This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using resolvent operators and KrasnoselskiSchaefer type fixed point theorem. An example is provided to illustrate the results.
AMS Mathematics Subject Classification: 35R12, 45K.
1. Introduction
The aim of this paper is to establish the existence results for the following neutral functional integrodifferential equations with infinite delays:
where
A
is the infinitesimal generator of a compact, analytic resolvent operator
R
(
t
),
t
≥ 0 in a Banach space
X
,
a
:
D
×
B_{h}
→
X
,
g
:
J
×
B_{h}
×
X
→
X
,
k
:
D
×
B_{h}
→
X
and
f
:
J
×
B_{h}
×
X
→
X
are given functions, where
B_{h}
is a phase space defined later and
D
= {(
t, s
) ∈
J
×
J
:
s
≤
t
}.0 <
t
_{1}
<
t
_{2}
< ... <
t_{m}
<
b
are bounded functions.
B
(
t
),
t
∈
J
is a bounded linear operator.
The histories
x_{t}
: (−∞, 0] →
X
,
x_{t}
(
s
) =
x
(
t
+
s
),
s
≤ 0, belong to an abstract phase space
B_{h}
.
Neutral differential and integrodifferential equations arise in many areas of applied mathematics and for this reason these equations have been investigated extensively in the last decades. There are many contributions relative to this topic and we refer the reader to
[1
,
2
,
3
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15]
.
The theory of nonlinear functional differential or integrodifferential equations with resolvent operators is an important branch of differential equations, which has an extensive physical background, see for instance
[16
,
17
,
18]
.
Since many control systems arsing from realistic models depend heavily on histories ( that is, the effect of infinite delay on the state equations
[23]
), there is real need to discuss the existence results for partial neutral functional integrodifferential equations with infinite delay. The development of the theory of functional differential equations with infinite delays depends on a choice of a phase space. In fact, various phase spaces have been considered and each different phase space has required a separate development of the theory
[20]
. The common space is the phase space
B
proposed by Hale and Kato in
[19]
.
The main purpose of this paper is to deal with the existence of mild solutions for the problem (1)(2). Here, we use an abstract phase space adopted in
[6
,
24]
. Sufficient conditions for the existence results are derived by means of the KrasnoselskiSchaefer type fixed point theorem combined with theories of analytic resolvent operators. The results generalise the results of
[4
,
6
,
7
,
21]
.
2. Main results
Throughout this paper, we assume that (
X
, ∥·∥) is a Banach space, the notaion
L
(
X
,
Y
) stands for the Banach space of all linear bounded operators from
X
into
Y
, and we abbreviate this notation to
L
(
X
) when
X
=
Y
.
R
(
t
),
t
> 0 is compact, analytic resolvent operator generated by
A
.
Assume that

(A1)Ais a densely defined, closed linear operator in a Banach space (X, ∥ · ∥) and generates aC0semigroupT(t). HenceD(A) endowed with the graph norm x = ∥x∥ + ∥Ax∥ is a Banach space which will be denoted by (Y,  · ).

(A2) {B(t) :t∈J} is a family of continuous linear operator from (Y,  · ) into (X, ∥ · ∥) . Moreover, there is an integrable functionc: [0,b] →R+such that for anyy∈Y, the mapt→B(t)ybelongs toW1,1(J,X) and
Definition 2.1.
A family {
R
(
t
) :
t
≥ 0} of continuous linear operators on
X
is called a resolvent operator for
if:

(R1)R(0) =I( the identity operator onX),

(R2) For allx∈X, the mapt→R(t)xis continuous fromJtoX,

(R3) For allt∈J,R(t) is a continuous linear operator onY, and for ally∈Y, the mapt→R(t)ybelongs toC(J,Y) ∩C′(J,X) and satisfies
Theorem 2.1
(
[21]
).
Let the assumptions (A1) and (A2) be satisfied. Then there exists a constant H
=
H
(
b
)
such that
where L
(
X
)
denotes the Banach space of continuous linear operators on X.
Next, if the C
_{0}

