# Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

Authors: Alka Chadha; Dwijendra N. Pandey
Journal: International Journal of Differential Equations
 (2014)

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## Abstract

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach spaceX. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.


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## Body

## 1. Introduction

In recent years, fractional calculus has becomes an active area of research due to its demonstrated applications in widespread fields of science and engineering such as mechanics, electrical engineering, medicine, biology, ecology, and many others. The memory and hereditary properties of various materials and processes can be described by a differential equation with fractional order. The fractional differential equation also describes the efficiency of nonlinear oscillations of the earthquake. The details on the theory and its applications can be found in [1–4] and references given therein.On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, and medicine. In addition, the improvement of the hypothesis of the functional differential equation with infinite delay relies on a choice of phase space. There are various phase spaces which have been studied. Hale and Kato in [5] introduced a common phase space 𝒫. For more details on phase space, we refer to books by Hale and Kato [5], Hino et al. [6] and papers [7–10]. For the study of impulsive differential equation, we refer to papers [7, 8, 11–18] and references given therein.The purpose of this work is to establish the existence of mild solution for impulsive fractional differential equation with infinite delay:(1)
D
t
q
[
u
(
t
)
+
g
(
t
,
u
t
)
]
=
A
[
u
(
t
)
+
g
(
t
,
u
t
)
]
+
ℐ
t
1
-
q
f
(
t
,
u
t
)
,
t
∈
J
=
[
0
,
T
]
,
t
≠
t
i
u
0
=
φ
∈
𝒫
,
Δ
u
(
t
i
)
=
I
i
(
u
t
i
)
,
i
=
1,2
,
…
,
n
,
n
∈
ℕ
,
where 0
<
T
<
∞, 0
<
q
<
1, A
:
D
(
A
)
⊂
X
→
X is a closed and densely defined linear operator and infinitesimal generator of a solution (resolvent) operator {
S
q
(
t
)
}
t
≥
0 on Banach space X, and D
t
q denotes the fractional derivative in Caputo sense and ℐ
t
1
-
q denotes the Riemann-Liouville fractional integral operator. The history u
t
:
(
-
∞
,
0
]
→
X defined by u
t
(
s
)
=
u
(
t
+
s
) for s
∈
(
-
∞
,
0
] belongs to some abstract phase space 𝒫 defined axiomatically and I
i
∈
C
(
X
,
X
)
(
i
=
1
,
…
,
n
); 0
≤
t
0
<
t
1
<
⋯
<
t
n
≤
t
n
+
1
=
T are fixed numbers and Δ
u
(
t
) denotes the jump of the function u at the point t, given by Δ
u
(
t
)
=
u
(
t
+
)
-
u
(
t
-
). The functions f
,
g
:
J
×
𝒫
→
X are appropriate functions and satisfy some conditions to be specified later.In [13], authors have considered the following impulsive fractional differential equation in a Banach space of the form
(2)
D
t
q
u
(
t
)
=
A
u
(
t
)
+
f
(
t
,
u
(
t
)
)
,
q
∈
(
0,1
]
,
t
∈
J
=
[
0
,
T
]
,
t
≠
t
i
,
u
(
0
)
=
u
0
,
u
(
t
i
+
)
=
u
(
t
i
-
)
+
y
i
,
i
=
1,2
,
…
,
n
,
n
∈
ℕ
,
where A
:
D
(
A
)
⊂
X
→
X is the infinitesimal generator of a C
0-semigroup {
T
(
t
)
:
t
≥
0
} on a Banach space, f
:
J
×
X
→
X is continuous, and u
0, y
i are the element of X. Authors have established some existence and uniqueness results for system (2) under the different assumptions on initial conditions.In this work, we adopt the idea of Wang et al. [13] and establish the existence of a mild solution for the problem (1) by using the measure of noncompactness and solution operator. The tool of measure of noncompactness has been used in linear operator theory, theory of differential and integral equations, the fixed point theory, and many others. For an initial study of theory of the measure of noncompactness, we refer to book of Banaś and Goebel [19] and Akhmerov et al. [20] and papers [9, 21–25] and references given therein.This paper is organized as follows: In Section2 we recall some basic definitions, lemmas, and theorems. We will prove the existence of a mild solution for the system (1) in Section 3. In the last section, we shall discuss an example to illustrate the application of the abstract results.


## 2. Preliminaries

Now we provide some basic definitions, notations, theorems, lemmas, and preliminary facts which will be used throughout this paper.LetX be a Banach space and let C
(
[
0
,
T
]
;
X
) be the Banach space of continuous functions u
(
t
) from [
0
,
T
] to X equipped with the norm ∥
y
∥
C
=
sup
⁡
t
∈
[
0
,
T
]
∥
y
(
t
)
∥
X and L
p
(
(
0
,
T
)
;
X
) denotes the Banach space of all Bochner-measurable functions from (
0
,
T
) to X with the norm
(3)
∥
u
∥
L
p
=
(
∫
(
0
,
T
)
∥
u
(
s
)
∥
X
p
d
s
)
1
/
p
.Assume that0
∈
ρ
(
A
), that is, A is invertible. Then, this permits us to define the positive fractional power A
α as closed linear operator with domain D
(
A
α
)
⊆
H for α
∈
(
0,1
]. Moreover, D
(
A
α
) is dense in H with the norm
(4)
∥
y
∥
α
=
∥
A
α
y
∥
.
