]. Here yt (·)
represents the history of the state from time t − r up to the present time t. We
assume that the histories yt belongs to some abstract phase space B, to be specified
later.
2000 Mathematics Subject Classification. 34G20, 34K40.
Key words and phrases. Perturbed functional equation; neutral evolution equations;
mild solution; fixed-point theory; nonlinear alternative; Fréchet spaces; infinite delay.
c
2008
Texas State University - San Marcos.
Submitted April 15, 2008. Published May 13, 2008.
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S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
In Section 4, we consider the following perturbed neutral evolution equation with
infinite delay
d
[y(t) − h(t, yt )] = A(t)y(t) + f (t, yt ) + g(t, yt ),
dt
y0 = φ ∈ B,
a.e. t ∈ J,
(1.3)
(1.4)
where A(·), f, g and φ are as in (1.1)–(1.2) and h : J × B → E is a given function.
Finally in Section 5, we give two examples to demonstrate our results.
Functional and partial functional differential equations have been used for modelling the evolution of physical, biological and economic systems in which the response of the system depends not only on the current state, but also on the past
history of the system. For more details on this topic, see for example the books
of Kolmanovskii and Myshkis [34], Hale and Verduyn Lunel [26] and Wu [37], and
the references therein. In the literature devoted to equations with finite delay, the
state space is the space of all continuous functions on the finite interval [−r, 0] for
r > 0, endowed with the uniform norm topology. Some results in this case can be
found in the books by Ahmed [4, 5], Heikkila and Lakshmikantham [27], and Pazy
[35] and the references therein.
When the delay is infinite, the notion of the phase space B plays an important
role in the study of both qualitative and quantitative theory. A usual choice is a
seminormed space satisfying suitable axioms, introduced by Hale and Kato in [25];
see also Corduneanu and Lakshmikantham [21], Kappel and Schappacher [33] and
Schumacher [36]. For a detailed discussion and applications on this topic, we refer
the reader to the book by Hale and Verduyn Lunel [26], Hino et al. [32] and Wu
[37].
Many publications are developed for study of (1.1) with A(t) = A. We refer the
reader to the books by [27] and the pioneer Hino and Murakami paper [31] and
the papers by Adimy et al [1, 2, 3], Balachandran et al. [10, 11], Benchohra and
Gorniewicz [16], Benchohra et al [17, 18], Ezzinbi [23], Henriquez [28] and Hernandez
[29, 30], where existence and uniqueness, among other things, are derived. In a series
of papers, Belmekki et al [12, 13, 14, 15] considered some classes of semilinear
perturbed functional differential problems where existence of solutions are given
over a bounded interval [0, b].
When A depends on time, Arara et al [6] considered a control multivalued problem on the bounded interval [0, b]. Recently, Baghli and Benchohra [8, 9] provided
uniqueness results for some classes of partial and neutral functional differential evolution equations on the semiinfinite interval J = [0, +∞) with local and nonlocal
conditions when the delay is finite. Our main purpose in this paper is to extend
some results from finite delay and those considered on a bounded interval to partial
and neutral perturbed evolution equations.
Sufficient conditions are established to obtain the existence of mild solutions,
which are fixed points of the appropriate operators. We apply a recent nonlinear
alternative given by Avramescu in [7], combined with the semigroup theory [4, 35].
2. Preliminaries
In this section, we introduce notation, definitions and theorems to be used later.
Let C([0, ∞); E) be the space of continuous functions from [0, ∞) to E and B(E)
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PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
3
be the space of all bounded linear operators from E to E, with the norm
kN kB(E) = sup{|N (y)| : |y| = 1}.
A measurable function y : [0, +∞) → E is Bochner integrable if |y| is Lebesgue
integrable. (For details on the Bochner integral properties, see Yosida [38]).
Let L1 ([0, +∞), E) be the Banach space of measurable functions y : [0, +∞) → E
which are Bochner integrable, equipped with the norm
Z +∞
kykL1 =
|y(t)| dt.
0
Consider the space
B+∞ = {y : (−∞, +∞) → E : y|J ∈ C(J, E), y0 ∈ B},
where y|J is the restriction of y to J.
In this paper, we will employ the axiomatic definition of the phase space B
introduced by Hale and Kato in [25] and follow the terminology used in [32]. Thus,
(B, k · kB ) will be a seminormed linear space of functions mapping (−∞, 0] to E,
and satisfying the following axioms:
(A1) If y : (−∞, b) → E with b > 0, is continuous on [0, b] and y0 ∈ B, then for
every t ∈ [0, b) the following conditions hold:
(i) yt ∈ B;
(ii) There exists a positive constant H such that |y(t)| ≤ Hkyt kB ;
(iii) There exist two functions K(·), M (·) : R+ → R+ independent of y(t)
with K continuous and M locally bounded such that
kyt kB ≤ K(t) sup{|y(s)| : 0 ≤ s ≤ t} + M (t)ky0 kB .
Denote Kb = sup{K(t) : t ∈ [0, b]} and Mb = sup{M (t) : t ∈ [0, b]}.
(A2) For the function y(.) in (A1), yt is a B-valued continuous function on [0, b].
(A3) The space B is complete.
Remark 2.1.
• Condition (ii) in (A1) is equivalent to |φ(0)| ≤ HkφkB for every φ ∈ B.
• Since k · kB is a seminorm, two elements φ, ψ ∈ B can verify kφ − ψkB = 0
without necessarily φ(θ) = ψ(θ) for all θ ≤ 0.
• From the equivalence of (ii), we can see that for all φ, ψ ∈ B such that
kφ − ψkB = 0. This implies necessarily that φ(0) = ψ(0).
Next we present some examples of phase spaces. For more details we refer to
the book by Hino et al [32].
Example 2.2. Let BC be the space of bounded continuous functions defined from
(−∞, 0] to E. Let BU C the space of bounded uniformly continuous functions
defined from (−∞, 0] to E. Let
C ∞ := {φ ∈ BC : lim φ(θ) exist in E}.
θ→−∞
0
C := {φ ∈ BC : lim φ(θ) = 0} .
θ→−∞
0
The space C is endowed with the uniform norm kφk = sup{|φ(θ)| : θ ≤ 0}.
Then the spaces BU C, C ∞ and C 0 satisfy conditions (A1)–(A3). BC satisfies
(A1), (A3) but not (A2).
