Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 49125, 13 pages
doi:10.1155/2007/49125
Research Article
Semigroup Approach to Semilinear Partial Functional
Differential Equations with Infinite Delay
Hassane Bouzahir
Received 6 November 2006; Revised 16 January 2007; Accepted 19 January 2007
Recommended by Simeon Reich
We describe a semigroup of abstract semilinear functional differential equations with
infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part
is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida
theorem. We clarify the properties of the phase space ensuring equivalence between the
equation under investigation and the nonlinear semigroup.
Copyright © 2007 Hassane Bouzahir. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Most of the existing results about functional differential equations with finite delay have
been recently under verification in the case of infinite delay. Our objective in this paper is
to study the solution semigroup generated by the following partial functional differential
equation with infinite delay:
d
x(t) = AT x(t) + F xt ,
dt
x0 = φ ∈ Ꮾ,
t ≥ 0,
(1.1)
where AT is a nondensely defined linear operator on a Banach space (E, | · |). The phase
space Ꮾ can be the space Cγ , γ being a positive real constant, of all continuous functions
φ : (−∞,0] → E such that limθ→−∞ eγθ φ(θ) exists in E, endowed with the norm φγ :=
supθ≤0 eγθ |φ(θ)|, φ ∈ Cγ . For every t ≥ 0, the function xt ∈ Ꮾ is defined by
xt (θ) = x(t + θ),
for θ ∈ (−∞,0].
(1.2)
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Journal of Inequalities and Applications
We assume the following.
(H1) F : Ꮾ → E is globally Lipschitz continuous; that is, there exists a positive constant
L such that |F(ψ1 ) − F(ψ2 )| ≤ Lψ1 − ψ2 Ꮾ for all ψ1 ,ψ2 ∈ Ꮾ.
A typical example that can be transformed into (1.1) is the following:
∂2
∂
w(t,ξ) = a 2 w(t,ξ) + bw(t,ξ) + c
∂t
∂ξ
+ f w(t − τ,ξ) ,
−∞
G(θ)w(t + θ,ξ)dθ
t ≥ 0, 0 ≤ ξ ≤ π,
w(t,0) = w(t,π) = 0,
w(θ,ξ) = w0 (θ,ξ),
0
(1.3)
t ≥ 0,
−∞ < θ ≤ 0, 0 ≤ ξ ≤ π,
where a, b, c, and τ are positive constants, f : R → R is a continuous function, G is a
positive integrable function on (−∞,0], and w0 : (−∞,0] × [0,π] → R is an appropriate
continuous function.
Effectively, in [1], an abstract treatment of (1.3) as (1.1) leads to a characterization of
exponential asymptotic stability near an equilibrium of (1.3) provided that the associated
linearized semigroup is exponentially stable.
For many quantitative studies of any problem of type (1.1) in a concrete space of functions mapping (−∞,0] into E, one should choose a space that verifies at least the fundamental axioms first introduced in [2]. That is, (Ꮾ, · Ꮾ ) is a (semi)normed abstract
linear space of functions mapping (−∞,0] into E, which satisfies the following.
(A) There is a positive constant H and functions K(·),M(·) : R+ → R+ , with K continuous and M locally bounded, such that for any σ ∈ R, a > 0, if x : (−∞,σ + a] → E,
xσ ∈ Ꮾ, and x(·) is continuous on [σ,σ + a], then for every t in [σ,σ + a] the following
conditions hold:
(i) xt ∈ Ꮾ;
(ii) |x(t)| ≤ H xt Ꮾ , which is equivalent to
(ii) for each ϕ ∈ Ꮾ, |ϕ(0)| ≤ H ϕᏮ ;
(iii) xt Ꮾ ≤ K(t − σ)supσ ≤s≤t |x(s)| + M(t − σ)xσ Ꮾ .
(A1) For the function x(·) in (A), t → xt is a Ꮾ-valued continuous function for t in
[σ,σ + a].