It is easy to see that D
(
A
α
) which is dense in X is a Banach space. Henceforth, we use X
α as notation of D
(
A
α
). Also, we have that X
κ
↪
X
α for 0
<
α
<
κ and, therefore, the embedding is continuous. Then, we define X
-
α
=
(
X
α
)
*, for each α
>
0. The space X
-
α, standing for the dual space of X
α, is a Banach space with the norm ∥
z
∥
-
α
=
∥
A
-
α
z
∥ for z
∈
X
-
α. For more details on the fractional powers of closed linear operators, we refer to book by Pazy [26].To consider the mild solution for the impulsive problem, we propose that the set𝒫
𝒞
(
[
0
,
T
]
;
X
)
=
{
u
:
[
0
,
T
]
→
X
:
u
is
continuous
at
t
≠
t
i
and
left
continuous
at
t = t
i
and
u
(
t
i
+
)
exists
,
for
all
i
=
1
,
…
,
m
}. Clearly, 𝒫
𝒞
(
[
0
,
T
]
;
X
) is a Banach space endowing the norm ∥
u
∥
𝒫
𝒞
=
sup
⁡
t
∈
[
0
,
T
]
∥
u
(
s
)
∥. For a function u
∈
𝒫
𝒞
(
[
0
,
T
]
;
X
) and i
∈
{
0,1
,
…
,
m
}, we define the function u
i
~
∈
C
(
[
t
i
,
t
i
+
1
]
,
X
) such that
(5)
u
i
~
(
t
)
=
{
u
(
t
)
,
for
t
∈
(
t
i
,
t
i
+
1
]
,
u
(
t
i
+
)
,
for
t
=
t
i
.
For W
⊂
𝒫
𝒞
(
[
0
,
T
]
;
X
) and i
∈
{
0,1
,
…
,
m
}, we have W
i
~
=
{
u
i
~
:
u
∈
W
} and following Accoli-Arzelà type criteria.Lemma 1.
A setB
⊂
𝒫
𝒞
(
[
0
,
T
]
;
X
) is relatively compact in 𝒫
𝒞
(
[
0
,
T
]
;
X
) if and only if each set B
j
~
(
j
=
1,2
,
…
,
m
) is relatively compact in C
(
[
t
j
,
t
j
+
1
]
,
X
)
(
j
=
0,1
,
…
,
m
).For the differential equation with infinite delay, Hale and Kato [5] proposed the phase space 𝒫 satisfying certain fundamental axioms.Definition 2 (see [6]).
A phase space𝒫 is a linear space which contains all the functions mapping (
-
∞
,
0
] into Banach space X with a seminorm ∥
·
∥
𝒫. The fundamental axioms assumed on 𝒫 are the following,(A)
Ifu
:
(
-
∞
,
a
+
T
]
→
X, T
>
0 is a continuous function on [
a
,
a
+
T
] such that u
a
∈
𝒫 and u
|
[
a
,
a
+
T
]
∈
𝒫
∈
𝒫
𝒞
(
[
a
,
a
+
T
]
;
X
), then for every t
∈
[
a
,
a
+
T
), the following conditions hold:
(i)
u
t
∈
𝒫,
(ii)
H
∥
u
t
∥
𝒫
≥
∥
u
(
t
)
∥,
(iii)
∥
u
t
∥
𝒫
≤
N
(
t
+
a
)
∥
u
a
∥
𝒫
+
K
(
t
-
a
)
sup
⁡
{
∥
u
(
s
)
∥
:
a
≤
s
≤
t
}.
WhereH is a positive constant, N
,
K
:
[
0
,
∞
)
→
[
0
,
∞
), N is a locally bounded, K is continuous, and H, N, K are independent of u
(
·
).
(A1)
For the functionu in (
A
1
), u
t is a 𝒫-valued continuous function for t
∈
[
a
,
a
+
T
].
(B)
The space𝒫 is complete.Now, we state some basic definitions and properties of fractional calculus.Mittag-Leffler. The definition of one parameter Mittag-Leffler function is given as
(6)
E
α
(
z
)
=
∑
k
=
0
∞
‍
z
k
Γ
(
α
k
+
1
)
,
and two-parameter Mittag-Leffler function is defined as
(7)
E
α
,
β
(
z
)
=
∑
k
=
0
∞
z
k
Γ
(
α
z
+
β
)
=
1
2
π
i
∫
C
μ
α
-
β
e
μ
μ
a
-
z
d
μ
,
α
,
β
>
0
,
z
∈
ℂ
,
where C is a contour which starts and ends at -
∞ and encircles the disc |
μ
|
≤
|
z
|
1
/
2 counter clockwise. The Laplace transform of the Mittag-leffler is defined as
(8)
L
(
t
β
-
1
E
α
,
β
(
-
ρ
α
t
α
)
)
=
λ
α
-
β
λ
α
+
ρ
α
,
Re
λ
>
ρ
1
/
α
,
ρ
>
0
.