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S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
Example 2.3. Let g be a positive continuous function on (−∞, 0]. We define:
φ(θ)
is bounded on(−∞, 0]},
g(θ)
φ(θ)
Cg0 := {φ ∈ Cg : lim
= 0}
θ→−∞ g(θ)
Cg := {φ ∈ C((−∞, 0], E) :
endowed with the uniform norm kφk = sup{ |φ(θ)|
g(θ) : θ ≤ 0}.
Also we assume that
(G1) For all a > 0, sup0≤t≤a sup{ g(t+θ)
g(θ) : −∞ < θ ≤ −t} < ∞.
Then the spaces Cg and Cg0 satisfy condition (A3). They satisfy conditions (A1)
and (A2) if (G1) holds.
Example 2.4. For each constant γ, we define the space
Cγ := {φ ∈ C((−∞, 0], E) : lim eγθ φ(θ) exist in E}
θ→−∞
γθ
endowed with the norm kφk = sup{e |φ(θ)| : θ ≤ 0}. Then in the space Cγ ,
assumptions (A1)–(A3) are satisfied.
Definition 2.5. A function f : J × B → E is said to be an L1 -Carathéodory
function if it satisfies:
(i) for each t ∈ J the function f (t, .) : B → E is continuous;
(ii) for each y ∈ B the function f (., y) : J → E is measurable;
(iii) for every positive integer k there exists hk ∈ L1 (J; R+ ) such that |f (t, y)| ≤
hk (t) for all kykB ≤ k and almost all t ∈ J.
In what follows, we assume that {A(t), t ≥ 0} is a family of closed densely defined linear unbounded operators on the Banach space E and with domain D(A(t))
independent of t.
Definition 2.6. A family of bounded linear operators {U (t, s)}(t,s)∈∆ : U (t, s) :
E → E for (t, s) ∈ ∆ := {(t, s) ∈ J × J : 0 ≤ s ≤ t < +∞} is called an evolution
system if the following properties are satisfied :
(1) U (t, t) = I where I is the identity operator in E,
(2) U (t, s)U (s, τ ) = U (t, τ ) for 0 ≤ τ ≤ s ≤ t < +∞,
(3) U (t, s) ∈ B(E) the space of bounded linear operators on E, where for every
(t, s) ∈ ∆ and for each y ∈ E, the mapping (t, s) → U (t, s)y is continuous.
(4) U (t, s) is a compact operator for 0 < s < t < +∞.
More details on evolution systems and their properties can be found in the books
by Ahmed [4], Engel and Nagel [22], and Pazy [35].
Let X be a Fréchet space with a family of semi-norms {k · kn }n∈N . Let Y ⊂ X,
we say that F is bounded if for every n ∈ N, there exists M n > 0 such that
kykn ≤ M n
for all y ∈ Y.
With X, we associate a sequence of Banach spaces {(X n , k · kn )} as follows: For
every n ∈ N, we consider the equivalence relation x ∼n y if and only if kx − ykn = 0
for all x, y ∈ X. We denote X n = (X|∼n , k · kn ) the quotient space, the completion
of X n with respect to k · kn . To every Y ⊂ X, we associate a sequence the {Y n }
of subsets Y n ⊂ X n as follows : For every x ∈ X, we denote [x]n the equivalence
class of x of subset X n and we defined Y n = {[x]n : x ∈ Y }. We denote Y n ,
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PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
5
intn (Y n ) and ∂n Y n , respectively, the closure, the interior and the boundary of Y n
with respect to k · k in X n . We assume that the family of semi-norms {k · kn }
verifies:
kxk1 ≤ kxk2 ≤ kxk3 ≤ ... for every x ∈ X.
Definition 2.7 ([7]). A function f : X → X is said to be a contraction if for each
n ∈ N there exists kn ∈ (0, 1) such that:
kf (x) − f (y)kn ≤ kn kx − ykn
for all x, y ∈ X.
We use the following nonlinear alternative, due to Avramescu, has been has a
version on Banach spaces by Burton-Kirk [19, 20].
Theorem 2.8 (Avramescu Nonlinear Alternative [7]). Let X be a Fréchet space
and let A, B : X → X be two operators satisfying:
(1) A is a compact operator,
(2) B is a contraction.
Then either one of the following statements holds:
(S1) The operator A + B has a fixed point;
(S2) The set {x ∈ X, x = λA(x) + λB( λx )} is unbounded for λ ∈ (0, 1).
3. Perturbed Evolution Equations
Before stating and proving the main result, we give the definition of mild solution
of the semilinear perturbed evolution (1.1)–(1.2).
Definition 3.1. We say that the function y(·) : R → E is a mild solution of
(1.1)–(1.2) if y(t) = φ(t) for all t ∈ (−∞, 0] and y satisfies the integral equation
Z t
y(t) = U (t, 0)φ(0) +
U (t, s)[f (s, ys ) + g(s, ys )] ds for each t ∈ [0, +∞). (3.1)
0
We introduce the following hypotheses:
c ≥ 1 such
(H1) U (t, s) is compact for t − s > 0 and there exists a constant M
that
c for every (t, s) ∈ ∆.
kU (t, s)kB(E) ≤ M
(H2) There exists a function p ∈ L1loc (J; R+ ) and a continuous nondecreasing
function ψ : R+ → (0, ∞) and such that:
|f (t, u)| ≤ p(t)ψ(kukB )
(H3) There exists a function η ∈
for a.e. t ∈ J and each u ∈ B.
L1loc (J, R+ )
|g(t, u) − g(t, v)| ≤ η(t)ku − vkB
such that:
for a.e. t ∈ J and all u, v ∈ B.
Theorem 3.2. Suppose that hypotheses (H1)–(H3) are satisfied and
Z n
Z +∞
ds
c
> Kn M
max(p(s), η(s))ds ds for each n ∈ N
s + ψ(s)
αn
0
with
Z
n
cH + Mn )kφkB .
|g(s, 0)|ds + (Kn M
c
αn = Kn M
0
Then (1.1)–(1.2) has a mild solution.
(3.2)
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S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
Proof. Let us fix τ > 1. For every n ∈ N, we define in B+∞ the semi-norms
∗
kykn := sup{e−τ Ln (t) |y(t)| : t ∈ [0, n]}
Rt
cKn η(t). Then C(B+∞ ; E) is a Fréchet
where L∗n (t) = 0 ln (s) ds and ln (t) = M
space with the family of semi-norms k · kn∈N .
