Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 310951, 26 pages
doi:10.1155/2010/310951
Research Article
Asymptotically Almost Periodic
Solutions for Abstract Partial Neutral
Integro-Differential Equation
José Paulo C. dos Santos,1 Sandro M. Guzzo,2
and Marcos N. Rabelo3
1
Departamento de Ciências Exatas, Universidade Federal de Alfenas, Rua Gabriel Monteiro da Silva,
700. 37130-000 Alfenas-MG, Brazil
2
Colegiado do curso de Matemática-UNIOESTE, Rua Universitária, 2069, Caixa Postal 711,
85819-110 Cascavel-PR, Brazil
3
Departamento de Matemática, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire,
Cidade Universitária, 50740-540 Recife-PE, Brazil
Correspondence should be addressed to José Paulo C. dos Santos, zepaulo@unifal-mg.edu.br
Received 24 November 2009; Revised 23 February 2010; Accepted 24 February 2010
Academic Editor: Toka Diagana
Copyright q 2010 José Paulo C. dos Santos et al. 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.
The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral
integro-differential equations with unbounded delay is studied.
1. Introduction
In this paper, we study the existence of asymptotically almost periodic mild solutions for a
class of abstract partial neutral integro-differential equations modelled in the form
d
xt ft, xt Axt dt
t
Bt − sxsds gt, xt ,
1.1
0
where A : DA ⊂ X → X and Bt : DBt ⊂ X → X, t ≥ 0, are closed linear operators;
X, · is a Banach space; the history xt : −∞, 0 → X, xt θ xt θ, belongs to some
abstract phase space B defined axiomatically f, g : I × B → X are appropriated functions.
The study of abstract neutral equations is motivated by different practical applications
in different technical fields. The literature related to ordinary neutral functional differential
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Advances in Difference Equations
equations is very extensive and we refer the reader to Chukwu 1, Hale and Lunel 2, Wu
3, and the references therein. As a practical application, we note that the equation
t
t
d
ut − λ
Ct − susds Aut λ
Bt − susds − pt qt
dt
−∞
−∞
1.2
arises in the study of the dynamics of income, employment, value of capital stock, and
cumulative balance of payment; see 1 for details. In the above system, λ is a real number,
the state ut ∈ Rn , C·, B· are n × n continuous functions matrices, A is a constant n × n
matrix, p· represents the government intervention, and q· the private initiative. We note
that by assuming the solution u is known on −∞, 0, we can transform this system into an
abstract system with unbounded delay described as 1.1.
Abstract partial neutral differential equations also appear in the theory of heat
conduction. In the classic theory of heat conduction, it is assumed that the internal energy
and the heat flux depend linearly on the temperature u and on its gradient ∇u. Under
these conditions, the classic heat equation describes sufficiently well the evolution of the
temperature in different types of materials. However, this description is not satisfactory in
materials with fading memory. In the theory developed in 4, 5, the internal energy and the
heat flux are described as functionals of u and ux . The next system, see for instance 6–9, has
been frequently used to describe this phenomenon,
t
t
d
ut, x k1 t − sus, xds cΔut, x k2 t − sΔus, xds,
dt
−∞
−∞
ut, x 0,
1.3
x ∈ ∂Ω.
In this system, Ω ⊂ Rn is open, bounded, and with smooth boundary; t, x ∈ 0, ∞ × Ω;
ut, x represents the temperature in x at the time t; c is a physical constant ki : R → R,
i 1, 2, are the internal energy and the heat flux relaxation, respectively. By assuming that
the solution u is known on −∞, 0 and k2 ≡ 0, we can transform this system into an abstract
system with unbounded delay described in the form 1.1.
Recent contributions on the existence of solutions with some of the previously
enumerated properties or another type of almost periodicity to neutral functional differential
equations have been made in 10, 11, for the case of neutral ordinary differential equations,
and in 12–15 for partial functional differential systems.
The purpose of this work is to study the existence of asymptotically almost periodic
mild solutions for the neutral system 1.1. To this end, we study the existence and qualitative
properties of an exponentially stable resolvent operator for the integro-differential system
dxt
Axt dt
t
Bt − sxsds, t ≥ 0,
0
1.4
x0 z ∈ X.
There exists an extensive literature related to the existence and qualitative properties of
resolvent operator for integro-differential equations. We refer the reader to the book by
Advances in Difference Equations
3
Gripenberg et al. 16 which contains an overview of the theory for the case where the
underlying space X has finite dimension. For abstract integro-differential equations described
on infinite dimensional spaces, we cite the Prüss book 17 and the papers 18–20, Da Prato
et al. 21, 22, and Lunardi 9, 23. To finish this short description of the related literature,
we cite the papers 24–26 where some of the above topics for the case of abstract neutral
integro-differential equations with unbounded delay are treated.
To the best of our knowledge, the study of the existence of asymptotically almost
periodic solutions of neutral integro-differential equations with unbounded delay described
in the abstract form 1.1 is an untreated topic in the literature and this is the main motivation
of this article.
To finish this section, we emphasize some notations used in this paper. Let Z, · Z and W, ·W be Banach spaces. We denote by LZ, W the space of bounded linear operators
from Z into W endowed with norm of operators, and we write simply LZ when Z W. By
RQ, we denote the range of a map Q, and for a closed linear operator P : DP ⊆ Z → W,
the notation DP represents the domain of P endowed with the graph norm, z1 zZ P zW , z ∈ DP . In the case Z W, the notation ρP stands for the resolvent set of P, and
Rλ, P λI − P −1 is the resolvent operator of P . Furthermore, for appropriate functions
denotes the Laplace transform of
K : 0, ∞ → Z and S : 0, ∞ → LZ, W, the notation K
t
K and S∗K the convolution between S and K, which is defined by S∗Kt 0 St−sKsds.
The notation Br x, Z stands for the closed ball with center at x and radius r > 0 in Z. As
usual, C0 0, ∞, Z represents the subspace of Cb 0, ∞, Z formed by the functions which
vanish at infinity.
2. Preliminaries
In this work, we will employ an axiomatic definition of the phase space B similar to that in
27. More precisely, B will denote a vector space of functions defined from −∞, 0 into X
endowed with a seminorm denoted by · B and such that the following axioms hold.