:= Ꮾ/ · Ꮾ = {ϕ
: ϕ ∈ Ꮾ} is
(B) The space Ꮾ or the space of equivalence classes Ꮾ
complete.
However, to obtain interesting qualitative results, a concrete choice should be made on
a space that verifies additional properties which are essential to investigate the equation.
A class of employed spaces is called uniform fading memory spaces. They verify that the
function K(·) is constant, limt→+∞ M(t) = 0, and the following extra property.
(C) If {φn }n≥0 is a Cauchy sequence in Ꮾ with respect to the (semi)norm and if φn
converges compactly to φ on (−∞,0], then φ is in Ꮾ and φn − φᏮ → 0 as n → +∞.
There are many examples of concrete spaces that verify the above properties. In [3],
it was proved, for instance, that if γ > 0, the above-defined space Cγ is a uniform fading
Hassane Bouzahir 3
memory space. Another example is given by
Cg0
:= φ ∈ C (−∞,0];E : lim
φ(θ)
θ →−∞
g(θ)
=0 ,
(1.4)
equipped with the norm
φ(θ)
φg := sup
−∞<θ ≤0
g(θ)
,
(1.5)
where g : (−∞,0] → [1,+∞) is a continuous function such that (g1): g is nonincreasing
and g(0) = 1, and (g2): the function G : [0,+∞) → [0,+∞) defined by G(t) :=
sup−∞<θ≤−t (g(t + θ)/g(θ)) tends to 0 as t tends to ∞.
In general, set for any positive continuous function g on (−∞,0],
φ(θ)
Cg := φ ∈ C (−∞,0];E :
LCg := φ ∈ Cg : lim
θ →−∞
UCg := φ ∈ Cg :
φ(θ)
g(θ)
g(θ)
φ(θ)
g(θ)
is bounded ,
exists in E ,
(1.6)
is uniformly continuous on (−∞,0] ,
such that (g3): G is locally bounded for t ≥ 0, (g4): g(θ) tends to ∞ as θ tends to −∞, and
(g2) are satisfied. Then LCg is a uniform fading memory space; the additional condition
(g5): logg(θ) is uniformly continuous on (−∞,0], ensuring that UCg is a uniform fading
memory space. Precisely, K(t) = sup−t≤θ≤0 (1/g(θ)) and M(t) = G(t). Note that for the
space Cγ , as defined above, g(θ) = e−γθ .
On the contrary, despite its consideration in some recent separate publications concerned with abstract stability investigations, unfortunately, the space C0 := {φ ∈ C((−∞,
0];E) : limθ→−∞ φ(θ) = 0} is not a uniform fading memory space. In [4] for instance,
some restrictive results about asymptotic behavior of solutions were obtained in the linear positive case on C0 . The followed method uses evolution semigroups, extrapolation
spaces, and critical spectrum on Banach lattices spaces. For the basic discussion about
the general phase space Ꮾ, we refer the reader to [3, especially Chapter 1] and [5, pages
401–406].
Although many authors have avoided repetitions by working on the abstract space Ꮾ,
the delicacy of some investigations restricts their work on a class of concrete spaces that
verify many properties such as Cγ with γ > 0. In [1, 6–8], we have considered (1.1) with
AT being nondensely defined and satisfying the Hille-Yosida condition. Precisely, since
D(AT ) is not densely defined, we have addressed the problems of existence, uniqueness,
regularity, existence of global attractor, existence of periodic solutions, and local stability
by means of the integrated semigroups theory. In this article, we use the Crandall Liggett
approach. We show the relation between a nonlinear semigroup and integral solutions to
(1.1).
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Journal of Inequalities and Applications
Notice that the most general results about functional differential equations with infinite delay are obtained notably in [9–15] and in [16] also in the situation where AT
depends on t. Our results extend earlier ones which require the delay to be finite and AT
to have a dense domain in E.