For more details we refer to [1].Laplace transform of integer order derivatives is defined as(9)
L
[
F
n
(
t
)
;
λ
]
=
λ
n
L
[
F
(
t
)
]
-
∑
k
=
0
n
-
1
‍
λ
n
-
k
-
1
F
k
(
0
)
=
λ
n
L
[
F
(
t
)
]
-
∑
k
=
0
n
-
1
‍
λ
k
F
n
-
k
-
1
(
0
)
.Definition 3.
The Riemann-Liouville fractional integral operatorℐ is defined as
(10)
ℐ
t
q
F
(
t
)
=
1
Γ
(
q
)
∫
0
t
(
t
-
s
)
q
-
1
F
(
s
)
d
s
,
where F
∈
L
1
(
(
0
,
T
)
;
X
) and q
>
0 is the order of the fractional integration.Definition 4.
The Riemann-Liouville fractional derivative is given as(11)
D
t
q
F
(
t
)
=
D
t
n
ℐ
t
n
-
q
F
(
t
)
,
n
-
1
<
q
<
n
,
n
∈
ℕ
,
where D
t
n
=
d
n
/
d
t
n, F
∈
L
1
(
(
0
,
T
)
;
X
), and ℐ
t
n
-
q
F
∈
W
n
,
1
(
(
0
,
T
)
;
X
). Here, the notation W
n
,
1
(
(
0
,
T
)
;
X
) stands for the Sobolev space defined as
(12)
W
n
,
1
(
(
0
,
T
)
;
X
)
=
{
∑
k
=
0
n
-
1
‍
y
∈
X
:
∃
z
∈
L
1
(
(
0
,
T
)
;
X
)
:
y
(
t
)
=
∑
k
=
0
n
-
1
‍
d
k
t
k
k
!
+
t
n
-
1
(
n
-
1
)
!
*
z
(
t
)
,
t
∈
(
0
,
T
)
}
.
Note that z
(
t
)
=
y
n
(
t
) and d
k
=
y
k
(
0
).Definition 5.
The Caputo fractional derivative is given as(13)
D
t
α
F
(
t
)
=
1
Γ
(
n
-
α
)
∫
0
t
(
t
-
s
)
n
-
α
-
1
F
n
(
s
)
d
s
,
n
-
1
<
α
<
n
,
where F
∈
C
n
-
1
(
(
0
,
T
)
;
X
)
∩
L
1
(
(
0
,
T
)
;
X
) and the following holds
(14)
ℐ
t
q
(
D
t
q
F
(
t
)
)
=
F
(
t
)
-
∑
k
=
0
n
-
1
‍
t
k
k
!
F
k
(
0
)
.Definition 6 (see [27]).
A family{
S
q
(
t
)
}
t
≥
0
⊂
ℒ
(
X
) of bounded linear operators in X is called a resolvent (or solution operator) generating by A if the following conditions are fulfilled:(1)
S
q
(
t
) is strongly continuous on ℝ
+ and S
q
(
0
)
=
I;
(2)
forx
∈
D
(
A
) and t
≥
0, S
q
(
t
)
D
(
A
)
⊂
D
(
A
) and A
S
q
(
t
)
x
=
S
q
(
t
)
A
x;
(3)
S
q
(
t
)
x is the solution of the equation
(15)
u
(
t
)
=
x
+
1
Γ
(
q
)
∫
0
t
(
t
-
s
)
q
-
1
A
u
(
s
)
d
s
,
∀
x
∈
D
(
A
)
,
t
≥
0
,where ℒ
(
X
) denotes the space of all bounded linear operators from X into X endowed with the norm of operators.Also, the solution operatorS
q
(
t
) for (15) is defined as (see [27])
(16)
λ
q
-
1
(
λ
q
I
-
A
)
-
1
x
=
∫
0
∞
e
-
λ
t
S
q
(
t
)
x
d
t
,
Re
λ
>
ω
,
x
∈
X
,
where ω
≥
0 and {
λ
q
:
Re
λ
>
ω
}
⊂
ρ
(
A
).Let(17)
∑
‍
(
ω
,
θ
)
=
{
λ
∈
ℂ
:
|
arg
(
λ
-
ω
)
|
<
θ
}
.Definition 7 (see [27]).
A solution operatorS
q
(
t
) is said to be analytic if S
q
(
·
)
:
ℝ
+
→
ℒ
(
X
) admits analytic extension to a sector ∑
‍
(
0
,
θ
0
) for some 0
<
θ
0
≤
π
/
2. Furthermore, An analytic resolvent S
q
(
t
) is said to be of analyticity type (
ω
0
,
θ
0
) if, for θ
0
>
θ and ω
0
<
ω, there exists M
=
M
(
ω
,
θ
) such that ∥
S
q
(
t
)
∥
≤
M
e
ω
Re
z for z
∈
∑
‍
(
0
,
θ
); here Re
z means the real part of z.In this work, we assume that solution operator{
S
q
(
t
)
}
t
≥
0 is analytic; that is, {
S
q
(
t
)
}
t
≥
0 satisfy the following property.(
HA
)
The mapt
↦
S
q
(
t
) is continuous from [
0
,
T
] to ℒ
(
X
) endowed with the uniform operator norm ∥
·
∥
ℒ
(
X
).Without loss of generality, we have that there exist a positive constantM such that ∥
S
q
(
t
)
∥
≤
M, for t
≥
0.Definition 8 (see [19]).