A If x : −∞, σ b → X, with b > 0, is continuous on σ, σ b and xσ ∈ B, then for
each t ∈ σ, σ b the following conditions hold:
i xt is in B,
ii xt ≤ Hxt B ,
iii xt B ≤ Kt − σ sup{xs : σ ≤ s ≤ t} Mt − σxσ B , where H > 0 is a
constant, and K, M : 0, ∞ → 1, ∞ are functions such that K· and M· are
respectively continuous and locally bounded, and H, K, M are independent of
x·.
A1 If x· is a function as in A, then xt is a B-valued continuous function on σ, σ b.
B The space B is complete.
C2 If ϕn n∈N is a sequence in Cb −∞, 0, X formed by functions with compact
support such that ϕn → ϕ uniformly on compact, then ϕ ∈ B and ϕn − ϕB → 0
as n → ∞.
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Advances in Difference Equations
Remark 2.1. In the remainder of this paper, L > 0 is such that
ϕ ≤ L supϕθ
B
2.1
θ≤0
for every ϕ : −∞, 0 → X continuous and bounded; see 27, Proposition 7.1.1 for details.
Definition 2.2. Let St : B → B be the C0 -semigroup defined by Stϕθ ϕ0 on −t, 0 and
Stϕθ ϕt θ on −∞, −t. The phase space B is called a fading memory if StϕB → 0
as t → ∞ for each ϕ ∈ B with ϕ0 0.
Remark 2.3. In this work, we suppose that there exists a positive K such that
max{Kt, Mt} ≤ K
2.2
for each t ≥ 0. Observe that this condition is verified, for example, if B is a fading memory,
see 27, Proposition 7.1.5.
Example 2.4 The phase space Cr × Lp ρ, X. Let r ≥ 0, 1 ≤ p < ∞, and let ρ : −∞, −r →
R be a nonnegative measurable function which satisfies the conditions g-5 and g-6 in
the terminology of 27. Briefly, this means that ρ is locally integrable, and there exists a
nonnegative, locally bounded function γ on −∞, 0 such that ρξ θ ≤ γξρθ, for all
ξ ≤ 0 and θ ∈ −∞, −r \ Nξ , where Nξ ⊆ −∞, −r is a set with Lebesgue measure zero.
The space Cr × Lp ρ, X consists of all classes of functions ϕ : −∞, 0 → X such that ϕ is
continuous on −r, 0, Lebesgue-measurable, and ρϕp is Lebesgue integrable on −∞, −r.
The seminorm in Cr × Lp ρ, X is defined by
ϕB : sup ϕθ : −r ≤ θ ≤ 0 −r
−∞
p
ρθϕθ dθ
1/p
.
2.3
The space B Cr × Lp ρ; X satisfies axioms A, A-1, and B. Moreover, when r 0 and
0
1/2
p 2, we can take H 1, Mt γ−t1/2 , and Kt 1 −t ρθdθ , for t ≥ 0; see 27,
Theorem 1.3.8 for details.
Now, we need to introduce some concepts, definitions, and technicalities on almost
periodical functions.
Definition 2.5. A function f ∈ CR, Z is almost periodic a.p. if for every ε > 0, there exists a
relatively dense subset of R, denoted by Hε, f, Z, such that
ft ξ − ft < ε,
Z
t ∈ R, ξ ∈ H ε, f, Z .
2.4
Definition 2.6. A function f ∈ C0, ∞, Z is asymptotically almost periodic a.a.p. if there
exists an almost periodic function g· and w ∈ C0 0, ∞, Z such that f· g· w·.
The next lemmas are useful characterizations of a.p and a.a.p functions.
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5
Lemma 2.7 see 28, Theorem 5.5. A function f ∈ C0, ∞, Z is asymptotically almost periodic
if and only if for every ε > 0 there exist Lε, f, Z > 0 and a relatively dense subset of 0, ∞, denoted
by Tε, f, Z, such that
ft ξ − ft < ε,
Z
t ≥ L ε, f, Z , ξ ∈ T ε, f, Z .
2.5
In this paper, AP Z and AAP Z are the spaces
AP Z f ∈ CR, Z : f is a.p. ,
AAP Z f ∈ C0, ∞, Z : f is a.a.p.
2.6
endowed with the norms |u|Z sups∈R usZ and uZ sups≥0 usZ , respectively. We
know from the result in 28 that AP Z and AAP Z are Banach spaces.
Next, Z, · Z and W, · W are abstract Banach spaces.
Definition 2.8. Let Ω be an open subset of W.
a A continuous function f ∈ CR × Ω, Z resp., f ∈ C0, ∞ × Ω; Z is called
pointwise almost periodic p.a.p., resp., pointwise asymptotically almost periodic
p.a.a.p. if f·, x ∈ AP Z resp., f·, x ∈ AAP Z for every x ∈ Ω.
b A continuous function F ∈ CR × Ω, Z is called uniformly almost periodic u.a.p.,
if for every ε > 0 and every compact K ⊂ Ω there exists a relatively dense subset of
R, denoted by Hε, f, K, Z, such that
f t ξ, y − f t, y ≤ ε,
Z
t, ξ, y ∈ R × H ε, f, K, Z × K.
2.7
c A continuous function f : C0, ∞×Ω, Z is called uniformly asymptotically almost
periodic u.a.a.p., if for every ε > 0 and every compact K ⊂ Ω there exists a
relatively dense subset of 0, ∞, denoted by Tε, f, K, Z, and Lε, f, K, Z > 0 such
that
ft ξ, y − ft, y ≤ ε,
Z
t ≥ L ε, f, K, Z , ξ, y ∈ T ε, f, K, Z × K.
2.8
The next lemma summarizes some properties which are fundamental to obtain our
results.
Lemma 2.9 see 29, Theorem 1.2.7. Let Ω ⊂ W be an open set. Then the following properties
hold.
a If f ∈ CR × Ω, Z is p.a.p. and satisfies a local Lipschitz condition at x ∈ Ω, uniformly at
t, then f is u.a.p.
b If f ∈ C0, ∞ × Ω, Z is p.a.a.p. and satisfies a local Lipschitz condition at x ∈ Ω,
uniformly at t, then f is u.a.a.p.
c If x ∈ AP X, then t → xt ∈ AP B. Moreover, if B is a fading memory space and
z ∈ CR, X is such that z0 ∈ B and z ∈ AAP X, then t → zt ∈ AAP B.