We transform (1.1)–(1.2) into a fixed-point problem. Consider the operator
N : B+∞ → B+∞ defined by


if t ∈ (−∞, 0];
φ(t),
Rt
N (y)(t) = U (t, 0)φ(0) + 0 U (t, s)f (s, ys ) ds
(3.3)

 Rt
+ 0 U (t, s)g(s, ys ) ds,
if t ∈ J.
Clearly, the fixed points of the operator N are mild solutions of (1.1)–(1.2).
For φ ∈ B, we define the function x(.) : R → E by
(
φ(t),
if t ∈ (−∞, 0];
x(t) =
U (t, 0)φ(0), if t ∈ J.
Then x0 = φ. For each function z ∈ B+∞ , set
y(t) = z(t) + x(t).
(3.4)
It is obvious that y satisfies (3.1) if and only if z satisfies z0 = 0 and
Z t
Z t
z(t) =
U (t, s)f (s, zs + xs ) ds +
U (t, s)g(s, zs + xs ) ds for t ∈ J.
0
Let
0
B+∞
0
0
0
by
→ B+∞
= {z ∈ B+∞ : z0 = 0}. Define the operators F, G : B+∞
Z t
F (z)(t) =
U (t, s)f (s, zs + xs ) ds for t ∈ J,
(3.5)
0
Z t
G(z)(t) =
U (t, s)g(s, zs + xs ) ds for t ∈ J.
(3.6)
0
Obviously the operator N having a fixed point is equivalent to F + G having a fixed
point. That F + G has a fixed point will be proved in several steps. First we show
that F is continuous and compact.
0
Step 1: F is continuous. Let (zk )k∈N be a sequence in B+∞
such that zk → z in
0
B+∞ . Then
Z t
|F (zk )(t) − F (z)(t)| = U (t, s)[f (s, zks + xs ) − f (s, zs + xs )] ds
0
Z t
≤
kU (t, s)kB(E) |f (s, zks + xs ) − f (s, zs + xs )| ds
0
Z t
c
≤M
|f (s, zks + xs ) − f (s, zs + xs )| ds → 0 as k → +∞.
0
Thus F is continuous.
0
Step 2: F maps bounded sets of B+∞
into bounded sets. For any d > 0, there
0
exists a positive constant ` such that for each z ∈ Bd = {z ∈ B+∞
: kzkn ≤ d} one
has kF (z)kn ≤ `.
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PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
7
Let z ∈ Bd . By the hypotheses (H1) and (H2), we have for each t ∈ J
Z
t
|F (z)(t)| = U (t, s)f (s, zs + xs ) ds
0
Z t
≤
kU (t, s)kB(E) |f (s, zs + xs )| ds
0
Z t
c
≤M
p(s)ψ(kzs + xs )kB ) ds.
0
Using the assumption (A1), we get
kzs + xs kB ≤ kzs kB + kxs kB
≤ K(s)|z(s)| + M (s)kz0 kB + K(s)|x(s)| + M (s)kx0 kB
≤ Kn |z(s)| + Kn kU (s, 0)kB(E) |φ(0)| + Mn kφkB
c|φ(0)| + Mn kφkB
≤ Kn |z(s)| + Kn M
cHkφkB + Mn kφkB
≤ Kn |z(s)| + Kn M
cH + Mn )kφkB .
≤ Kn |z(s)| + (Kn M
Set
cH + Mn )kφkB ,
cn := (Kn M
Dn := Kn d + cn .
Then
kzs + xs kB ≤ Kn |z(s)| + cn ≤ Dn .
Using the nondecreasing character of ψ, we get
Z t
c
|F (z)(t)| ≤ M ψ(Dn )
p(s) ds.
(3.7)
0
Thus
cψ(Dn )kpkL1 := `.
kF (z)k+∞ ≤ M
0
Step 3: F maps bounded sets into equicontinuous sets of B+∞
. We consider Bd
as in Step 2 and we show that F (Bd ) is equicontinuous. Let τ1 , τ2 ∈ J with τ2 > τ1
and z ∈ Bd . Then
Z τ1
|F (z)(τ2 ) − F (z)(τ1 )| ≤ [U (τ2 , s) − U (τ1 , s)]f (s, zs + xs ) ds
0
Z τ2
+
U (τ2 , s)|f (s, zs + xs )| ds
τ
Z τ1 1
≤
kU (τ2 , s) − U (τ1 , s)kB(E) |f (s, zs + xs )| ds
0
Z τ2
+
kU (τ2 , s)kB(E) |f (s, zs + xs )| ds.
τ1
Using kzs + xs kB ≤ Dn in (3.7) and the nondecreasing character of ψ, we get
|F (z)(τ2 ) − F (z)(τ1 )|
Z τ1
Z
cψ(Dn )
≤ ψ(Dn )
kU (τ2 , s) − U (τ1 , s)kB(E) p(s)ds + M
0
τ2
p(s)ds.
τ1
The right-hand of the above inequality tends to zero as τ2 − τ1 → 0, since U (t, s)
is a strongly continuous operator and the compactness of U (t, s) for t > s implies
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S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
the continuity in the uniform operator topology (see [5, 35]). As a consequence of
Steps 1 to 3 together with the Arzelá-Ascoli theorem it suffices to show that the
operator F maps Bd into a precompact set in E.
Let t ∈ J be fixed and let be a real number satisfying 0 < < t. For z ∈ Bd
we define
Z
t−
F (z)(t) = U (t, t − )
U (t − , s)f (s, zs + xs ) ds.
0
Since U (t, s) is a compact operator, the set Z (t) = {F (z)(t) : z ∈ Bd } is precompact in E for every , 0 < < t. Moreover
Z t
kU (t, s)kB(E) |f (s, zs + xs )|ds.
|F (z)(t) − F (z)(t)| ≤
t−
Using kzs + xs kB ≤ Dn in (3.7) and the nondecreasing character of ψ, we get
Z t
c
|F (z)(t) − F (z) (t)| ≤ M ψ(Dn )
p(s)ds.
t−
Therefore the set Z(t) = {F (z)(t) : z ∈ Bd } is totally bounded. Hence the set
{F (z)(t) : z ∈ Bd } is relatively compact E. So we deduce from Steps 1, 2 and 3
that F is a compact operator.