The Hausdorff measure of noncompactnessχ
Z is defined as
(18)
χ
Z
(
F
)
=
inf
⁡
{
2
3
ϵ
>
0
:
F
can
be
covered
by
finite
number
of
balls
with
radius
ϵ
2
3
}
,
for bounded set F
⊂
Z, where Z is a Banach space.Lemma 9 (see [19]).
For any bounded setU
,
V
⊂
Y, where Y is a Banach space. Then, the following properties are fulfilled:(i)
χ
Y
(
U
)
=
0 if and only if U is pre-compact;
(ii)
χ
Y
(
U
)
=
χ
Y
(
conv
U
)
=
χ
Y
(
U
¯
), where conv
U and U
¯ denotes the convex hull and closure of U, respectively;
(iii)
χ
Y
(
U
)
⊂
χ
Y
(
V
), when U
⊂
V;
(iv)
χ
Y
(
U
+
V
)
≤
χ
Y
(
U
)
+
χ
Y
(
V
), where U
+
V
=
{
u
+
v
:
u
∈
U
,
v
∈
V
};
(v)
χ
Y
(
U
∪
V
)
≤
max
⁡
{
χ
Y
(
U
)
,
χ
Y
(
V
)
};
(vi)
χ
Y
(
λ
U
)
=
λ
·
χ
Y
(
U
), for any λ
∈
ℝ;
(vii)
if the mapP
:
D
(
P
)
⊂
Y
→
𝒵 is continuous and satisfies the Lipschitsz condition with constant κ, then we have that χ
𝒵
(
P
U
)
≤
κ
χ
Y
(
U
) for any bounded subset U
⊂
D
(
P
), where Y and 𝒵 are Banach space.The details on the measure of noncompactness and its applications can be found in a book by Banaś and Goebel [19] and papers [9, 10, 21, 23, 24].Lemma 10 (see [19]).
A bounded and continuous mapQ
:
D
⊂
Z
→
Z is a χ
Z-contraction if there exists a constant 0
<
κ
<
1 such that χ
Z
(
Q
(
U
)
)
≤
κ
χ
Z
(
U
), for any bounded closed subset U
⊂
D, where Z is a Banach space.Lemma 11 (see [28]).
LetD
⊂
Z be closed and convex with 0
∈
D and let the continuous map Q
:
D
→
D be a χ
Z-contraction. If the set {
u
∈
D
:
u
=
λ
Q
u
,
for
0
<
λ
<
1
} is bounded, then the map Q has a fixed point in D.Lemma 12 ((Darbo-Sadovskii) [19]).
LetD
⊂
Z be bounded, closed, and convex. If the continuous map Q
:
D
→
D is a χ
Z-contraction, then the map Q has a fixed point in D.In this work, we consider thatχ denotes the Hausdorff measure of noncompactness in X, χ
C denotes the Hausdorff measure in noncompactness of C
(
[
0
,
T
]
;
X
) and χ
𝒫
𝒞 denotes the Hausdorff measure of noncompactness in 𝒫
𝒞
(
[
0
,
T
]
;
X
).Lemma 13 (see [19, 21]).
IfU is bounded subset of C
(
[
0
,
T
]
;
X
). Then, one has that χ
(
U
(
t
)
)
≤
χ
C
(
U
), for
all
t
∈
[
0
,
T
], where U
(
t
)
=
{
u
(
t
)
;
u
∈
U
}
⊆
X. Furthermore, if U is equicontinuous on [
0
,
T
], then χ
(
U
(
t
)
) is continuous on the interval [
0
,
T
] and
(19)
χ
C
(
U
)
=
sup
⁡
t
∈
[
0
,
T
]
{
χ
(
U
(
t
)
)
}
.Lemma 14 (see [19]).
IfU
⊂
C
(
[
0
,
T
]
;
X
) is bounded and equicontinuous set, then χ
(
U
(
t
)
) is continuous and
(20)
χ
(
∫
0
t
U
(
s
)
d
s
)
≤
∫
0
t
χ
(
U
(
s
)
)
d
s
,
∀
t
∈
[
0
,
T
]
,
where ∫
0
t
‍
U
(
s
)
d
s
=
{
∫
0
t
‍
u
(
s
)
d
s
,
u
∈
U
}.Lemma 15 (see [29]).
(1) IfU
⊂
𝒫
𝒞
(
[
0
,
T
]
;
X
) is bounded, then χ
(
U
(
t
)
)
≤
χ
𝒫
𝒞
(
U
), for
all
t
∈
[
0
,
T
], where U
(
t
)
=
{
u
(
t
)
:
u
∈
U
}
⊂
X.
(2) IfU is piecewise equicontinuous on [
0
,
T
], then χ
(
U
(
t
)
) is piecewise continuous for t
∈
[
0
,
T
] and
(21)
χ
𝒫
𝒞
(
U
)
=
sup
⁡
{
χ
(
U
(
t
)
)
:
t
∈
[
0
,
T
]
}
.
(3) IfU
⊂
𝒫
𝒞
(
[
0
,
T
]
;
X
) is bounded and equicontinuous, then χ
(
U
(
t
)
) is piecewise continuous for t
∈
[
0
,
T
] and
(22)
χ
(
∫
0
t
U
(
s
)
d
s
)
≤
∫
0
t
χ
(
U
(
s
)
)
d
s
,
∀
t
∈
[
0
,
T
]
,
where ∫
0
t
‍
U
(
s
)
d
s
=
{
∫
0
t
‍
u
(
s
)
d
s
:
u
∈
U
}.