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Advances in Difference Equations
d If f ∈ CR × Ω, Z is u.a.p. and y ∈ AP W is such that {yt : t ∈ R}
ft, yt ∈ AP Z.
W
⊂ Ω, then t →
e If f ∈ C0, ∞ × Ω, Z is u.a.a.p and y ∈ AAP W is such that {yt : t ∈ R}
then t → ft, yt ∈ AAP Z.
W
⊂ Ω,
3. Resolvent Operators
In this section, we study the existence and qualitative properties of an exponentially resolvent
operator for the integro-differential abstract Cauchy problem
dxt
Axt dt
t
Bt − sxsds,
0
3.1
x0 x ∈ X.
The results obtained for the resolvent operator in this section are similar to those that
can be found, for instance, in the papers 21, 23, 30. In this paper, we prove the necessary
estimates for the proof of an existence theorem of asymptotically almost periodic solutions for
1.1. For the better comprehension of the subject, we will introduce the following definitions,
hypothesis, and results.
We introduce the following concept of resolvent operator for integro-differential
problem 3.1.
Definition 3.1. A one-parameter family of bounded linear operators Rtt≥0 on X is called a
resolvent operator of 3.1 if the following conditions are verified.
a Function R· : 0, ∞ → LX is strongly continuous and R0x x for all x ∈ X.
b For x ∈ DA, R·x ∈ C0, ∞, DA C1 0, ∞, X, and
dRtx
ARtx dt
dRtx
RtAx dt
t
Bt − sRsxds,
3.2
Rt − sBsxds,
3.3
0
t
0
for every t ≥ 0,
c There exist constants M > 0, β such that Rt ≤ Meβt for every t ≥ 0.
Definition 3.2. A resolvent operator Rtt≥0 of 3.1 is called exponentially stable if there exist
positive constants M, α such that Rt ≤ Me−αt .
In this work, we always assume that the following conditions are verified.
H1 The operator A : DA ⊆ X → X is the infinitesimal generator of an analytic
semigroup T tt≥0 on X, and there are constants M0 > 0, ω ∈ R, and ϑ ∈ π/2, π
ω, | argλ−ω| < ϑ} and Rλ, A ≤ M0 /|λ−ω|
such that ρA ⊇ Λω,ϑ {λ ∈ C : λ /
for all λ ∈ Λω,ϑ .
Advances in Difference Equations
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H2 For all t ≥ 0, Bt : DBt ⊆ X → X is a closed linear operator, DA ⊆ DBt,
and B·x is strongly measurable on 0, ∞ for each x ∈ DA. There exists b· ∈
exists for Reλ > 0 and Btx ≤ btx1 for all t > 0 and
L1loc R such that bλ
x ∈ DA. Moreover, the operator valued function B : Λω,π/2 → LDA, X has
to Λω,ϑ such that Bλx
≤ Bλ
x1
an analytical extension still denoted by B
for all x ∈ DA, and Bλ O1/|λ| as |λ| → ∞.
H3 There exist a subspace D ⊆ DA dense in DA and positive constants Ci , i 1, 2,
such that AD ⊆ DA, BλD
⊆ DA, and ABλx
≤ C1 x for every x ∈ D
and all λ ∈ Λω,ϑ .
In the sequel, for r > 0, θ ∈ π/2, ϑ, and w ∈ R, set
ω, |λ| > r, argλ − ω < θ ,
Λr,ω,θ λ ∈ C : λ /
3.4
and for ω Γir,θ , i 1, 2, 3, the paths
ω Γ1r,θ ω teiθ : t ≥ r ,
ω Γ2r,θ ω reiξ : −θ ≤ ξ ≤ θ ,
3.5
ω Γ3r,θ ω te−iθ : t ≥ r ,
with ω Γr,θ 3
i1
ω Γir,θ are oriented counterclockwise. In addition, ΨG is the set
−1
ΨG λ ∈ C : Gλ : λI − A − Bλ
∈ LX .
3.6
We next study some preliminary properties needed to establish the existence of a
resolvent operator for the problem 3.1.
Lemma 3.3. There exists r1 > 0 such that Λr1 ,ω,ϑ ⊆ ΨG and the function G : Λr1 ,ω,ϑ → LX is
analytic. Moreover,
−1
A ,
Gλ Rλ, A I − BλRλ,
3.7
and there exist constants Mi for i 1, 2, 3 such that
Gλ ≤
AGλx ≤
M1
,
|λ − ω|
M2
x1 ,
|λ − ω|
AGλ ≤ M3 ,
for every λ ∈ Λr1 ,ω,ϑ .
x ∈ DA,
3.8
3.9
3.10
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Advances in Difference Equations
Proof. Since
Rλ, A1
BλRλ, A ≤ Bλ
⎛
≤⎝
M0 Bλ
|λ − ω|
M0 |λ|Bλ
|λ − ω|
⎞
⎠
Bλ ,
3.11
fixed ε < 1, there exists a positive number r1 such that BλRλ,
A ≤ ε for λ ∈
A has a continuous inverse with I −
Λr1 ,ω,ϑ . Consequently, the operator I − BλRλ,
BλRλ,
A−1 ≤ 1/1 − ε. Moreover, for x ∈ X, we have
−1
A x x,
λI − Bλ
− A Rλ, A I − BλRλ,
3.12
−1 A
λI − Bλ
− A x x,
Rλ, A I − BλRλ,
3.13
and for x ∈ DA,
which shows 3.7 and that Λr1 ,ω,ϑ ⊆ ΨG. Now, from 3.7 we obtain RGλ ⊆ DA and
−1
A .
AGλ λRλ, A − I I − BλRλ,
3.14
Consequently,
1
λRλ, A − I
1−ε
M0 |ω|
1
1
≤
M0 1−ε
|λ − ω|
AGλ ≤
3.15
≤ M3 ,
the functions G, AG : Λr1 ,ω,ϑ → LX are analytic, and estimates 3.8, and 3.10 are valid.