In this paper, we proceed as follows. In Section 2, we recall some basic results on existence, uniqueness, and properties of integral solutions to (1.1). Then, in Section 3, we
establish properties of the solution operator in nonlinear case. Next, we rely upon the
well-known Crandall and Liggett theorem in order to compute the nonlinear solution
semigroup by an exponential formula. Finally, we give the link between the semigroup
given by the Crandall and Liggett theorem and the integral solution to (1.1).
2. Basic results
Throughout, we assume that AT satisfies the Hille-Yosida condition:
(H2) there exist two constants β ≥ 1 and ω0 ∈ R with (ω0 ,+∞) ⊂ ρ(AT ) and
sup{(λ − ω0 )n R(λ,AT )n : λ > ω0 , n ∈ N} ≤ β,
where ρ(AT ) is the resolvent set of AT and R(λ,AT ) = (λI − AT )−1 .
Definition 2.1. A function x : (−∞,a] → E, a > 0, is an integral solution of (1.1) in (−∞,a]
if the following conditions hold:
(i) x is continuous on [0, a];
t
(ii) 0 x(s)ds ∈ D(AT ), for t ∈ [0,a];
(iii)
⎧
t
t
⎪
⎨φ(0) + A
x(s)ds + F s,xs ds,
T
x(t) = ⎪
0
0
⎩
φ(t),
0 ≤ t ≤ a,
(2.1)
−∞ < t ≤ 0.
It follows from (ii) of the above definition that for an integral solution x, one has
x(t) ∈ D(AT ) for all t ≥ 0. In particular, φ(0) ∈ D(AT ).
Define the part A0 of AT in D(AT ) by
D A0 = x ∈ D AT : AT x ∈ D AT ,
A0 x = AT x,
for x ∈ D A0 .
(2.2)
Recall (cf. [17]) that A0 generates a C0 -semigroup (T0 (t))t≥0 on D(AT ).
It is known (see [1, 6]) that under (H1) and (H2), for φ ∈ Ꮾ such that φ(0) ∈ D(AT ),
(1.1) admits a unique integral solution x(·,φ) given by the following formula:
⎧
t
⎪
⎨T (t)φ(0) + lim
T0 (t − s)Bλ F xs (·,φ) ds,
0
λ→+∞ 0
x(t,φ) = ⎪
⎩
φ(t),
where Bλ = λR(λ,AT ).
for t ≥ 0,
for t ∈ (−∞,0],
(2.3)
Hassane Bouzahir 5
Set
ᐄ := ϕ ∈ Ꮾ : ϕ(0) ∈ D AT
.
(2.4)
Define ᐁ(t) on ᐄ for t ≥ 0 by
ᐁ(t)ϕ = xt (·,ϕ),
(2.5)
where x(·,φ) is the integral solution of (1.1). The point of departure for our results is [1]
where we have proved that (ᐁ(t))t≥0 is a strongly continuous semigroup satisfying the
following properties:
(i) (ᐁ(t))t≥0 satisfies, for t ≥ 0 and θ ∈ (−∞,0], the following translation property
⎧
⎨ ᐁ(t + θ)ϕ (0),
if t + θ ≥ 0,
ᐁ(t)ϕ (θ) = ⎩
ϕ(t + θ),
(2.6)
if t + θ ≤ 0,
(ii) there exist two positive locally bounded functions m(·),n(·) : R+ → R+ such that,
for all ϕ1 ,ϕ2 ∈ X, and t ≥ 0,
ᐁ(t)ϕ1 − ᐁ(t)ϕ2 ≤ m(t)en(t) ϕ1 − ϕ2 .
Ꮾ
Ꮾ
(2.7)
Moreover, if F is a bounded linear operator and Ꮾ is a subspace of C((−∞,0];E) satisfying
axioms (A1), (A2), (B) and the following axiom which was introduced in [13]:
(D) for a sequence (ϕn )n≥0 in Ꮾ, if ϕn Ꮾ → 0, then |ϕn (s)| → 0 for each s ∈ (−∞,0],
then Aᐁ : D(Aᐁ ) ⊆ ᐄ → ᐄ such that Aᐁ ϕ = ϕ for any ϕ ∈ D(Aᐁ ), where
D Aᐁ = ϕ ∈ ᐄ : ϕ is continuously differentiable, ϕ(0) ∈ D AT ,
ϕ ∈ ᐄ, ϕ (0) = AT ϕ(0) + F(ϕ) ,
(2.8)
is the infinitesimal generator of (ᐁ(t))t≥0 .