semigroup T
(·)
generated by A is compact ( that is, T
(
t
)
is a compact operator for all t
> 0
), then the corresponding resolvent operator R
(·)
is also compact ( that is, R
(
t
)
is a compact operator for all t
> 0
) and is operator norm continuous ( or continuous in the uniform operator topology) for t
> 0.
Proof
. Now, we define the abstract phase space
B_{h}
as given in
[24
,
7]
.
Assume that
h
: (−∞, 0] → (0,+∞) is a continuous function with
For any
a
> 0, we define
and equip the space
B
with the norm
Let us define
If
B_{h}
is endowed with the norm
then it is clear that (
B_{h}
, ∥ · ∥
_{Bh}
) is a Banach space.
Now we consider the space
Set ∥ · ∥
_{b}
be a semi norm in
B′_{h}
defined by
Next, we introduce the basic definitions and lemmas which are used throughout this paper.
Let
A
:
D
(
A
) →
X
be the infinitesimal generator of a compact, analytic resolvent operator
R
(
t
),
t
≥ 0. Let 0 ∈
ρ
(
A
), then it is possible to define the fractional power (−
A
)
^{α}
, for 0 <
α
≤ 1, as closed linear operator on its domain
D
(−
A
)
^{α}
. Further more, the subspace
D
(−
A
)
^{α}
is dense in
X
, and the expression
defines a norm on
D
(−
A
)
^{α}
.
Furthermore, we have the following properties appeared in
[22]
. □
Lemma 2.2.
The following properties hold:
(i) If
0 <
β
<
α
≤ 1,
then X_{α}
⊂
X_{β} and the imbedding is compact whenever the resolvent operator of A is compact.
(ii) For every
0 <
α
≤ 1
there exists C_{α}
> 0
such that
Lemma 2.3
(
[5]
).
Let
Փ
_{1}
,Փ
_{2}
be two operators satisfying
Փ
_{1}
is contraction and
Փ
_{2}
is completely continuous. Then either
(i) the operator equation
Փ
_{1}
x
+ Փ
_{2}
x
=
x has a solution, or
(ii) the set
is unbounded for λ
∈ (0, 1).
Lemma 2.4
(
[12]
).
Let v
(·),
w
(·) : [0,
b
] → [0,∞)
be continuous functions. If w
(·)
is nondecreasing and there are constants θ
> 0, 0 <
α
< 1
such that
then
for every t
∈ [0,
b
]
and every n
∈
N such that nα
> 1,
and
Γ(·)
is the Gamma function.
Lemma 2.5
(
[6]
).
Assume x
∈
B′_{h}
,
then for t
∈
J, x_{t}
∈
B_{h}. Moreover
,
where
Definition 2.2.
A function
x
: (−∞,
b
] →
X
is called a mild solution of problem (1)(2) if the following holds:
x
_{0}
=
ϕ
∈
B_{h}
on (−∞, 0]; the restriction of
x
(·) to the interval
J
is continuous, and for each
s
∈ [0,
t
), the function
is integrable and the integral equation
is satisfied.
Definition 2.3.
A map
f
:
J
×
B_{h}
×
X
→
X
is said to be an
L
^{1}
Caratheodory if
(i) For each
t
∈
J
, the function
f
(
t
, ·, ·) :
B_{h}
×
X
→
X
is continuous.
(ii) For each (
ϕ, x
) ∈
B_{h}
×
X
; the function
f
(·,
ϕ, x
) :
J
→
X
is strongly measurable.
(iii) For every positive integer
q
> 0, there exists
α_{q}
∈
L
^{1}
(
J,R
_{+}
) such that
3. Existence Results
In this section, we shall present and prove our main result. For the proof of the main result, we will use the following hypotheses:
(H1) ([see Lemma 2.2])
A
is the infinitesimal generator of a compact analytic resolvent operator
R
(
t
),
t
> 0 and 0 ∈
ρ
(
A
) such that
(H2) There exist a constant
N
_{1}
> 0 such that
(H3) There exist constants 0 <
β
< 1,
C
_{0}
,
c
_{1}
,
c
_{2}
,
N
_{2}
such that
g
is
X_{β}
valued, (−
A
)
^{β}g
is continuous, and

(i)(t, x) ∈J×Bh,

(ii)∥(−A)βg(t,x1,y1) − (−A)βg(t,x2,y2)∥ ≤N2[∥x1−x2∥Bh+ ∥y1−y2∥] fort∈J,x1,x2∈Bh,y1,y2∈X, with
(H4) (i) For each (
t, s
) ∈
D
, the function
k
(
t, s
, ·) :
B_{h}
→
X
is continuous and for each
x
∈
B_{h}
, the function
k
(·, ·,
x
) :
D
→
X
is strongly measurable.