In addition, for x ∈ DA, we can write
−1
Rλ, AAx
I
−
BλRλ,
A
ARλ,
A
AGλx ≤ x
−
ARλ,
Ax
−1 I
−
BλRλ,
A
I
−
I
−
BλRλ,
A
ARλ,
A
x
Rλ, AAx
AGλBλRλ,
Ax Rλ, AAx
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Ax1 Rλ, AAx
≤ M3 BλRλ,
≤
M2
x1 ,
|λ − ω|
3.16
for |λ| sufficiently large. This proves 3.9 and completes the proof.
Observation 1. If R· is a resolvent operator for 3.1, it follows from 3.3 that RλλI
−
A − Bλx x for all x ∈ DA. Applying Lemma 3.3 and the properties of the Laplace
transform, we conclude that R· is the unique resolvent operator for 3.1.
In the remainder of this section, r and θ are numbers such that r > r1 and θ ∈ π/2, ϑ.
Moreover, we denote by C a generic constant that represents any of the constants involved
in the statements of Lemma 3.3 as well as any other constant that arises in the estimate that
follows. We now define the operator family Rtt≥0 by
⎧
⎪
⎨ 1
eλt Gλdλ,
Rt 2πi ωΓr,θ
⎪
⎩I,
t > 0,
3.17
t 0.
We will next establish that Rtt≥0 is a resolvent operator for 3.1.
Lemma 3.4. The function R· is exponentially bounded in LX.
Proof. If t > 1, from 3.17 and estimate 3.8, we get
C
Rt ≤
π
≤
∞
C θ tωr cos ξ
e
e
dξ
s
2π −θ
r
C
Cθ rt ωt
e e .
πr|cos θ|
π
tωs cos θ ds
3.18
On the other hand, using that G· is analytic on Λr,ω,θ , for t ∈ 0, 1, we obtain
1 λt
Rt e Gλdλ
2πi ωΓr/t,θ
C ∞ tωs cos θ ds
C θ tωr cos ξ
≤
e
e
dξ
π r/t
s
2π −θ
"
! ∞
C θ r cos ξ
C
u cos θ du
e
e
dξ eωt
≤
π r
u
2π −θ
Cθ r ωt
C
e e .
≤
πr|cos θ|
π
This complete the proof.
3.19
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Lemma 3.5. The operator function R· is exponentially bounded in LDA.
Proof. It follows from 3.9 that the integral in
St 1
2πi
eλt AGλdλ,
3.20
t > 0,
ωΓr,θ
is absolutely convergent in LDA, X and defines a linear operator St ∈ LDA, X.
Using that A is closed, we can affirm that St ARt. From Lemma 3.3, G : Λr,ω,ϑ →
LDA is analytic and Gλ1 ≤ C|λ − ω|−1 . If t > 1 and x ∈ DA, we have
"
! ∞
C θ tωr cos ξ
C
tωs cos θ ds
e
e
dξ x1
ARtx ≤
π r
s
2π −θ
Cθ rt ωt
C
e e x1 .
≤
πr|cos θ|
π
3.21
For t ∈ 0, 1 and x ∈ DA, we get
1 λt
e AGλxdλ
ARtx 2πi ωΓr/t,θ
! ∞
"
C
C θ tωr cos ξ
tωs cos θ ds
≤
e
e
dξ x1
π r/t
s
2π −θ
Cθ r ωt
C
e e x1 .
≤
πr|cos θ|
π
3.22
From before and Lemma 3.4, we infer that R· is exponentially bounded in LDA. The
proof is finished.
Lemma 3.6. The function R : 0, ∞ → LX is strongly continuous.
Proof. It is clear from 3.17 that R·x is continuous at t > 0 for every x ∈ X. We next establish
the continuity at t 0. Let ω ≥ 0 and N be sufficiently large, using that
1
2πi
1
N → ∞ 2πi
λ−1 eλt dλ lim
ωΓr,θ
{ωΓr,θ : |r|≤N}∪ ωCN,θ
λ−1 eλt dλ 1,
3.23
where ω CN,θ represent the curve ω Neiξ for θ ≤ ξ ≤ 2π − θ.
For x ∈ DA and 0 < t ≤ 1, we get
1
Rtx − x 2πi
1
2πi
eλt Gλx − λ−1 eλt x dλ
ωΓr,θ
xdλ.
e λ Gλ A Bλ
λt −1
ωΓr,θ
3.24
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Furthermore, it follows from 3.8, and assumption H2 that
λt −1
x ≤
e λ Gλ A Bλ
eωr C
Φλ,
|λ||λ − ω|
3.25
where Φ· is integrable for λ ∈ ω Γr,θ . From the Lebesgue dominated convergence theorem,
we infer that
1
lim Rtx − x
t → 0
2πi
xdλ.
λ−1 Gλ A Bλ
ωΓr,θ
3.26
Let now ω CL,θ be the curve ω Leiξ for θ ≤ ξ ≤ 2π − θ. Turning to apply Cauchy’s theorem
combining with the estimate
CθL
xdλ ≤
λ−1 Gλ A Bλ
,
L − ω2
ωCL,θ
3.27
we obtain
1
2πi
xdλ
λ−1 Gλ A Bλ
ωΓr,θ
1
L → ∞ 2πi
lim
{ωΓr,θ : |r|≤L}∪ ωCL,θ
xdλ 0,
λ−1 Gλ A Bλ
3.28
we can affirm that limt → 0 Rtx − x 0 for all x ∈ DA, which completes and the proof
since DA is dense in X and R· is bounded on 0, 1.
Notice that ω < 0, the sectors Λr,0,ϑ ⊆ Λr,ω,ϑ , from Lemma 3.3, G : Λr,ω,ϑ → LX is
analytic. Consider the contours
γ1 {λ s − iN sinθ : cosθ ≤ s ≤ ω cosθ},
ω ΓN
r,θ
γ2 {λ s iN sinθ : ω cosθ ≤ s ≤ cosθ},
3.29
ω teiθ : r ≤ t ≤ N ∪ ω reiξ : −θ ≤ ξ ≤ θ ∪ ω te−iθ : r ≤ t ≤ N ,
iθ
iξ
−iθ
te
:
r
≤
t
≤
N
∪
re
:
−θ
≤
ξ
≤
θ
∪
te
:
r
≤
t
≤
N
,
0 ΓN
r,θ
∪γ2 ∪0ΓN
∪γ1 , oriented counterclockwise. By Cauchy theorem for 0 < t ≤ 1,
and RN ωΓN
r,θ
r,θ
we obtain
eλt Gλdλ 0.