3. Main results
Our first main result can be considered as an extension of the above result to the case
where F is nonlinear. The concrete choice of Ꮾ (LCg0 , LCg , or UCg ) seems to be best
adapted to obtain our results. Here, we suppose sufficient conditions on g. The proof
combines the ideas of [1, 18] or [19].
Proposition 3.1. Let Ꮾ = LCg0 , Ꮾ = LCg (with (g1)), or Ꮾ = UCg (with (g5)). Then Conditions (H1) and (H2) imply that Aᐁ is the infinitesimal generator of (ᐁ(t))t≥0 .
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Journal of Inequalities and Applications
Proof. Let ϕ ∈ ᐄ be continuously differentiable such that
ϕ ∈ ᐄ, ϕ(0) ∈ D AT ,
ϕ (0) = AT ϕ(0) + F(ϕ).
(3.1)
Let x(·,ϕ) : (−∞,+∞) → E be the unique integral solution of (1.1). We have to show that
limt→0+ (1/t)(ᐁ(t)ϕ − ϕ) exists in ᐄ and is equal to ϕ . By definition of xt (·,ϕ) and ᐁ,
⎧
⎨ ᐁ(t)ϕ (0),
x(t,ϕ) = ⎩
ϕ(t),
if t ≥ 0,
(3.2)
if t ∈ (−∞,0],
then
⎧
1
⎪
⎪
⎪
⎨ x(t + θ,ϕ) − ϕ(θ) (0),
1
t
ᐁ(t)ϕ − ϕ (θ) = ⎪
1
t
⎪
⎪ ϕ(t + θ) − ϕ(θ),
⎩
t
t + θ > 0,
(3.3)
t + θ ∈ (−∞,0].
If t + θ ≤ 0, (1/t)(ᐁ(t)ϕ − ϕ)(θ) tends to D+ ϕ(θ) as t → 0+ , where D+ ϕ(θ) is the right
derivative of ϕ in θ.
If t + θ > 0, we have
1
ᐁ(t)ϕ − ϕ (θ) − ϕ (θ)
t
=
1
T0 (t + θ)ϕ(0) + lim
t
λ→∞
t+θ
0
(3.4)
T0 (t + θ − s)Bλ F xs ds − ϕ(θ) − ϕ (θ).
Let S(t), t ≥ 0, be the integrated semigroup associated with T0 (t), t ≥ 0. We obtain
from the last equality
1
ᐁ(t)ϕ − ϕ (θ) − ϕ (θ)
t
=
1
ϕ(0) + S(t + θ)AT ϕ(0) + lim
t
λ→∞
t+θ
+ lim
λ→∞ 0
t+θ
0
T0 (t + θ − s)Bλ F xs − F(ϕ) ds
(3.5)
T0 (t + θ − s)Bλ F(ϕ)ds − ϕ(θ) − ϕ (θ).