## 3. Main Results

In this section, we will establish the existence results of solution for (1) by using solution operator and Hausdorff's measure of noncompactness.From [13], we adopt the following concept of solution for impulsive differential problem (1).Definition 16.
A piecewise continuous functionu
:
(
-
∞
,
T
]
→
X is said to be a mild solution for impulsive problem (1) if u
0
=
φ, u
(
·
)
|
J
∈
𝒫
𝒞 and
(23)
u
(
t
)
=
{
S
q
(
t
)
[
φ
(
0
)
+
g
(
0
,
φ
)
]
-
g
(
t
,
u
t
)
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
)
d
s
,
t
∈
[
0
,
t
1
]
,
S
q
(
t
)
[
φ
(
0
)
+
g
(
0
,
φ
)
]
-
g
(
t
,
u
t
)
+
S
q
(
t
-
t
1
)
I
1
(
u
t
1
)
+
S
q
(
t
-
t
1
)
×
[
g
(
t
1
,
u
t
1
+
I
1
(
u
t
1
)
)
-
g
(
t
1
,
u
t
1
)
]
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
)
d
s
,
t
∈
(
t
1
,
t
2
]
,
⋮
S
q
(
t
)
[
φ
(
0
)
+
g
(
0
,
φ
)
]
-
g
(
t
,
u
t
)
+
∑
i
=
1
n
‍
S
q
(
t
-
t
i
)
I
1
(
u
t
i
)
+
∑
i
=
1
n
‍
S
q
(
t
-
t
i
)
×
[
g
(
t
i
,
u
t
i
+
I
i
(
u
t
i
)
)
-
g
(
t
i
,
u
t
i
)
]
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
)
d
s
,
t
∈
(
t
n
,
T
]
,
where
(24)
S
q
(
t
)
=
1
2
π
i
∫
Γ
‍
e
λ
t
λ
q
-
1
R
(
λ
q
,
A
)
d
λ
.Now we list the following assumptions which are required to establish main results.(Hf)
The functionf
:
J
×
𝒫
→
X satisfies the following conditions:
(1)
the functionf
(
·
,
u
)
:
J
→
X is strongly measurable for every u
∈
𝒫 and u
0
∈
𝒫, u
|
J
∈
𝒫
𝒞;
(2)
f
(
t
,
·
)
:
𝒫
→
X is a continuous function for each t
∈
J;
(3)
there exists an integrable functionm
f
:
[
0
,
T
]
→
[
0
,
∞
) and a nondecreasing continuous function Ω
:
[
0
,
∞
)
→
(
0
,
∞
) such that
(25)
∥
f
(
τ
,
x
)
∥
≤
m
f
(
τ
)
Ω
(
∥
x
∥
𝒫
)
,
τ
∈
J
,
x
∈
𝒫
;
(4)
there exists an integrable functionη
:
[
0
,
T
]
→
[
0
,
∞
) such that, for any bounded set B
⊂
𝒫, we have
(26)
χ
(
S
q
(
τ
)
f
(
τ
,
B
)
)
≤
η
(
τ
)
sup
⁡
-
∞
≤
θ
≤
0
χ
(
B
(
θ
)
)
;
for almost everywhereτ
∈
J, where B
(
θ
)
=
{
u
(
θ
)
:
u
∈
B
}.
(Hg)
(1) For0
<
β
<
1, A
β
g
(
·
,
·
) is Lipschitz continuous function for all (
t
,
v
)
∈
J
×
𝒫 and there exist positive constants C
1 and C
2 such that
(27)
∥
A
β
g
(
t
,
u
)
∥
≤
C
1
∥
u
∥
𝒫
+
C
2
;
(2)
there exists a constantL
g
>
0 such that
(28)
∥
A
β
g
(
t
,
u
)
-
A
β
g
(
t
,
v
)
∥
≤
L
g
∥
u
-
v
∥
𝒫
,
for allu
,
v
∈
𝒫.
(HI)
The functionsI
i
:
X
→
X
,
(
i
=
1
,
…
,
n
) are continuous functions and satisfy the following conditions:
(1)
There is a constantL
I
>
0 such that
(29)
∥
I
i
(
u
)
-
I
i
(
v
)
∥
≤
L
I
∥
u
-
v
∥
𝒫
,
∀
u
,
v
∈
𝒫
.
(2)
There exist positive constantsL
j
(
j
=
1,2
) such that
(30)
∥
I
i
(
u
)
∥
=
L
1
∥
u
∥
𝒫
+
L
2
,
for allu
∈
𝒫.