0
, then using (H1) and (H3)
Step 4: G is a contraction mapping. Let z, z ∈ B+∞
for each t ∈ [0, n] and n ∈ N
Z t
|G(z)(t) − G(z)(t)| ≤
kU (t, s)kB(E) |g(s, zs + xs ) − g(s, z s + xs )| ds
0
Z t
cη(s)kzs + xs − z s − xs kB ds
M
≤
0
Z t
cη(s)kzs − z s kB ds.
M
≤
0
Using (A1), we obtain
Z
t
cη(s)(K(s)|z(s) − z(s)| + M (s)kz0 − z 0 kB ) ds
M
|G(z)(t) − G(z)(t)| ≤
0
Z
t
cKn η(s)|z(s) − z(s)| ds
M
≤
0
Z
t
∗
∗
[ln (s)eτ Ln (s) ][e−τ Ln (s) |z(s) − z(s)|] ds
≤
0
Z
≤
0
t
∗
eτ Ln (s) 0
dskz − zkn
τ
∗
1
≤ eτ Ln (t) kz − zkn .
τ
Therefore,
1
kz − zkn .
τ
So, the operator G is a contraction for all n ∈ N.
Step 5: For applying Theorem 2.8, we must check (S2): i.e. it remains to show
that the set
z
0
E = {z ∈ B+∞
: z = λF (z) + λG
for some 0 < λ < 1}
λ
kG(z) − G(z)kn ≤
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PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
9
is bounded. Let z ∈ E. By (H1)–(H3), we have for each t ∈ [0, n]
Z t
|z(t)| ≤
kU (t, s)kB(E) |f (s, zs + xs )|ds
0
Z t
+
kU (t, s)kB(E) |g(s, zs + xs ) − g(s, 0) + g(s, 0)|ds
0
Z t
c
≤M
p(s)ψ (kzs + xs kB ) ds
0
Z t
Z t
c
c
+M
η(s)kzs + xs kB ds + M
|g(s, 0)|ds.
0
0
Using (3.7) we get
kzs + xs kB ≤ Kn |z(s)| + cn .
The nondecreasing character of ψ gives
Z t
c
|z(t)| ≤ M
p(s)ψ(Kn |z(s)| + cn )ds
0
Z t
Z t
c
c
+M
η(s)(Kn |z(s)| + cn )ds + M
|g(s, 0)|ds.
0
0
Then
t
Z
c
Kn |z(t)| + cn ≤ Kn M
p(s)ψ(Kn |z(s)| + cn )ds
0
Z
t
Z
c
η(s)(Kn |z(s)| + cn )ds + Kn M
c
+ Kn M
0
t
|g(s, 0)|ds + cn .
0
Set
Z
t
|g(s, 0)|ds + cn ,
c
αn := Kn M
0
thus
Z
t
c
Kn |z(t)| + cn ≤Kn M
p(s)ψ(Kn |z(s)| + cn )ds
Z t
c
+ Kn M
η(s)(Kn |z(s)| + cn )ds + αn .
0
0
We consider the function µ defined by
µ(t) := sup{Kn |z(s)| + cn : 0 ≤ s ≤ t},
0 ≤ t < +∞.
?
Let t ∈ [0, t] be such that
µ(t) = Kn |z(t? )| + cn ,
by the previous inequality, we have
Z t
Z t
c
c
µ(t) ≤ Kn M
p(s)ψ(µ(s))ds + Kn M
η(s)µ(s)ds + αn
0
0
for t ∈ [0, n]. Let us denote the right-hand side of the above inequality as v(t).
Then, we have
µ(t) ≤ v(t) for all t ∈ [0, n].
From the definition of v, we have v(0) = αn and
cp(t)ψ(µ(t)) + Kn M
cη(t)µ(t)
v 0 (t) = Kn M
a.e. t ∈ [0, n].
10
S. BAGHLI, M. BENCHOHRA
Using the nondecreasing character of ψ, we get
cp(t)ψ(v(t)) + Kn M
cη(t)v(t)
v 0 (t) ≤ Kn M
EJDE-2008/69
a.e. t ∈ [0, n].
This implies that for each t ∈ [0, n] and using (3.2), we get
Z v(t)
Z t
ds
c
≤ Kn M
max(p(s), η(s))ds
s + ψ(s)
αn
0
Z n
c
max(p(s), η(s))ds
≤ Kn M
0
Z +∞
ds
.
<
s
+
ψ(s)
αn
Thus, for every t ∈ [0, n], there exists a constant Nn such that v(t) ≤ Nn and hence
µ(t) ≤ Nn . Since kzkn ≤ µ(t), we have kzkn ≤ Nn . This shows that the set E
is bounded. Then statement (S2) in Theorem 2.8 does not hold. The nonlinear
alternative of Avramescu implies that (S1) holds, we deduce that the operator
F + G has a fixed-point z ? . Then y ? (t) = z ? (t) + x(t), t ∈ (−∞, +∞) is a fixed
point of the operator N , which is the mild solution of (1.1)–(1.2).
4. Perturbed Neutral Evolution Equations
In this section, we give an existence result for the perturbed neutral evolution
problem with infinite delay (1.3)–(1.4). Firstly we define the mild solution.
Definition 4.1. We say that the function y(·) : R → E is a mild solution of
(1.3)–(1.4) if y(t) = φ(t) for all t ∈ (−∞, 0] and y satisfies the integral equation
Z t
y(t) = U (t, 0)[φ(0) − h(0, φ)] + h(t, yt ) +
U (t, s)A(s)h(s, ys )ds
0
(4.1)
Z t
+
U (t, s)[f (s, ys ) + g(s, ys )] ds for each t ∈ [0, +∞).
0
In what follows we need the following assumptions:
(H4) There exists a constant M 0 > 0 such that
kA−1 (t)kB(E) ≤ M 0
(H5) There exists a constant 0 < L <
for all t ∈ J.
1
M 0 Kn
|A(t)h(t, φ)| ≤ L(kφkB + 1)
such that
for all t ∈ J, φ ∈ B.
(H6) There exists a constant L∗ > 0 such that
|A(s)h(s, φ) − A(s)h(s, φ)| ≤ L∗ (|s − s| + kφ − φkB )
for all s, s ∈ J and φ, φ ∈ B.