RN
3.30
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The following estimate:
ωcosθ
C
λt
eRes−iN sinθt
ds
e Gλdλ ≤
γ1
|s − iN sinθ − ω|
cosθ
≤
ωcosθ
est
cosθ
≤
ωt
C
C
e − 1 cosθt
ds ≤
e
N|sinθ|
N|sinθ|
t
3.31
C
e
N|sinθ|
is the one responsible for the fact that the integral γ1 eλt Gλdλ tends to 0 as N tend to ∞,
in a similar way the integral γ2 eλt Gλdλ, tend to 0 as N tend to ∞, so that
1
2πi
eλt Gλdλ
ωΓr,θ
1
2πi
eλt Gλdλ.
0Γr,θ
3.32
For x ∈ DA, we obtain
Rtx − x 1
2πi
1
2πi
eλt Gλx − λ−1 eλt x dλ
0Γr,θ
xdλ,
e λ Gλ A Bλ
λt −1
3.33
0Γr,θ
and proceeding as before, we obtain limt → 0 Rtx − x 0 for all x ∈ X, which ends the
proof.
The following result can be proved with an argument similar to that used in the proof
of the preceding lemma with changing DA by D.
Lemma 3.7. The function R : 0, ∞ → LDA is strongly continuous.
We next set δ min{ϑ − π/2, π − ϑ}.
Lemma 3.8. The function R : 0, ∞ → LX has an analytic extension to Λδ,0 , and
1
dRz
dz
2πi
λeλz Gλdλ,
ωΓr,θ
z ∈ Λδ,0 .
3.34
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13
Proof. For λ ∈ ω Γr,θ and z ∈ Λδ,0 , we can write λz ω|z|ei argz s|z|eiargzξ , where
π/2 < argz ξ < π, −θ ≤ ξ ≤ θ and s ≥ r. If |z| > 1, from 3.8 and 3.17, we obtain
C
1
eReλz
Rz ≤
|dλ|
2πi ωΓr,θ
|λ − ω|
C ∞ ω|z| cosargzs|z| cosargzθ ds
≤
e
π r
s
θ
C
eω|z| cosargzr|z| cosargzξ dξ
2π −θ
!
"
Cθ r|z| ω|z| cosargz
C
≤
e
e
.
π
πr cos argz θ 3.35
Using that G· is analytic on Λr,ω,θ , for z ∈ Λδ,0 , 0 < |z| < 1, we get
1 λz
e Gλdλ
Rz 2πi ωΓr/|z|,θ
C
≤
π
∞
r/|z|
eω|z| cosargzs|z| cosargzθ
ds
s
C θ ω|z| cosargzr cosargzξ
e
dξ
2π −θ
! ∞
"
C
C θ r cosargzξ
u cosargzθ du
≤
e
e
dξ eω|z| cosargz
π r
u
2π −θ
"
!
Cθ r
C
e eω|z| cosargz .
≤
π
πr cos argz θ 3.36
This property allows us to define the extension Rz by this integral.
Similarly, the integral on the right hand side of 3.34 is also absolutely convergent in
LX and strong, continuous on X for z ∈ Λδ,0 . For λ ∈ ω Γr,θ ,
eλzh − eλz
eλzh − eλz
λz
λz C
Gλ − λe Gλ ≤ − λe −→ 0 as |h| −→ 0,
r
h
h
eλzh − eλz
C
λz
Gλ − λe Gλ ≤ eReλz
Σλ,
h
|λ − ω|
3.37
where Σ· is integrable for λ ∈ ω Γr,θ . From the Lebesgue dominated convergence theorem,
we obtain that R z verifies 3.34. The proof is ended.
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Advances in Difference Equations
Lemma 3.9. For every λ ∈ C with Reλ > max{0, ω r}, Rλ
Gλ.
Proof. Using that G· is analytic on Λr,ω,θ and that the integrals involved in the calculus are
absolutely convergent, we have
Rλ
∞
e
−λt
Rtdt 0
∞
0
1
2πi
1
2πi
e−λ−γt G γ dγdt
ωΓr,θ
λ−γ
−1 G γ dγ
ωΓr,θ
!
lim
L→∞
1
2πi
{ωΓr,θ : |r|≤L}∪ ωCL,θ
λ−γ
−1 G γ dγ
"
3.38
Gλ.
Theorem 3.10. The function R· is a resolvent operator for the system 3.1.
Proof. Let x ∈ DA. From Lemma 3.9, for Reλ > max{0, ω r},
x x,
λI − A − Bλ
Rλ
3.39
which implies
1
1
1
Rλx
Bλx.
x RλA
Rλ
λ
λ
λ
3.40
Applying 31, Proposition 1.6.4, Corollary 1.6.5, we get
Rtx x t
RsAxds t s
0
0
Rs − ξBξxdξds
3.41
0
which in turn implies that
dRt
x RtAx dt
t
Rt − sBsxds.
3.42
0
Arguing as above but using the equality λI − A − Bλ
Rλx
x, we obtain that 3.2
holds.
On the other hand, by Lemma 3.8 we infer that R·x ∈ C1 0, ∞, X. Next, we analyze
the differentiability on t 0. Let a > 0 and x ∈ DA, for all ε > 0; we can choose δ ∈ 0, a
such that
sup RtAx t∈0,δ
t
0
Rt − sBsds − Ax < ε.
3.43
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15
For ζ ∈ X and t ∈ 0, δ, there exists cζ,t ∈ 0, t such that
cζ,t
ζ ◦ Rtx − ζ ◦ R0x
ζ R cζ,t Ax R cϕ,t − s Bsxds .
t
0
3.44
Consequently, for t ∈ 0, δ we have that
ζ ◦ Rtx − ζ ◦ R0x
Rtx − R0x
− Ax sup − ζAx
t
t
ζ≤1
s
≤ sup RsAx Rs − τBτxdτ − Ax
,
3.45
0
s∈0,δ
which proves the existence of the right derivative of R· at zero and that d/dtRt|t0 Ax.
This proves that resolvent equation 3.3 is valid for every t ≥ 0 and R·x ∈ C1 0, ∞; X for
every x ∈ DA. This completes the proof.
Corollary 3.11. If ω r < 0, then the function R· is an exponentially stable resolvent operator for
the system 3.1.