(H1)
(31)
K
T
M
1
-
ζ
2
∫
0
t
m
f
(
s
)
d
s
<
∫
d
∞
‍
d
s
Ω
(
s
)
,
where(32)
ζ
1
=
(
M
K
T
H
+
N
T
+
M
C
1
∥
A
-
β
∥
K
T
)
∥
φ
∥
𝒫
+
K
T
C
2
∥
A
-
β
∥
(
M
+
1
)
+
K
T
(
n
M
L
2
(
1
+
L
g
)
)
,
ζ
2
=
[
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
]
K
T
<
1
,
d
=
ζ
1
1
-
ζ
2
;
(H2)
(33)
K
T
[
∥
A
-
β
∥
L
g
+
n
M
L
I
+
M
n
L
g
(
2
+
L
I
)
]
+
∫
0
t
‍
η
(
s
)
d
s
<
1
.Now, letz
:
(
-
∞
,
T
]
→
X be a function given by z
0
=
φ and z
(
t
)
=
S
q
(
t
)
φ
(
0
) on J. It is easy to see that ∥
z
t
∥
𝒫
≤
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫, where K
T
=
sup
⁡
0
≤
t
≤
T
K
(
t
), N
T
=
sup
⁡
0
≤
t
≤
T
N
(
t
).Theorem 17.
Suppose that hypotheses(
H
A
), (
H
f
), (
H
g
), (
H
I
), and (
H
1
)
-
(
H
2
) are satisfied. Then, there exists a mild solution for the impulsive problem (1).Proof.
Consider the spaceS
(
T
)
=
{
u
:
(
-
∞
,
T
]
→
X
:
u
0
=
0
,
u
|
J
∈
𝒫
} endowed with supremum norm ∥
·
∥
𝒫. Define the operator Q
:
S
(
T
)
→
S
(
T
) by
(34)
(
Q
x
)
(
t
)
=
{
0
,
t
∈
(
-
∞
,
0
]
S
q
(
t
)
g
(
0
,
φ
)
-
g
(
t
,
u
t
+
z
t
)
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
+
z
s
)
d
s
,
t
∈
[
0
,
t
1
]
,
S
q
(
t
)
g
(
0
,
φ
)
-
g
(
t
,
u
t
+
z
t
)
+
S
q
(
t
-
t
1
)
I
1
(
u
t
1
+
z
t
1
)
+
S
q
(
t
-
t
1
)
×
[
g
(
t
1
,
u
t
1
+
z
t
1
+
I
1
(
u
t
1
+
z
t
1
)
)
-
g
(
t
1
,
u
t
1
+
z
t
1
)
]
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
+
z
s
)
d
s
,
t
∈
(
t
1
,
t
2
]
,
⋮
S
q
(
t
)
g
(
0
,
φ
)
-
g
(
t
,
u
t
+
z
t
)
+
∑
i
=
1
n
‍
S
q
(
t
-
t
i
)
I
1
(
u
t
i
+
z
t
i
)
+
∑
i
=
1
n
‍
S
q
(
t
-
t
i
)
×
[
g
(
t
i
,
u
t
i
+
z
t
i
+
I
i
(
u
t
i
+
z
t
i
)
)
-
g
(
t
i
,
u
t
i
+
z
t
i
)
]
+
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
+
z
s
)
d
s
,
t
∈
(
t
n
,
T
]
,
and we have that
(35)
∥
u
t
+
z
t
∥
𝒫
≤
∥
u
t
∥
𝒫
+
∥
z
t
∥
𝒫
,
≤
K
(
t
)
sup
⁡
{
∥
u
(
s
)
∥
:
0
≤
s
≤
t
}
+
N
(
t
)
∥
u
0
∥
𝒫
+
K
(
t
)
sup
⁡
{
∥
z
(
s
)
∥
:
0
≤
s
≤
t
}
+
N
(
t
)
∥
z
0
∥
𝒫
,
≤
K
(
t
)
∥
u
∥
t
+
K
(
t
)
M
H
∥
φ
∥
𝒫
+
N
(
t
)
∥
φ
∥
𝒫
,
≤
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
∥
u
∥
t
,
where ∥
u
∥
t
=
sup
⁡
0
≤
s
≤
t
∥
u
(
s
)
∥. Thus, Q is well defined and with the values in S
(
T
) by our assumptions. By Lebesgue dominated convergence theorem, axioms of phase space and assumptions (
Hf
), (
Hg
), and (
HI
), it is clear that Q is continuous map. Furthermore, by uniformly continuity of the map t
↦
S
q
(
t
) on (
0
,
T
], we obtain that set Q
(
S
(
T
)
) is equicontinuous. We prove the result in following steps.
Step
1. The set {
u
∈
𝒫
𝒞
:
u
=
λ
Q
u
,
for
0
<
λ
<
1
} is bounded.
Letu
λ be a solution of u
=
λ
Q
u for 0
<
λ
<
1. Therefore, we have that
(36)
∥
u
λ
t
+
z
t
∥
𝒫
≤
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
∥
u
λ
∥
t
.
Take v
λ
(
t
)
=
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
∥
u
λ
∥
t, for t
∈
J. Then, we get that for t
∈
[
0
,
t
1
]
(37)
∥
u
λ
(
t
)
∥
=
∥
λ
Q
u
λ
(
t
)
∥
≤
∥
Q
u
λ
(
t
)
∥
,
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
(
C
1
v
λ
(
t
)
+
C
2
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
.