Theorem 4.2. Suppose that hypotheses (H1)–(H6) are satisfied and
Z +∞
Z n
cKn
M
ds
>
max(L, η(s), p(s))ds for each n ∈ N
s + ψ(s)
1 − M 0 LKn 0
ζn
(4.2)
with
Z t
i
h
Kn
c
c
c
c
ζn :=
M 0 L 1 + M + cn + M kφkB + M Ln + M
|g(s, 0)| ds + cn ,
1 − M 0 LKn
0
c
and cn := (Kn M H + Mn )kφkB . Then (1.3)–(1.4) has a mild solution.
EJDE-2008/69
PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
11
e : B+∞ → B+∞ defined by
Proof. Consider the operator N

φ(t),
if t ∈ (−∞, 0];



U (t, 0)[φ(0) − h(0, φ)] + h(t, y )
t
e (y)(t) =
Rt
N

+
U
(t,
s)A(s)h(s,
y
)ds
s


 R0t
+ 0 U (t, s)[f (s, ys ) + g(s, ys )]ds, if t ∈ J.
(4.3)
e are mild solutions of (1.3)–(1.4).
Note that the fixed points of the operator N
For φ ∈ B, we define the function x : R → E by
(
φ(t),
if t ∈ (−∞, 0];
x(t) =
U (t, 0)φ(0), if t ∈ J.
Then x0 = φ. For each function z ∈ B+∞ , set
y(t) = z(t) + x(t).
(4.4)
It is obvious that y satisfies (4.1) if and only if z satisfies z0 = 0. For t ∈ J, we get
Z
t
z(t) = h(t, zt + xt ) − U (t, 0)h(0, φ) +
U (t, s)A(s)h(s, zs + xs )ds
0
Z t
Z t
+
U (t, s)f (s, zs + xs )ds +
U (t, s)g(s, zs + xs )ds.
0
0
e : B 0 → B 0 by
Define the operators Fe, G
+∞
+∞
Z
Fe(z)(t) =
t
U (t, s)f (s, zs + xs )ds
(4.5)
0
and
Z
e
G(z)(t)
= h(t, zt + xt ) − U (t, 0)h(0, φ) +
Z t
+
U (t, s)g(s, zs + xs )ds.
t
U (t, s)A(s)h(s, zs + xs )ds
0
(4.6)
0
e having a fixed point is equivalent to Fe + G
e having a fixed
Obviously the operator N
e
e
point. The proof that F + G has a fixed point is done in several steps.
Step 1: Fe is continuous and compact. This can be shown as we did for F in
Section 3.
12
S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
0
e is a contraction mapping. Let z, z ∈ B+∞
Step 2: G
, then using (H1), (H3)–(H6)
for each t ∈ [0, n] and n ∈ N,
e
e
|G(z)(t)
− G(z)(t)|
≤ |h(t, zt + xt ) − h(t, z t + xt )|
Z t
+
kU (t, s)kB(E) |A(s)[h(s, zs + xs ) − h(s, z s + xs )]| ds
0
Z t
+
kU (t, s)kB(E) |g(s, zs + xs ) − g(s, z s + xs )| ds
0
≤ kA−1 (s)k|A(t)h(t, zt + xt ) − A(t)h(t, z t + xt )|
Z t
c|A(s)h(s, zs + xs ) − A(s)h(s, z s + xs )| ds
M
+
0
Z t
c|g(s, zs + xs ) − g(s, z s + xs )| ds
+
M
0
Z t
cL∗ kzs + xs − z s − xs kB ds
M
≤ M 0 L∗ kzt + xt − z t − xt kB +
0
Z t
cη(s)kzs + xs − z s − xs kB ds
M
+
0
Z t
c[L∗ + η(s)]kzs − z s kB ds.
M
≤ M 0 L∗ kzt − z t kB +
0
Using (A1), we obtain
e
e
|G(z)(t)
− G(z)(t)|
≤ M 0 L∗ K(t)|z(t) − z(t)| + M (t)kz0 − z 0 kB
Z t
c[L∗ + η(s)] K(s)|z(s) − z(s)| + M (s)kz0 − z 0 kB ds
+
M
0
Z t
cKn [L∗ + η(s)]|z(s) − z(s)| ds.
≤ M 0 L∗ Kn |z(t) − z(t)| +
M
0
cKn [L∗ + η(t)] for the family seminorms {k · kn }n∈N . Then
Let ln (t) = M
Z t
e
e
|G(z)(t)
− G(z)(t)|
≤ M 0 L∗ Kn |z(t) − z(t)| +
ln (s)]|z(s) − z(s)| ds
0
∗
∗
≤ M 0 L∗ Kn eτ Ln (t) e−τ Ln (t) |z(t) − z(t)|
Z t
∗
∗
+
ln (s)eτ Ln (s) e−τ Ln (s) |z(s) − z(s)| ds
0
τ L∗
n (t)
≤ M 0 L∗ Kn e
Z
kz − zkn +
0
τ L∗
n (t)
≤ M 0 L∗ Kn e
t
∗
eτ Ln (s) 0
dskz − zkn
τ
∗
1
kz − zkn + eτ Ln (t) kz − zkn
τ
1 ∗
≤ M 0 L∗ Kn + eτ Ln (t) kz − zkn .
τ
Therefore,
e
e
kG(z)
− G(z)k
n ≤ [M 0 L∗ Kn +
1
]kz − zkn .
τ
EJDE-2008/69
PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
13
e is a contraction for all n ∈ N.
So, for [M 0 L∗ Kn + τ1 ] < 1, the operator G
Step 3: The set
0
e z
Ee = z ∈ B+∞
: z = λFe(z) + λG
λ
for some 0 < λ < 1
e Then, we have
is bounded. Let z ∈ E.
|z(t)| ≤ |h(t, zt + xt )| + kU (t, 0)kB(E) |h(0, φ)|
Z t
kU (t, s)kB(E) |A(s)h(s, zs + xs )| ds
+
0
Z t
+
kU (t, s)kB(E) |f (s, zs + xs )| ds
0
Z t
+
kU (t, s)kB(E) |g(s, zs + xs ) − g(s, 0) + g(s, 0)| ds.