In the next result, we denote by −Aϑ the fractional power of the operator −A see
32 for details.
Theorem 3.12. Suppose that the conditions H1–H3 are satisfied. Then there exists a positive
number C such that
⎧
⎨Cerωt ,
−Aϑ Rt ≤
⎩Cerωt t−ϑ ,
t ≥ 1,
t ∈ 0, 1,
3.46
for all ϑ ∈ 0, 1.
Proof. Let ϑ ∈ 0, 1. From 32, Theorem 6.10, there exists Cϑ > 0 such that
−Aϑ x ≤ Cϑ Axϑ x1−ϑ ,
x ∈ DA.
3.47
Since G· is a DA valued function, for all x ∈ X
−Aϑ Gλx ≤ Cϑ AGλxϑ Gλx1−ϑ
≤ Cϑ M3ϑ xϑ
≤
C
|λ − ω|1−ϑ
M11−ϑ
1−ϑ
|λ − ω|
x,
x1−ϑ
3.48
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Advances in Difference Equations
where C is independent of λ. From 3.48, we get for t ≥ 1
1 ϑ
ϑ
λt
e −A Gλdλ
−A Rt ≤ 2πi ωΓr,θ
≤
≤
C
π
!
∞
etωs cos θ
r
ds
C
s1−ϑ 2π
θ
−θ
etωr cos ξ
rdξ
r 1−ϑ
3.49
"
C
Cθr ϑ rt ωt
e ≤ Cerωt .
e
π
πr 1−ϑ |cos θ|
On the other hand, using that G· is analytic on Λr,ω,θ , for t ∈ 0, 1, we get
1 ϑ
ϑ
λt
e −A Gλdλ
−A Rt 2πi ωΓr/t,θ
C ∞ tωs cos θ ds
C θ tωr cos ξ rt−1 dξ
e
e
π r/t
s1−ϑ 2π −θ
r 1−ϑ tϑ−1
"
! ∞
θ
−1
−1
C
C
u cos θ t du
r cos ξ rt dξ
eωt
≤
e
e
π r
u1−ϑ tϑ−1 2πr −ϑ −θ
r 1−ϑ tϑ−1
!
"
C
Cθr ϑ r eωt
e
≤
.
π
πr 1−ϑ |cos θ|
tϑ
≤
3.50
From the previous facts, we conclude that
−Aϑ Rt ≤ Cerωt t−ϑ ,
t ∈ 0, 1,
3.51
which ends the proof.
Corollary 3.13. If ω r < 0 and ϑ ∈ 0, 1, then there exists φ ∈ L1 0, ∞ such that
−Aϑ Rt ≤ φt.
3.52
In the remainder of this section, we discuss the existence and regularity of solutions of
dxt
Axt dt
t
Bt − sxsds ft,
t ∈ 0, a,
3.53
0
x0 z ∈ X,
3.54
where f ∈ L1 0, a, X. In the sequel, R· is the operator function defined by 3.17. We
begin by introducing the following concept of classical solution.
Advances in Difference Equations
17
Definition 3.14. A function x : 0, b → X, 0 < b ≤ a, is called a classical solution of 3.533.54 on 0, b if x ∈ C0, b, DA ∩ C1 0, b, X, the condition 3.54 holds and 3.53 is
verified on 0, a.
The next result has been established in 30.
Theorem 3.15 30, Theorem 2. Let z ∈ X. Assume that f ∈ C0, a, X and x· is a classical
solution of 3.53-3.54 on 0, a. Then
xt Rtz t
Rt − sfsds,
t ∈ 0, a.
3.55
0
An immediate consequence of the above theorem is the uniqueness of classical
solutions.
Corollary 3.16. If u, v are classical solutions of 3.53-3.54 on 0, b, then u v on 0, b.
Motivated by 3.55, we introduce the following concept.
Definition 3.17. A function u ∈ C0, a, X is called a mild solution of 3.53-3.54 if
ut Rtz t
Rt − sfsds,
t ∈ 0, a.
3.56
0
4. Existence Result of Asymptotically Almost Periodic Solutions
In this section, we study the existence of asymptotically almost periodic mild solutions for
the abstract integro-differential system 1.1. To establish our existence result, motivated by
the previous section we introduce the following assumptions.
P1 There exists a Banach space Y, · Y continuously included in X such that the
following conditions are verified.
a For every t ∈ 0, ∞, Rt ∈ LX ∩ LY, DA and Bt ∈ LY, X. In
addition, AR·x, B·x ∈ C0, ∞, X for every x ∈ Y .
b There are positive constants M, β such that
max Rs, BsLY,X ≤ Me−βt ,
s ≥ 0.
4.1
c There exists φ ∈ L1 0, ∞ such that ARtLY,X ≤ φt, t ≥ 0.
P2 The continuous function f : R × B → Y is p.a.a.p, and there exists a continuous
function Lf : 0, ∞ → 0, ∞, such that
f t, ψ1 − f t, ψ2 Y ≤ Lf rψ1 − ψ2 B ,
t, ψj ∈ R × Br 0, B.
4.2
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Advances in Difference Equations
P3 The continuous function g : R × B → X is p.a.a.p, and there exists a continuous
function Lg : 0, ∞ → 0, ∞ such that
g t, ψ1 − g t, ψ2 ≤ Lg rψ1 − ψ2 B ,
t, ψj ∈ R × Br 0, B.
4.3
Motivated by the theory of resolvent operator, we introduce the following concept of
mild solution for 1.1.
Definition 4.1. A function u : −∞, b → X, 0 < b ≤ a, is called a mild solution of 1.1 on
b, if u0 ϕ ∈ B; u|0,b ∈ C0, b : X; the functions τ → ARt − τfτ, uτ and τ →
0,
τ
Bτ
− ξfξ, uξ dξ are integrable on 0, t for every t ∈ 0, b and
0
ut Rt ϕ0 f 0, ϕ − ft, ut −
−
t
0
Rt − s
s
t
ARt − sfs, us ds
0
Bs − ξf ξ, uξ dξds 0
t
4.4
Rt − sgs, us ds,
t ∈ 0, b.
0
Lemma 4.2. Let condition P1 —c hold and let v be a function in AAP Y . If u : 0, ∞ → X is
t
the function defined by ut 0 ARt − svsds, then u· ∈ AAP X.