For t
∈
(
t
1
,
t
2
], we have
(38)
∥
u
λ
(
t
)
∥
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
(
C
1
v
λ
(
t
)
+
C
2
)
+
M
∥
I
1
(
v
λ
(
t
1
)
)
∥
+
M
∥
g
(
t
1
,
v
λ
(
t
1
)
+
I
1
(
v
λ
(
t
1
)
)
)
-
g
(
t
1
,
v
λ
(
t
1
)
)
∥
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
,
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
(
C
1
v
λ
(
t
)
+
C
2
)
+
M
(
L
1
(
v
λ
(
t
)
)
+
L
2
)
+
M
L
g
(
L
1
v
λ
(
t
)
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
.
For t
∈
(
t
n
,
T
],
(39)
∥
u
λ
(
t
)
∥
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
(
C
1
v
λ
(
t
)
+
C
2
)
+
n
M
(
L
1
(
v
λ
(
t
)
)
+
L
2
)
+
n
M
L
g
(
L
1
v
λ
(
t
)
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
.
Therefore, for all t
∈
[
0
,
T
]
=
J, we have
(40)
∥
u
λ
(
t
)
∥
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
(
C
1
v
λ
(
t
)
+
C
2
)
+
n
M
L
1
(
1
+
L
g
)
v
λ
(
t
)
+
n
M
L
2
(
1
+
L
g
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
,
≤
M
∥
A
-
β
∥
(
C
1
∥
φ
∥
𝒫
+
C
2
)
+
∥
A
-
β
∥
C
2
+
n
M
L
2
(
1
+
L
g
)
+
(
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
)
v
λ
(
t
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
.
From v
λ
(
t
)
=
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
∥
u
λ
∥
t, it implies that
(41)
v
λ
(
t
)
≤
(
M
K
T
H
+
N
T
+
M
∥
A
-
β
∥
C
1
K
T
)
∥
φ
∥
𝒫
+
K
T
∥
A
-
β
∥
C
2
(
M
+
1
)
+
K
T
(
n
M
L
2
(
1
+
L
g
)
)
+
K
T
(
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
)
v
λ
(
t
)
+
M
K
T
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
,
and consequently,
(42)
v
λ
(
t
)
≤
ζ
1
1
-
ζ
2
+
M
K
T
1
-
ζ
2
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
,
=
d
+
M
K
T
1
-
ζ
2
∫
0
t
m
f
(
s
)
Ω
(
v
λ
(
s
)
)
d
s
,
where
(43)
ζ
1
=
(
M
K
T
H
+
N
T
+
M
C
1
∥
A
-
β
∥
K
T
)
∥
φ
∥
𝒫
+
K
T
∥
A
-
β
∥
C
2
(
M
+
1
)
+
n
K
T
M
L
2
(
1
+
L
g
)
,
ζ
2
=
K
T
[
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
]
.
Let ξ
λ
(
t
)
=
d
+
(
M
K
T
/
(
1
-
ζ
2
)
)
∫
0
t
‍
m
f
(
s
)
Ω
(
ξ
λ
(
s
)
)
d
s. Thus ξ
λ
(
0
)
=
d. Therefore, we get
(44)
ξ
λ
′
(
t
)
Ω
(
ξ
λ
(
t
)
)
≤
K
T
M
1
-
ζ
2
m
f
(
t
)
.
Integrating above inequality we have that
(45)
∫
d
ξ
λ
(
t
)
‍
d
s
Ω
(
s
)
≤
K
T
M
1
-
ζ
2
∫
0
t
‍
m
f
(
s
)
d
s
<
∫
d
∞
‍
d
s
Ω
(
s
)
.
It gives that the functions ξ
λ
(
t
) are bounded on interval J. Therefore, the functions v
λ
(
t
) are bounded and u
λ
(
·
) are also bounded on J.
Step
2. The map Q is a χ-contraction.
Firstly, we introduce decomposition ofQ into Q
=
Q
1
+
Q
2, for t
≥
0 such that
(46)
Q
1
u
(
t
)
=
{
S
q
(
t
)
g
(
0
,
φ
)
-
g
(
t
,
u
t
+
z
t
)
,
t
∈
[
0
,
t
1
]
,
S
q
(
t
)
g
(
0
,
φ
)
-
g
(
t
,
u
t
+
z
t
)
+
∑
i
=
1
m
‍
S
q
(
t
-
t
i
)
I
i
(
u
t
i
+
z
t
i
)
+
∑
i
=
1
m
‍
S
q
(
t
-
t
i
)
×
[
g
(
t
i
,
u
t
i
+
z
t
i
+
I
i
(
u
t
i
+
z
t
i
)
)
-
g
(
t
i
,
u
t
i
+
z
t
i
)
]
,
t
∈
(
t
m
,
t
m
+
1
]
,
m
=
1,2
,
3
,
…
,
n
;
Q
2
u
(
t
)
=
∫
0
t
‍
S
q
(
t
-
s
)
f
(
s
,
u
s
+
z
s
)
d
s
,
t
∈
[
0
,
T
]
.
To prove the result, we firstly show that Q
1 is Lipschitz continuous. For x
1
,
x
2
∈
S
(
T
) and t
∈
[
0
,
t
1
], we have that
(47)
∥
Q
1
x
1
(
t
)
-
Q
1
x
2
(
t
)
∥
=
∥
g
(
t
,
x
1
t
+
z
t
)
-
g
(
t
,
x
2
t
+
z
t
)
∥
,
≤
∥
A
-
β
∥
L
g
∥
x
1
t
-
x
2
t
∥
.