0
By (A1) and (H1)–(H6), we have
ckA−1 (s)k|A(t)h(0, φ)|
|z(t)| ≤ kA−1 (s)k|A(t)h(t, zt + xt )| + M
Z t
Z t
c
c
+M
|A(s)h(s, zs + xs )| ds + M
f (s, zs + xs ) ds
0
0
Z t
Z t
c
c
+M
|g(s, zs + xs ) − g(s, 0)| ds + M
|g(s, 0)| ds
0
0
cM 0 L(kφkB + 1)
≤ M 0 L(kzt + xt kB + 1) + M
Z t
Z t
cL
c
+M
(kzs + xs kB + 1) ds + M
p(s)ψ(kzs + xs )kB ) ds
0
0
Z t
Z t
c
c
+M
η(s)kzs + xs kB ds + M
|g(s, 0)| ds
0
0
cM 0 L + M
cLn + M
cM 0 LkφkB
≤ M 0 Lkzt + xt kB + M 0 L + M
Z t
Z t
c
cL
+M
|g(s, 0)| ds + M
kzs + xs kB ds
0
0
Z t
Z t
c
c
+M
p(s)ψ(kzs + xs )kB ) ds + M
η(s)kzs + xs kB ds.
0
0
Using kzs + xs kB ≤ Kn |z(s)| + cn in (3.7) and the nondecreasing character of ψ,
we get
cM 0 L + M
cLn + M
cM 0 LkφkB
|z(t)| ≤ M 0 L(Kn |z(t)| + cn ) + M 0 L + M
Z t
Z t
c
cL
+M
|g(s, 0)| ds + M
(Kn |z(s)| + cn ) ds
0
0
Z t
Z t
c
c
+M
p(s)ψ(Kn |z(s)| + cn ) ds + M
η(s)(Kn |z(s)| + cn ) ds.
0
0
14
S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
Then
c[1 + kφkB ]) + M
cLn
(1 − M 0 LKn )|z(t)| ≤ M 0 L(cn + 1 + M
Z t
Z t
c
cL
+M
|g(s, 0)| ds + M
(Kn |z(s)| + cn ) ds
0
0
Z t
c
+M
p(s)ψ(Kn |z(s)| + cn ) ds
0
Z t
c
+M
η(s)(Kn |z(s)| + cn ) ds.
0
Set
h
Kn
c + cn + M
ckφkB + M
cLn + M
c
ζn :=
M 0L 1 + M
1 − M 0 LKn
Z
t
i
|g(s, 0)| ds + cn .
0
Thus
cLKn Z t
M
(Kn |z(s)| + cn ) ds
1 − M 0 LKn 0
Z t
cKn
M
+
p(s)ψ(Kn |z(s)| + cn ) ds
1 − M 0 LKn 0
Z t
cKn
M
η(s)(Kn |z(s)| + cn ) ds.
+
1 − M 0 LKn 0
Kn |z(t)| + cn ≤ ζn +
Consider the function µ defined by
µ(t) := sup{Kn |z(s)| + cn : 0 ≤ s ≤ t},
0 ≤ t < +∞.
Let t? ∈ [0, t] be such that µ(t) = Kn |z(t? )| + cn , by the previous inequality, we
have
Z t
cLKn Z t
cKn
M
M
µ(s) ds +
η(s)µ(s) ds
µ(t) ≤ ζn +
1 − M 0 LKn 0
1 − M 0 LKn 0
Z t
cKn
M
+
p(s)ψ(µ(s)) ds for t ∈ [0, n].
1 − M 0 LKn 0
Let us denote the right-hand side of the above inequality as v(t). Then, we have
µ(t) ≤ v(t)
for all t ∈ [0, n].
From the definition of v, we have v(0) = ζn and
v 0 (t) =
cKn
M
[Lµ(t) + η(t)µ(t) + p(t)ψ(µ(t))]
1 − M 0 LKn
a.e. t ∈ [0, n].
Using the nondecreasing character of ψ, we get
v 0 (t) ≤
cKn
M
[Lv(t) + η(t)v(t) + p(t)ψ(v(t))] a.e. t ∈ [0, n].
1 − M 0 LKn
EJDE-2008/69
PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
15
This implies that for each t ∈ [0, n] and using the condition (4.2), we get
Z
v(t)
ζn
Z t
cKn
ds
M
max(L, η(s), p(s))ds
≤
s + ψ(s)
1 − M 0 LKn 0
Z n
cKn
M
≤
max(L, η(s), p(s))ds
1 − M 0 LKn 0
Z +∞
ds
<
.
s
+
ψ(s)
ζn
en such that v(t) ≤ N
en and
Thus, for every t ∈ [0, n], there exists a constant N
en . Since kzkn ≤ µ(t), we have kzkn ≤ N
en . This shows that the
hence µ(t) ≤ N
e
set E is bounded. Then the statement (S2) in Theorem 2.8 does not hold. The
nonlinear alternative of Avramescu implies that (S1) holds, we deduce that the
e has a fixed-point z ? . Then y ? (t) = z ? (t) + x(t), t ∈ (−∞, +∞) is
operator Fe + G
e , which is the mild solution of (1.3)–(1.4).
a fixed point of the operator N
5. Applications
To illustrate the previous results, we give in this section two applications.
Example 5.1. Consider the model
Z 0
∂2v
∂v
(t, ξ) = a(t, ξ) 2 (t, ξ) +
P (θ)r(t, v(t + θ, ξ))dθ
∂t
∂ξ
−∞
Z 0
+
Q(θ)s(t, v(t + θ, ξ))dθ, t ∈ [0, +∞), ξ ∈ [0, π]
(5.1)
−∞
v(t, 0) = v(t, π) = 0
v(θ, ξ) = v0 (θ, ξ)
t ∈ [0, +∞)
− ∞ < θ ≤ 0, ξ ∈ [0, π],
where a(t, ξ) is a continuous function and is uniformly Hölder continuous in t;
P, Q : (−∞, 0] → R; r, s : [0, +∞) × R → R and v0 : (−∞, 0] × [0, π] → R are
continuous functions.
Consider E = L2 ([0, π], R) and define A(t) by A(t)w = a(t, ξ)w00 with domain
D(A) = {w ∈ E : w, w0 are absolutely continuous, w00 ∈ E, w(0) = w(π) = 0}
Then A(t) generates an evolution system U (t, s) satisfying assumption (H1) (see
[24]).
For the phase space B, we choose the well known space BU C(R− , E), the space
of uniformly bounded continuous functions endowed with the norm
kϕk = sup |ϕ(θ)| for ϕ ∈ B.