Since T0 (t)ϕ(0) = ϕ(0) + AT S(t)ϕ(0) and
t+θ
lim
λ→∞ 0
T0 (t + θ − s)Bλ F(ϕ)ds = lim S(t + θ)Bλ F(ϕ)
λ→∞
= lim Bλ S(t + θ)F(ϕ)
λ→∞
= S(t + θ)F(ϕ),
(3.6)
Hassane Bouzahir 7
we deduce that
1
ᐁ(t)ϕ − ϕ (θ) − ϕ (θ)
t
t+θ
1
=
ϕ(0) + S(t + θ)ϕ (0) + lim
T0 (t + θ − s)Bλ F xs − F(ϕ) ds − ϕ(θ)
t
λ→∞ 0
− ϕ (θ)
t+θ
t+θ
1
S(t + θ)ϕ (0) −
ϕ (0)ds + lim
T0 (t + θ − s)Bλ F xs − F(ϕ) ds
t
λ→∞ 0
0
1
1
t+θ + ϕ(0) +
ϕ (0) − ϕ(θ) − ϕ (θ)
t
t
t
t+θ
t+θ
1
=
T0 (s)ϕ (0) − ϕ (0) ds + lim
T0 (t + θ − s)Bλ F xs − F(ϕ) ds
t 0
λ→∞ 0
0
1 0 1
+
ϕ (s)ds − ϕ (θ) + ϕ (0) −
ϕ (0)ds
t θ
t θ
t+θ
1 t+θ =
T0 (s)ϕ (0) − ϕ (0) ds + lim
T0 (t + θ − s)Bλ F xs − F(ϕ) ds
t 0
λ→∞ 0
1 0 +
ϕ (s) − ϕ (0) ds + ϕ (0) − ϕ (θ).
t θ
(3.7)
=
Hence
t+θ
1
T0 (s)ϕ (0) − ϕ (0)ds
ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) ≤ 1
t
t
0
t+θ
1
ds
T
(t
+
θ
−
s)B
F
x
F(ϕ)
−
+ lim 0
λ
s
t λ→∞ 0
0
1 ϕ (s) − ϕ (0)ds + ϕ (0) − ϕ (θ).
+
t
(3.8)
θ
Let ε > 0 and choose α > 0 small enough such that if 0 < t < α, −∞ < θ ≤ 0, and t + θ >
0, then
1
t
t+θ
0
T0 (s)ϕ (0) − ϕ (0)ds < ε ,
4
t+θ
ε
1
< ,
T
(t
+
θ
−
s)B
F
x
F(ϕ)
ds
lim −
0
λ
s
4
t λ→∞ 0
0
1 ϕ (s) − ϕ (0)ds + ϕ (0) − ϕ (θ) < ε ,
t
4
θ
and if 0 < t < α, −∞ < θ ≤ 0, and t + θ ≤ 0, then
(3.9)
1 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) ≤ 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ)
t
g(θ) t
1
ε
< .
ϕ(t
+
θ)
−
ϕ(θ)
ϕ
(θ)
−
=
t
4
(3.10)
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Journal of Inequalities and Applications
Consequently, if 0 < t < α, then
1 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) < 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) < ε.
t
g(θ) t
Hence
1
ᐁ(t)ϕ − ϕ − ϕ < ε.
t
(3.11)
(3.12)
Ꮾ
This proves that limt→0+ (1/t)(ᐁ(t)ϕ − ϕ) exists and is equal to ϕ . Then, ϕ ∈ D(Aᐁ ).
Conversely, let ϕ ∈ ᐄ such that
lim
t →0+
1
1
ᐁ(t)ϕ − ϕ = lim+ xt (·,ϕ) − ϕ = ψ = Aᐁ ϕ
t →0 t
t
exists in ᐄ.
(3.13)
We can easily see that axiom (D) is verified by Cγ , LCg , and UCg which implies that
lim+
t →0
1
xt (θ,ϕ) − ϕ(θ)
t
exists for all θ ≤ 0 and is equal to ψ(θ).
(3.14)
Then, for θ ∈ (−∞,0), we have
ψ(θ) = lim+
t →0
1
ϕ(t + θ) − ϕ(θ) = D+ ϕ(θ);
t
(3.15)
that is, D+ ϕ = ψ in (−∞,0). Since ψ is continuous, D+ ϕ is also continuous in (−∞,0).
Let us recall the following result.
Lemma 3.2 [20]. Let ϕ be continuous and differentiable on the right on [a,b). If D+ ϕ is
continuous on [a,b), then ϕ is continuously differentiable on [a,b).