(ii) There exist an integrable functionm:J→ [0,∞) and a constantγ> 0, such that

where Ω : [0,∞) → (0,∞) is a continuous nondecreasing function. Assume that the finite bound ofisL0.
(H5) The function
f
:
J
×
B_{h}
×
X
→
X
satisfies the following caratheodory conditions:

(i)t→f(t, x, y) is measurable for each (x, y) ∈Bh×X,

(ii) (x, y) →f(t, x, y) is continuous for almost allt∈J.
(H6) ∥
f
(
t, x, y
)∥ ≤
p
(
t
)
Ψ
(∥
x
∥
_{Bh}
+∥
y
∥) for almost all
t
∈
J
and all
x
∈
B_{h}
,
y
∈
X
, where
p
∈
L
^{1}
(
J,R
_{+}
) and
Ψ
:
R
_{+}
→ (0,∞) is continuous and increasing with
where
with
lM
_{0}
N
_{2}
(1 +
N
_{1}
) < 1,
We consider the operator Փ :
B_{h}′
→
B_{h}′
defined by
From hypothesis (
H
1), (
H
2) and Lemma 2.3, the following inequality holds:
Then from Bochner theorem
[25]
, it follows that
is integrable on [0,
t
).
For
ϕ
∈
B_{h}
, we defined by
by
and then
It is easy to see that
x
satisfies (3) if and only if
y
satisfies
y
_{0}
= 0 and
Let
thus
is a Banach space. Set
for some
q
≥ 0, then
is uniformly bounded, and for
y
∈
B_{q}
, from Lemma 2.5, we have
Define the operator
Now we decompose
where
Obviously the operator Փ has a fixed point is equivalent to
has one. Now, we shall show that the operators
satisfy all the conditions of Lemma 2.3.
Lemma 3.1.
If assumptions (H1)(H6) hold, then
is a contraction and
is completely continuous
.
Proof
. First we show that
is a contraction on
From (H1)(H3) and Lemma 2.5, we have
Since ∥
u
_{0}
∥
_{Bh}
= 0, ∥
v
_{0}
∥
_{Bh}
= 0. Taking supremum over
t
,
where
Thus
is a contraction on
□
Next we show that the operator
is completely continuous. First we prove that
maps bounded sets into bounded sets in
Indeed, it is enough to show that there exists a positive constant Λ such that for each
Now for each
t
∈
J
,
By (H1)(H6) and (5), we have for
t
∈
J
,
Then for each
Next we show that
maps bounded sets into equicontinuous sets of
Let 0 <
r
_{1}
<
r
_{2}
≤
b
, for each
Let
r
_{1}
,
r
_{2}
∈
J
− {
t
_{2}
,
t
_{2}
, ...,
t_{m}
}. Then we have
The righthand side from Theorem 2.1 of the above inequality tends to zero as
r
_{2}
→
r
_{1}
and for ϵ sufficiently small. Thus the set
is equicontinuous. Here we consider only the case 0 <
r
_{1}
<
r
_{2}
≤
b
, since the other cases
r
_{1}
<
r
_{2}
≤ 0 or
r
_{1}
≤ 0 ≤
r
_{2}
≤
b
are very simple.
Next, we show that
is continuous.
Let
Then there is a number
q
> 0 such that 
y
^{(n)}
(
t
) ≤
q
for all
n
and a.e.
t
∈
J
, so
y
^{(n)}
∈
B_{q}
and
y
∈
B_{q}
. In view of (5), we have
By (
H
3), (
H
5) and Definition 2.2,
We have by the dominated convergence theorem that
Thus
is continuous.
Next we show that
maps
B_{q}
into a precompact in
X
. Let 0 <
t
≤
b
be fixed and
ϵ
be a real number satisfying 0 <
ϵ
<
t
. For
y
∈
B_{q}
, we define the operators
From Theorem 2.1 and the compactness of the operator
R
(
ϵ
), the set
is precompact in
X
, for every
ϵ
, 0 <
ϵ
<
t
. Moreover, by Theorem 2.1 and for each
y
∈
B_{q}
, we have
So the set
is precompact in
X
by using the total boundedness. Applying this idea again and observing that
Therefore,
and there are precompact sets arbitrarily close to the set
. Thus the set
is precompact in