∞
Proof. Let η max{2vY , 0 φsds}.
Let Tε/3η−1 , v, Y , T T ε/3η−1 , v, Y be as in
∞
Lemma 2.7 and T1 > 1 such that T1 φsds ≤ ε/3η−1 . For t ≥ T T1 and ξ ∈ Tε/3η−1 , v, Y ,
we get
ut ξ − ut ≤
0
−ξ
≤
t
0
−ξ
ARt − sLY,X vs ξY ds
0
ARt − sLY,X vs ξ − vsY ds
φt − svs ξY ds
T
0
t
T
≤ vY
≤ vY
4.5
φt − svs ξ − vsY ds
φt − svs ξ − vsY ds
tξ
t
∞
T1
φsds 2vY
φsds 2vY
t
t−T
∞
φsds ε
t−T
φsds
0
ε
φsds η−1
3
T1
∞
φsds
0
Advances in Difference Equations
19
which implies that
ut ξ − ut ≤ ε,
t≥T
ε −1
η , v, Y
3
ε
T1 , ξ ∈ T η−1 , v, Y .
3
4.6
Now, from inequality 4.6 and Lemma 2.7, we conclude that u· is a.a.p. The proof is
complete.
Lemma 4.3. Assume that the condition P1 is fulfilled. Let v ∈ AAP Y and let w· : 0, ∞ → X
be the function defined by
wt t
Rt − s
0
s
Bs − ξvξdξds,
4.7
t ≥ 0.
0
Then w· ∈ AAP X.
Proof. Let Tε/3η−1 , v, Y , T T ε/3η−1 , v, Y be as in Lemma 2.7 and T1 > 1 such that
∞
e−βs ds ≤
T1
ε −1
η ,
3
te−βt−T ≤
ε −1
η
3
4.8
for t ≥ T1 , where η max{3vY M2 /β, supt≥T1 M2 /βte−βt−T }. For t ≥ T T1 and ξ ∈
Tε/3η−1 , v, Y , we get
wt ξ − wt tξ
Rt ξ − s
s
0
t
Bs − uvudu ds
0
0
ξ
Rt ξ − s
0
Rt − s
0
s
s
Rt − s Bs − uvu ξ − vudu ds
0
Bs − uvudu ds −
0
t
t
0
s
Bs − uvudu ds
0
Rt − s
ξ
0
Bs ξ − uvudu ds.
4.9
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Advances in Difference Equations
We obtain
wt ξ − wt
≤
t
0
T
Rt − s Bs − uLY,X vu ξ − vuY du ds
0
t
0
ξ
s
Rt − s Bs − uLY,X vu ξ − vuY du ds
T
Rt ξ − s
s
0
t
0
≤
t
0
ξ
Rt − s Bs ξ − uLY,X vuY du ds.
0
Me−βt−s
T
0
0
t
Me
−βt−s
Me−βs−u vu ξ − vuY du ds
t
0
T
ξ
Me
−βtξ−s
t
Me−βs−u vu ξ − vuY du ds
s
0
Bs − uLY,X vuY du ds
0
Me−βt−s
ξ
0
≤ 2vY
0
t
Me
0
vY
vY
≤ 2vY
Me−βsξ−u vuY du ds.
T
Me du ds ε
βu
t
0
ξ
0
Me−βtξ−s
s
0
t
Me
−βt
t
Meβu du ds
T
Me−βs−u du ds
0
Me−βt−s
0
ξ
Meβsξ−u du ds
0
M2 −βt−T M2 −βt−T te
te
ε
β
β
vY
≤ 3vY
−βt
Me−βs−u vuY du ds
M2
β
tξ
t
e−βs ds vY
M2 −βt−T te
β
M2 −βt−T M2
M2 −βt−T te
te
ε
vY
β
β
β
∞
e−βs ds,
T1
4.10
Advances in Difference Equations
21
which implies that
wt ξ − wt ≤ ε,
t≥T
ε −1
η , v, Y
3
T1 , ξ ∈ T
ε −1
η , v, Y .
3
4.11
From inequality 4.11 and Lemma 2.7, we conclude that w· is a.a.p., which ends the proof.
Now, we can establish our existence result.
Theorem 4.4. Assume that B is a fading memory space and P1 , P2 , and P3 are held. If Lf 0 Lg 0 0 and ft, 0 gt, 0 0 for every t ∈ R, then there exists ε > 0 such that for each
ϕ ∈ Bε 0, B, there exists a mild solution, u·, ϕ, of 1.1 on 0, ∞ such that u·, ϕ ∈ AAP X and
u0 ·, ϕ ϕ.
Proof. Let r > 0 and 0 < λ < 1 be such that
!
M2
Θ MHλ MLf λrλ Lf λ 1Krλ 1K ic LY,X φL1 2
β
"
4.12
M
Lg λ 1Krλ 1K
< 1,
β
where K is the constant introduced in Remark 2.3. We affirm that the assertion holds for ε λr. Let ϕ ∈ Bε 0, B. On the space
D x ∈ AAP X : x0 ϕ0, xt ≤ r, t ≥ 0
4.13
endowed with the metric du, v u − v, we define the operator Γ : D → C0, ∞; X by
Γut Rt ϕ0 f 0, ϕ − ft, u
#t −
−
t
0
Rt − s
s
0
t
ARt − sfs, u
# s ds
0
Bs − ξf ξ, u
# ξ dξds t
4.14
Rt − sgs, u
# s ds,
t ≥ 0,
0
# u on 0, ∞. From the
where u
# : R → X is the function defined by the relation u
# 0 ϕ and u
hypothesis P1 , P2 , and P3 , we obtain that Γu is well defined and that Γu ∈ C0, ∞; X.
Moreover, from Lemmas 4.2 and 4.3 it follows that Γu ∈ AAP X.