From the above lemma, we deduce that the function ϕ is continuously differentiable in
(−∞,0) and ϕ = ψ. On the other hand, for θ = 0, one has limθ→0− ϕ (θ) exists and equals
ψ(0). From this we infer that the function ϕ is continuously differentiable in (−∞,0] and
ϕ = ψ ∈ ᐄ. We also deduce that t → ᐁ(t)ϕ is continuously differentiable. On the other
hand, we have
x(t) = ϕ(0) + AT
t
0
t
x(s)ds +
t
0
F xs ds.
(3.16)
t
This implies that limt→0+ AT [((1/t) 0 x(s)ds) + (1/t) 0 F(xs )ds] exists and hence
t
t
limt→0+ AT ((1/t) 0 x(s)ds) exists. From the closedness of AT and the fact that (1/t) 0 x(s)ds
t
∈ D(AT ) for t > 0, we deduce that limt→0+ (1/t) 0 x(s)ds exists in D(AT ) and is equal to
ϕ(0). Consequently, ϕ(0) ∈ D(AT ) and ϕ (0) = AT ϕ(0) + F(ϕ). This completes the proof
of the proposition.
The next result may be considered as an extension of a similar one in [19]. Our goal is
to establish the Crandall and Liggett exponential formula
lim
n→+∞
t
I − Aᐁ
n
−n
ϕ = ᐁ(t)ϕ,
∀ϕ ∈ ᐄγ , t ≥ 0.
(3.17)
Hassane Bouzahir 9
We restrict our choice to ᐄγ := {φ ∈ Cγ : φ(0) ∈ D(AT )} with γ > 0. Recall that this
specified space is a uniform fading memory one.
Proposition 3.3. Let Ꮾ = Cγ with γ > 0. Suppose that (H1) and (H2) are satisfied. Then,
the operator Aᐁ given by Proposition 3.1 satisfies the following Crandall Liggett conditions.
(a) Im(I − λAᐁ ) = ᐄγ for all λ ∈ (0,1/(L + ω0 )).
(b) For all ψ1 ,ψ2 ∈ ᐄγ and λ ∈ (0,1/(L + ω0 )),
I − λAᐁ −1 ψ1 − I − λAᐁ −1 ψ2 ≤
γ
1
ψ1 − ψ 2 γ .
1 − λ L + ω0
(3.18)
(c) D(Aᐁ ) is dense in ᐄγ .
Proof. (a) It is well known that one can suppose without loss of generality that ω0 >
−L and T0 (t) ≤ eω0 t . To prove (a), it is clear from the definition of Aᐁ that (I −
λAᐁ )(D(Aᐁ )) ⊆ ᐄγ for λ > 0. On the other hand, for ψ ∈ ᐄγ and λ > 0, let us solve the
following equation:
I − λAᐁ ϕ = ψ,
ϕ ∈ D Aᐁ .
(3.19)
Recall that with γ > 0 and λ > 0, the function W(1/λ)ψ(0) : θ → e(1/λ)θ ψ(0), θ ≤ 0, belongs
to ᐄγ . Also, the fact that Ꮿγ with γ > 0 is a uniform fading memory space implies that the
0
function Mλ ψ : θ → (1/λ) θ e(1/λ)(θ−s) ψ(s)ds, θ ≤ 0, belongs to ᐄγ (see [21, 22]). Moreover, we can see that the solution of (3.19) is
1
1
ϕ(0) (θ) + Mλ ψ (θ) = e(1/λ)θ ϕ(0) +
ϕ(θ) = W
λ
λ
0
θ
e(1/λ)(θ−s) ψ(s)ds.
(3.20)
Next, we suppose that 0 < λω0 < 1. From (3.19) evaluated at 0 and the definition of Aᐁ
we get
ϕ(0) = I − λAT
−1 ψ(0) + λF(ϕ) .
(3.21)
Introduce the following mapping Gλψ : E → E defined by
Gλψ (x) = I − λAT
−1
ψ(0) + λF W
1
x + Mλ ψ
λ
∀x ∈ E.