θ≤0
16
S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
If we put for ϕ ∈ BU C(R− , E) and ξ ∈ [0, π],
y(t)(ξ) = v(t, ξ),
t ∈ [0, +∞), ξ ∈ [0, π],
φ(θ)(ξ) = v0 (θ, ξ), −∞ < θ ≤ 0, ξ ∈ [0, π],
Z 0
f (t, ϕ)(ξ) =
P (θ)r(t, ϕ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π],
Z
−∞
0
Q(θ)s(t, ϕ(θ)(ξ))dθ,
g(t, ϕ)(ξ) =
−∞ < θ ≤ 0, ξ ∈ [0, π].
−∞
Then, (5.1) takes the abstract partial perturbed evolution form (1.1)–(1.2). To
show the existence of mild solutions to (5.1), we assume the following hypotheses:
• The function s is Lipschitz continuous with respect to its second argument.
Let lip(s) denote the Lipschitz constant of s.
• There exist p ∈ L1 ([0, +∞), R+ ) and a nondecreasing continuous function
ψ : [0, +∞) → (0, ∞) such that
|r(t, u)| ≤ p(t)ψ(|u|),
for t ∈ [0, +∞), u ∈ R.
• P and Q are integrable on (−∞, 0].
By the dominated convergence theorem, one can show that f is a continuous
function from B to E. Moreover the mapping g is Lipschitz continuous in its
second argument, in fact, we have
Z 0
|g(t, ϕ1 ) − g(t, ϕ2 )| ≤ lip(s)
|Q(θ)| dθ|ϕ1 − ϕ2 |, for ϕ1 , ϕ2 ∈ B.
−∞
On the other hand, for ϕ ∈ B and ξ ∈ [0, π] we have
Z 0
|f (t, ϕ)(ξ)| ≤
|p(t)P (θ)|ψ(|(ϕ(θ))(ξ)|)dθ.
−∞
Since the function ψ is nondecreasing, it follows that
Z 0
|f (t, ϕ)| ≤ p(t)
|P (θ)|dθψ(|ϕ|),
for ϕ ∈ B.
−∞
Proposition 5.2. Under the above assumptions, if we assume that condition (3.2)
in Theorem 3.2 is true, ϕ ∈ B, then the problem (5.1) has a mild solution which is
defined in (−∞, +∞).
Example 5.3. Consider the model
Z 0
∂
v(t, ξ) −
T (θ)u(t, v(t + θ, ξ))dθ
∂t
−∞
Z 0
2
∂ v
= a(t, ξ) 2 (t, ξ) +
P (θ)r(t, v(t + θ, ξ))dθ
∂ξ
−∞
Z 0
+
Q(θ)s(t, v(t + θ, ξ))dθ t ∈ [0, +∞), ξ ∈ [0, π]
−∞
v(t, 0) = v(t, π) = 0,
v(θ, ξ) = v0 (θ, ξ),
t ∈ [0, +∞)
−∞ < θ ≤ 0, ξ ∈ [0, π],
(5.2)
EJDE-2008/69
PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
17
where a(t, ξ) is a continuous function and is uniformly Hölder continuous in t;
T, P, Q : (−∞, 0] → R; u, r, s : [0, +∞) × R → R and v0 : (−∞, 0] × [0, π] → R are
continuous functions.
Consider E = L2 ([0, π], R) and define A(t) by A(t)w = a(t, ξ)w00 with domain
D(A) = {w ∈ E : w, w0 are absolutely continuous , w00 ∈ E, w(0) = w(π) = 0}
Then A(t) generates an evolution system U (t, s) satisfying assumptions (H1) and
(H4) (see [24]).
For the phase space B, we choose the well known space BU C(R− , E) : the space
of uniformly bounded continuous functions endowed with the norm
kϕk = sup |ϕ(θ)| for ϕ ∈ B.
θ≤0
−
If we put for ϕ ∈ BU C(R , E) and ξ ∈ [0, π],
t ∈ [0, +∞), ξ ∈ [0, π],
y(t)(ξ) = v(t, ξ),
φ(θ)(ξ) = v0 (θ, ξ), −∞ < θ ≤ 0, ξ ∈ [0, π],
Z 0
h(t, ϕ)(ξ) =
T (θ)u(t, ϕ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π],
−∞
0
Z
f (t, ϕ)(ξ) =
P (θ)r(t, ϕ(θ)(ξ))dθ,
−∞ < θ ≤ 0, ξ ∈ [0, π]
Q(θ)s(t, ϕ(θ)(ξ))dθ,
−∞ < θ ≤ 0, ξ ∈ [0, π].
−∞
Z 0
g(t, ϕ)(ξ) =
−∞
Then, (5.2) takes the abstract neutral perturbed evolution form (1.3)–(1.4). To
show the existence of the mild solution to (5.2), we assume the following hypotheses:
• the functions u and s are Lipschitz with respect to its second argument,
and constants lip(u) and lip(s) respectively.
• There exist p ∈ L1 ([0, +∞), R+ ) and a nondecreasing continuous function
ψ : [0, +∞) → (0, ∞) such that
|r(t, u)| ≤ p(t)ψ(|u|),
for t ∈ [0, +∞), u ∈ R.
• T , P and Q are integrable on (−∞, 0].
By the dominated convergence theorem, one can show that f is a continuous
function from B to E. Moreover the mapping h and g are Lipschitz continuous in
its second argument, in fact, we have
Z 0
|g(t, ϕ1 ) − g(t, ϕ2 )| ≤ lip(s)
|Q(θ)|dθ|ϕ1 − ϕ2 |, for ϕ1 , ϕ2 ∈ B,
−∞
Z
0
|T (θ)|dθ|ϕ1 − ϕ2 |,
|h(t, ϕ1 ) − h(t, ϕ2 )| ≤ M 0 L∗ lip(u)
for ϕ1 , ϕ2 ∈ B.
−∞
On the other hand, for ϕ ∈ B and ξ ∈ [0, π] we have
Z 0
|f (t, ϕ)(ξ)| ≤
|p(t)P (θ)|ψ(|(ϕ(θ))(ξ)|)dθ.
−∞
Since the function ψ is nondecreasing, it follows that
Z 0
|f (t, ϕ)| ≤ p(t)
|P (θ)|dθψ(|ϕ|), for ϕ ∈ B.
−∞
18
S. BAGHLI, M. BENCHOHRA
EJDE-2008/69
Proposition 5.4. Under the above assumptions, if we assume that condition (4.2)
in Theorem 4.2 is true, ϕ ∈ B, then (5.2) has a mild solution which is defined in
(−∞, +∞).