X
.
Therefore from ArzelaAscoli theorem, we can conclude that the operator
is completely continuous. In order to study the existence results for the problem (1)(2), we introduce a parameter
λ
∈ (0, 1) and consider the following nonlinear operator equation
where Փ is already defined. The following lemma proves that an a priori bound exists for the solution of the above equation.
Lemma 3.2.
If hypotheses (H1)(H6) are satisfied, then there exists an a priori bound K
> 0
such that
∥
x_{t}
∥
_{Bh}
≤
K, t
∈
J, where K depends only on b and on the functions
Proof
. From the equation (6), we have
Thus from this proof and Lemma 2.4 it follows that
Let
μ
(
t
) = sup{∥
x_{s}
∥
_{Bh}
: 0 ≤
s
≤
t
}, then the function
μ
(
t
) is nondecreasing in
J
, and we have
By using lemma 2.5, we have
where
Let us take the right hand side of the above inequality as
v
(
t
). Then
v
(0) =
B
_{0}
K
_{1}
,
μ
(
t
) ≤
v
(
t
), 0 ≤
t
≤
b
and
Since Ψ and Ω are nondecreasing.
Let
Then
w
(0) =
v
(0) and
v
(
t
) ≤
w
(
t
).
This implies that
This implies that
v
(
t
) < ∞. So there is a constant
K
such that
v
(
t
) ≤
K, t
∈
J
. So ∥
x_{t}
∥
_{Bh}
≤
μ
(
t
) ≤
v
(
t
) ≤
K, t
∈
J
, where
K
depends only on
b
and on the functions
□
Theorem 3.3.
Assume that the hypotheses (H1)(H6) hold. Then the problem (1)(2) has at least one mild solution on J.
Proof
. Let us take the set
Then for any
we have by Lemma 3.2 that ∥
x_{t}
∥
_{Bh}
≤
K, t
∈
J
, and we have
which implies that the set
G
is bounded on
J
.
Consequently, by KrasnoselskiSchaefer type fixed point theorem and Lemma 3.2 the operator
has a fixed point
Then
x
is a fixed point of the operator Փ which is a mild solution of the problem (1)(2). □
4. Example
Consider the following partial neutral integrodifferential equation of the form
where
ϕ
∈
B_{h}
. We take
X
=
L
^{2}
[0,
π
] with the norm  · 
_{L2}
and define
A
:
X
→
X
by
Aw
=
w′′
with the domain
D
(
A
) = {
w
∈
X
:
w,w′
are absolutely continuous,
w′′
∈
X, w
(0) =
w
(
π
) = 0}.Then
where
n
= 1, 2, . . . .. is the orthogonal set of eigen vectors of
A
. It is well known that
A
generates a strongly continuous semigroup that is analytic, and resolvent operator
R
(
t
) can be extracted this analytic semigroup and given by
Since the analytic semigroup
R
(
t
) is compact, there exists a constant
M
_{1}
> 0 such that ∥
R
(
t
)∥ ≤
M
_{1}
. Especially, the operator (−
A
)
^{½}
is given by
with the domain
Let
and define
Hence for (
t, ϕ
) ∈ [0,
b
] ×
B_{h}
, where
ϕ
(
θ
)(
x
) =
ϕ
(
θ, x
), (
θ, x
) ∈ (−∞, 0] × [0,
π
]. Set
and
where
Then, the system (7)(9) is the abstract formulation of the system (1)(2). Further, we can impose some suitable conditions on the above defined functions to verify the assumptions on Theorem 3.3. We can conclude that system (7)(9) has at least one mild solution on
J
.
BIO
S. Chandrasekaran
Department of Mathematics, SNS College of Technology, Coimbatore  641 035, Tamil Nadu, India.
email: chandrusavc@gmail.com
S. Karunanithi
Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore  641 029, Tamil Nadu, India.
email: sknithi1957@yahoo.co.in
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