(3.22)
For x, y ∈ E, we have
λ
G (x) − Gλ (y) ≤ R 1 ,AT F W 1 x + Mλ ψ − F W 1 y + Mλ ψ λ
ψ
ψ
λ
λ
λL W 1 x − W 1 y ≤
1 − λω0
λ
λ γ
(3.23)
λL
≤
sup eγθ e(1/λ)θ (x − y)
1 − λω0 θ≤0
≤
λL
|x − y |.
1 − λω0
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Journal of Inequalities and Applications
Next, we suppose that λ ∈ (0,1/(L + ω0 )). Then, Gλψ is a strict contraction and it has a
unique fixed point x in E. Knowing that (I − λAT )−1 (E) ⊆ D(AT ), we deduce that this
fixed point belongs to D(AT ). Consequently, Im(I − λAᐁ ) = ᐄγ for all λ ∈ (0,1/(L + ω0 )).
(b) Let λ ∈ (0,1/(L + ω0 )) be fixed. Set ᏶λ := (I − λAᐁ )−1 which is well defined from
ᐄγ to D(Aᐁ ). We prove that ᏶λ is Lipschitz continuous with Lipschitz constant less
than (1 − λω0 )/(1 − λ(L + ω0 )). In fact, let λ > 0 with λ ∈ (0,1/(L + ω0 )) and ᏶λ ψ1 := ϕ1
᏶λ ψ2 := ϕ2 for ψ1 ,ψ2 ∈ ᐄγ . Given ε > 0, by definition, there exists θ ∈ (−∞,0] such that
eγθ ϕ1 (θ) − ϕ2 (θ) > ϕ1 − ϕ2 γ − ε.
(3.24)
Using (3.21) and (H2), we get
(1/λ)θ
1
1
R ,AT
ψ1 (0) − ψ2 (0) + F ϕ1 − F ϕ2 eγθ ϕ1 (θ) − ϕ2 (θ) ≤ eγθ e
λ
λ
0
1
+
e(1/λ)(θ−s) ψ1 (s) − ψ2 (s) ds
λ
θ
1
ψ1 − ψ2 + λLϕ1 − ϕ2 ≤ e(1/λ)θ
γ
γ
1 − λω0
0
1 (1/λ)θ
+
e(γ−1/λ)s ds
λe
ψ1 − ψ 2 γ
θ
(1/λ)θ
λLe(1/λ)θ e
ϕ1 − ϕ2 + 1 − e(1/λ)θ ψ1 − ψ2 γ +
≤
γ
1 − λω0
1 − λω0
(3.25)
which implies that
(1/λ)θ
1 − λω0 − λLe(1/λ)θ (1/λ)θ
ϕ1 − ϕ2 ≤ ε + e
ψ1 − ψ 2 ,
+
1
−
e
γ
γ
1 − λω0
1 − λω0
(1/λ)θ
1 − λω0
e
+ 1 − λω0 − e(1/λ)θ + λω0 e(1/λ)θ ψ1 − ψ 2 γ
1 − λω0 − λL
1 − λω0
1
ψ1 − ψ 2 γ .
≤
1 − λ ω0 + L
ϕ1 − ϕ2 ≤
γ
(3.26)
To prove (c), using similar arguments as in [23, the proof of Proposition 3.5], we can
verify that for all λ > 0 with λ ∈ (0,1/(L + ω0 )) and ψ ∈ ᐄγ ,
ψ − ᏶ λ ψ ≤
γ
λ λL
F(0)
ψ γ +
1 − λω
1 − λω
−1
+ ψ − ψ(0) − Mλ ψ − ψ(0) γ + I − λAT ψ(0) − ψ(0).
(3.27)
Since by its definition ᏶λ ψ belongs to D(Aᐁ ), assertion (c) follows from the fact that
limλ→0+ |(I − λAT )−1 x − x| = 0 for all x ∈ D(AT ) (see [24]) and the following result.
Lemma 3.4. Set ᐄγ0 := {φ ∈ ᐄγ such that φ(0) = 0}. Then for all λ > 0 and ψ ∈ ᐄγ0 the
function Mλ ψ tends to ψ in ᐄγ0 as λ tends to zero.