References
[1] M. Adimy, H. Bouzahir and K. Ezzinbi; Existence for a class of partial functional differential
equations with infinite delay, Nonlinear Anal., 46 (2001), 91-112.
[2] M. Adimy, H. Bouzahir and K. Ezzinbi; Local existence and stability for some partial functional differential equations with infinite delay, Nonlinear Anal., 48 (2002), 323-348.
[3] M. Adimy, H. Bouzahir and K. Ezzinbi; Existence and stability for some partial neutral
functional differential equations with infinite delay, J. Math. Anal. Appl., 294 (2004), 438461.
[4] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research
Notes in Mathematics Series, 246. Longman Scientific & Technical, Harlow John Wiley &
Sons, Inc., New York, 1991.
[5] N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing
Co. Pte. Ltd., Hackensack, NJ, 2006.
[6] A. Arara, M. Benchohra, L. Górniewicz and A. Ouahab; Controllability results for semilinear
functional differential inclusions with unbounded delay, Math. Bulletin, 3 (2006), 157-183.
[7] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electronic J. Qual.
The. Diff. Equa., 5 (2003), 1-15.
[8] S. Baghli and M. Benchohra; Uniqueness results for partial functional differential equations
in Fréchet spaces, Fixed Point Theory (to appear).
[9] S. Baghli and M. Benchohra; Existence results for semilinear neutral functional differential
equations involving evolution operators in Fréchet spaces, Georgian Math. J. (to appear).
[10] K. Balachandran and E. R. Anandhi; Controllability of neutral functional integrodifferential
infinite delay systems in Banach spaces, Taiwanese J. Math., 8 (2004), 689-702.
[11] K. Balachandran and A. Leelamani; Null controllability of neutral integrodifferential systems
with infinite delay, Math. Problems Engineering, 2006 (2006), 1-18.
[12] M. Belmekki, M. Benchohra, K. Ezzinbi and S. K. Ntouyas; Existence results for some partial
functional differential equations with infinite delay, Nonlinear Stud. (to appear).
[13] M. Belmekki, M. Benchohra and S. K. Ntouyas; Existence results for semilinear perturbed
functional differential equations with nondensely defined operators, Fixed Point Th. Appl.,
2006, (2006), 1-13.
[14] M. Belmekki, M. Benchohra and S. K. Ntouyas; Existence results for densely defined semilinear functional differential inclusions, J. Contemporary Math., 1 (1) (2008), 25-44.
[15] M. Belmekki, M. Benchohra and S. K. Ntouyas; Existence results for nondensely defined semilinear functional differential inclusions, J. Nonlinear Func. Anal. Differ. Equ 1 (2) (2007),
261-279.
[16] M. Benchohra and L. Gorniewicz; Existence results of nondensely defined impulsive semilinear
functional differential inclusions with infinite delay, J. Fixed Point Th. Appl., 2 (1) (2007),
11-51.
[17] M. Benchohra, L. Górniewicz and S. K. Ntouyas; Controllability of Some Nonlinear Systems
in Banach spaces : The fixed point theory approach, Pawel Wlodkowicz University College,
Plock, 2003.
[18] M. Benchohra and S. K. Ntouyas; Existence of mild solutions on semiinfinite interval for first
order differential equation with nonlocal condition, Comment. Math. Univ. Carolinae, 41 (3)
(2000), 485-491.
[19] T. A. Burton, A fixed-point theorem of Krasnoselskii; Appl. Math. Letters, 11 (1) (1998),
85-88.
[20] T. A. Burton and C. Kirk; A fixed point theorem of Krasnoselskii type, Math. Nachrichten,
189 (1998), 23-31.
[21] C. Corduneanu and V. Lakshmikantham; Equations with unbounded delay, Nonlinear Anal.,
4 (1980), 831-877.
[22] K.J. Engel and R. Nagel; One-Parameter Semigroups for Linear Evolution Equations,
Springer-Verlag, New York, 2000.
EJDE-2008/69
PERTURBED EVOLUTION EQUATIONS WITH INFINITE DELAY
19
[23] K. Ezzinbi; Existence and stability for some partial functional differential equations with
infinite delay, Electronic J. Diff Equa., 2003 (116) (2003), 1-13.
[24] A. Freidman; Partial Differential Equations, Holt, Rinehat and Winston, New York, 1969.
[25] J. Hale and J. Kato; Phase space for retarded equations with infinite delay, Funkcial. Ekvac.,
21 (1978), 11-41.
[26] J. K. Hale and S. M. Verduyn Lunel; Introduction to Functional Differential Equations,
Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.
[27] S. Heikkila and V. Lakshmikantham; Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994.
[28] H. R. Henriquez; Existence of periodic solutions of neutral functional differential equations
with unbounded delay, Proyecciones, 19 (3) (2000), 305-329.
[29] E. Hernandez; Regularity of solutions of partial neutral functional differential equations with
unbounded delay, Proyecciones, 21 (1) (2002), 65-95.
[30] E. Hernandez; A Massera type criterion for a partial neutral functional differential equation,
Electronic J. Differ. Equa., 2002 (40) (2002), 1-17.
[31] Y. Hino and S. Murakami; Total stability in abstract functional differential equations with infinite delay, Electronic J. of Qualitative Theory of Diff. Equa., Lecture Notes in Mathematics,
1473, Springer-Verlag, Berlin, 1991.
[32] Y. Hino, S. Murakami, and T. Naito; Functional Differential Equations with Unbounded
Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.
[33] F. Kappel and W. Schappacher; Some considerations to the fundamental theory of infinite
delay equations, J. Differential Equations, 37 (1980), 141-183.
[34] V. Kolmanovskii, and A. Myshkis; Introduction to the Theory and Applications of FunctionalDifferential Equations. Mathematics and its Applications, 463. Kluwer Academic Publishers,
Dordrecht, 1999.
[35] A. Pazy; Semigroups of linear operators and applications to partial differential equations,
Springer-Verlag, New York, 1983.
[36] K. Schumacher; Existence and continuous dependence for differential equations with unbounded delay, Arch. Rational Mech. Anal., 64 (1978), 315-335.
[37] J. Wu; Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996.
[38] K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin, 1980.
Selma Baghli
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, BP 89, 22000 Sidi BelAbbès, Algérie
E-mail address: selma baghli@yahoo.fr
Mouffak Benchohra
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, BP 89, 22000 Sidi BelAbbès, Algérie
E-mail address: benchohra@yahoo.com