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Advances in Difference Equations
Next, we prove that Γ· is a contraction from D into D. If u ∈ D and t ≥ 0, we get
Γut ≤ MHλr MLf λrλr ic LY,X Lf λ 1Krλ 1Kr
t
φt − sLf λ 1Krλ 1Kr ds
0
t
Me−βt−s
0
t
s
Me−βs−ξ Lf λ 1Krλ 1Kr dξds
0
Me−βt−s Lg λ 1Krλ 1Kr ds
0
≤ MHλr MLf λrλr Lf λ 1Krλ 1Kr
∞
φsds Lf λ 1Krλ 1Kr
0
∞
Me
0
∞
−βs
4.15
∞
−βξ
ds
Me dξ Lf λ 1Krλ 1Kr
0
−βs
e ds Lg λ 1Kλ 1Kr
0
≤ Θr,
where the inequality #
ut ≤ λ 1Kr has been used and ic : Y → X represent the continuous
inclusion of Y on X. Thus, ΓD ⊂ D. On the other hand, for u, v ∈ D we see that
Γut − Γvt
# t − ft, v#t ≤ ic LY,X ft, u
t
0
t
0
Rt − s
s
0
t
0
# s − fs, v#s Y ds
ARt − sLY,X fs, u
# ξ − fξ, v#ξ Y dξds
Bs − ξLY,X fξ, u
4.16
# s − gs, v#s ds
Rt − sgs, u
!
M2
≤ Lf λ 1KrK ic LY,X φL1 2
β
Lg λ 1KK
"
u − v
M
u − v ≤ Θu − v,
β
which shows that Γ· is a contraction from D into D. The assertion is now a consequence of
the contraction mapping principle. The proof is complete.
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23
5. Applications
In this section, we study the existence of asymptotically almost periodic solutions of the
partial neutral integro-differential system
t π
∂
ut, ξ b s − t, η, ξ u s, η dηds
∂t
−∞
0
"
!
t
t
∂2
−γt−s
μ ut, ξ e
us, ξds a0 s − tus, ξds,
∂ξ2
0
−∞
5.1
ut, 0 ut, π 0,
uθ, ξ ϕθ, ξ,
for t, ξ ∈ 0, a × 0, π, θ ≤ 0, μ < 0, and γ > 0. Moreover, we have identified ϕθξ ϕθ, ξ.
To represent this system in the abstract form 1.1, we choose the spaces X L2 0, π
and B C0 × L2 ρ, X; see Example 2.4 for details. We also consider the operators A, Bt :
DA ⊆ X → X, t ≥ 0, given by Ax x μx, Btx e−γt Ax for x ∈ DA {x ∈ X :
x ∈ X, x0 xπ 0}. Moreover, A has discrete spectrum, the eigenvalues are −n2 μ,
n ∈ N, with corresponding eigenvectors zn ξ 2/π1/2 sinnξ, and the set of functions
$
−n2 −μt
< x, zn > zn for x ∈ X.
{zn : n ∈ N} is an orthonormal basis of X and T tx ∞
n1 e
For α ∈ 0, 1, from 32 we can define the fractional power −Aα : D−Aα ⊂ X → X of
$
α
α
α
2
A is given by −Aα x ∞
n1 n − μ x, zn zn , where D−A {x ∈ X : −A x ∈ X}. In
the next theorem, we consider Y D−A1/2 . We observe that ρA ⊃ {λ ∈ C : Reλ ≥ μ}
and λRλ, A ≤ M1 for Reλ ≥ μ; from 33, Proposition 2.2.11, we obtain that A is a
sectorial operator satisfying Rλ, A ≤ M/|λ − μ|, M > 0. Moreover, it is easy to see that
conditions H2-H3 in Section 3 are satisfied with bt e−γt , and D C0∞ 0, π is the
space of infinitely differentiable functions that vanish at ξ 0 and ξ π. Under the above
conditions, we can represent the system
∂ut, ξ
∂t
!
∂2
μ
∂ξ2
"
ut, ξ t
e
−γt−s
us, ξds ,
0
5.2
ut, π ut, 0 0,
in the abstract form
dxt
Axt dt
t
Bt − sxsds,
0
x0 z ∈ X.
5.3
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We define the functions f, g : B → X by
f ψ ξ 0 π
−∞
g ψ ξ b s, η, ξ ψ s, η dηds,
0
0
−∞
5.4
a0 sψs, ξds,
where
i the functions a0 : R → R are continuous and Lg : 0
−∞
1/2
a0 s2 /ρsds
< ∞.;
ii the functions b·, ∂bs, η, ξ/∂ξ are measurable, bs, η, π bs, η, 0 0 for all
s, η and
⎧⎛
⎞1/2
!
"2
⎪
⎨ π0 π
i
∂
Lf : max ⎝
ρ−1 θ
bθ, η, ξ dηdθdξ⎠ : i 0, 1
⎪
∂ξi
⎩
0 −∞ 0
⎫
⎪
⎬
⎪
⎭
< ∞.
5.5
Moreover, f, g are bounded linear operators, fLB,X ≤ Lf , gLB,X ≤ Lg , and a
straightforward estimation using ii shows that fI × B ⊂ D−A1/2 and
−A1/2 ft, ·
LB,X
≤ Lf
5.6
for all t ∈ I. This allows us to rewrite the system 5.1 in the abstract form 1.1 with
u0 ϕ ∈ B.
Theorem 5.1. Assume that the previous conditions are verified. Let 2 < K < γ and μ < 0 such that
|μ| > max{MK 1 γ, γ}, then there exists a mild solution u· ∈ AAP X of 5.1 with u0 ϕ.
Proof. For λ μ seiθ , from |λ γ| ≥ s − |γ μ|, we obtain
1
BλRλ, A ≤ λ γ
1
≤
λ γ 1
≤
λ γ !
"
M
M|λ|
1 λ − μ λ − μ
"
!
M
1
|λ|
λ γ λ γ λ − μ
!
"
γ M
1
1 λ γ λ γ λ − μ
Advances in Difference Equations
25
"
!
γ M
1
1
1
≤
λ − μ
s− γ μ
s− γ μ
s− γ μ
1Kγ
1
M
≤
λ − μ
K
K
≤
1
1
,
K K
5.7
since s ≥ r max{MK 1 γ, K |γ μ|}. By using a similar procedure as in the
proofs of Lemma 3.3 and Theorem 3.10, we obtain the existence of resolvent operator for 5.2.
From the hypothesis, we obtain μ r < 0; by the Lemma 3.3, Corollaries 3.11 and 3.13, the
assumption P1 is satisfied. From Theorem 4.4, the proof is complete.
Acknowledgment
José Paulo C. dos Santos is partially supported by FAPEMIG/Brazil under Grant CEX-APQ00476-09.
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