Hassane Bouzahir 11
Proof. Set ᐄγ0 := {φ ∈ ᐄγ such that φ(0) = 0}. We can see that the operator B0 : D(B0 ) ⊆
ᐄγ0 → ᐄγ0 such that B0 φ = φ for any φ ∈ D(B0 ) where
D B0 = φ ∈ ᐄγ0 : φ is continuously differentiable and φ ∈ ᐄγ0
(3.28)
is the infinitesimal generator of a C0 semigroup on ᐄγ0 . Moreover,
Mλ ψ = I − λB0
−1
for any λ > 0, ψ ∈ ᐄγ0 .
ψ
(3.29)
This implies that the assertion of the lemma is a direct consequence of a basic result that
can be found, for instance, in [24, page 248].
Corollary 3.5. Let ᐄ = ᐄγ with γ > 0. Suppose that (H1) and (H2) are satisfied. Then,
for all ϕ ∈ ᐄγ and t ≥ 0,
lim
n→+∞
t
I − Aᐁ
n
−n
ϕ = ᐁ(t)ϕ.
(3.30)
The proof of the above result is based on the following special case of the well-known
Crandall and Liggett theorem (see [25]).
Theorem 3.6 [25]. Let (Y , · ) be a Banach space and B a nonlinear operator with dense
domain D(B) in Y . Suppose that there exists a positive real constant ω such that
(a) Im(I − λB) = Y for all λ ∈ (0,1/ω),
(b) for all y1 , y2 ∈ Y and λ ∈ (0,1/ω),
(I − λB)−1 y1 − (I − λB)−1 y2 ≤
1 y 1 − y 2 .
1 − λω
(3.31)
Then for all y ∈ Y , the limit
W0 (t)y := lim
n→+∞
t
I− B
n
−n
y
(3.32)
exists in Y . Moreover, the family of operators (W0 (t))t≥0 satisfies
(i) W0 (0) = I,
(ii) W0 (t1 + t2 ) = W0 (t1 )W0 (t2 ) for all t1 ,t2 ≥ 0,
(iii) W0 (t)y1 − W0 (t)y2 ≤ eωt y1 − y2 for all y1 , y2 ∈ Y and t ≥ 0.
Let us now give the link between the semigroup given by the Crandall and Liggett
theorem and the integral solution to (1.1). The proof will be omitted because it is very
similar to the case of finite delay (see [18] or [19]).
Proposition 3.7. Let ᐄ = ᐄγ with γ > 0. Suppose that (H1) and (H2) are satisfied and the
operator A : D(A) ⊆ ᐄγ → ᐄγ such that Aϕ = ϕ for any ϕ ∈ D(A), where
D(A) = ϕ ∈ ᐄγ : ϕ is continuously differentiable, ϕ(0) ∈ D AT ,
ϕ ∈ ᐄγ , ϕ (0) = AT ϕ(0) + F(ϕ) ,
(3.33)
12
Journal of Inequalities and Applications
satisfies the hypotheses of Crandall and Liggett theorem in ᐄγ . Let (U(t))t≥0 be the nonlinear
semigroup given by
U(t)ϕ = lim
n→+∞
−n
t
I− A
n
ϕ
∀ϕ ∈ ᐄγ , t ≥ 0.
(3.34)
Then for all ϕ ∈ ᐄγ , the function y := y(·,ϕ) : (−∞,+∞) → R defined by
⎧
⎨ U(t)ϕ (0),
y(t,ϕ) = ⎩
ϕ(t),
t ≥ 0,
t ≤ 0,
(3.35)
is an integral solution of (1.1).
Acknowledgments
The author would like to thank Professors M. Adimy and K. Ezzinbi for helpful discussions. This research was supported by TWAS under contract no. 04-150 RG/MATHS/
AF/AC.
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Hassane Bouzahir: National Engineering School of Applied Sciences (ENSA), University Ibn Zohr,
P.O. Box 1136, Agadir 80000, Morocco
Email address: bouzahir@ensa-agadir.ac.ma