For t
∈
(
t
m
,
t
m
+
1
], m
=
1,2
,
…
,
n we have
(48)
∥
Q
1
x
1
(
t
)
-
Q
1
x
2
(
t
)
∥
≤
∥
A
-
β
∥
L
g
∥
x
1
t
-
x
2
t
∥
𝒫
+
m
M
L
I
∥
x
1
t
-
x
2
t
∥
𝒫
+
M
m
L
g
(
2
+
L
I
)
∥
x
1
t
-
x
2
t
∥
𝒫
,
=
K
T
[
∥
A
-
β
∥
L
g
+
m
M
L
I
+
M
m
L
g
(
2
+
L
I
)
]
×
∥
x
1
-
x
2
∥
T
.
Thus for t
∈
[
0
,
T
] we have
(49)
∥
Q
1
x
1
(
t
)
-
Q
1
x
2
(
t
)
∥
≤
K
T
[
∥
A
-
β
∥
L
g
+
n
M
L
I
+
M
n
L
g
(
2
+
L
I
)
]
×
∥
x
1
-
x
2
∥
T
.
Taking supremum on [
0
,
T
], we get
(50)
∥
Q
1
x
1
-
Q
1
x
2
∥
T
≤
K
T
[
∥
A
-
β
∥
L
g
+
n
M
L
I
+
M
n
L
g
(
2
+
L
I
)
]
×
∥
x
1
-
x
2
∥
T
.
Hence, it implies that Q
1 satisfies the Lipschitz condition with Lipschitz constant L, where L
=
K
T
[
∥
A
-
β
∥
L
g
+
n
M
L
I
+
M
n
L
g
(
2
+
L
I
)
].
Therefore, from Lemma9 (vii), we have that for any bounded set B
∈
𝒫
𝒞
(51)
χ
𝒫
𝒞
(
Q
1
B
)
≤
L
χ
𝒫
𝒞
(
B
)
.
Next, we show thatQ
2 is a χ-contraction. Let B be an arbitrary bounded subset S
(
T
). Since S
q
(
t
) is equicontinuous solution operator, therefore S
q
(
t
-
s
)
f
(
s
,
u
s
+
z
s
) is piecewise continuous. From Lemma 9, we have that for any bounded set B
∈
𝒫
𝒞,
(52)
χ
(
Q
2
B
(
t
)
)
=
χ
(
∫
0
t
S
q
(
t
-
s
)
f
(
s
,
B
s
+
z
s
)
d
s
)
≤
∫
0
t
η
(
s
)
sup
⁡
-
∞
<
ω
≤
0
χ
(
B
(
s
+
ω
)
+
z
(
s
+
ω
)
)
d
s
≤
∫
0
t
η
(
s
)
sup
⁡
0
≤
τ
≤
s
χ
(
B
(
τ
)
)
d
s
≤
χ
𝒫
𝒞
(
B
)
∫
0
t
η
(
s
)
d
s
,
t
∈
[
0
,
T
]
.
Thus for any bounded set B
∈
𝒫
𝒞
(53)
χ
𝒫
𝒞
(
Q
B
)
=
χ
𝒫
𝒞
(
Q
1
B
+
Q
2
B
)
≤
χ
𝒫
𝒞
(
Q
1
B
)
+
χ
𝒫
𝒞
(
Q
2
B
)
≤
(
L
+
∫
0
t
η
(
s
)
d
s
)
χ
𝒫
𝒞
(
B
)
.
By the assumption (H2), we obtain that χ
𝒫
𝒞
(
Q
B
)
<
χ
𝒫
𝒞
(
B
); that is, Q is a contraction. Therefore, Q has at least one fixed point in B by Darbo fixed point theorem. Let u be a fixed point of Q on S
(
T
), then y
=
u
+
z is a mild solution for (1).Theorem 18.
Suppose that (Hf), (Hg), and (HI) are satisfied and(54)
K
T
[
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
]
+
∫
0
T
m
f
(
s
)
d
s
lim
⁡
τ
→
∞
sup
⁡
Ω
(
τ
)
τ
<
1
.
Then, there exists a mild solution for the impulsive problem (1).Proof.
Proceeding as in the proof of Theorem17, we infer that Q defined by (34) is continuous from S
(
T
) into S
(
T
). Next we indicate that there exists r
>
0 such that Q
(
B
r
)
⊂
B
r, where B is defined by B
r
=
{
u
∈
S
(
T
)
:
∥
u
∥
T
≤
r
}. To this end, let us assume that assertion is false, then for any r
>
0 there exists u
r
∈
B
r and t
r
∈
J such that r
<
∥
Q
u
r
(
t
r
)
∥. Therefore, for t
r
∈
[
0
,
t
1
] and u
r
∈
B
r,
(55)
r
<
∥
Q
u
r
(
t
r
)
∥
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
g
(
t
,
u
r
t
r
+
z
t
r
)
∥
+
M
∫
0
t
m
f
(
s
)
Ω
(
∥
u
r
s
+
z
s
∥
)
d
s
,
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
(
C
1
∥
u
r
t
r
+
z
t
r
∥
𝒫
+
C
2
)
+
M
∫
0
t
‍
m
f
(
s
)
Ω
(
∥
u
r
s
+
z
s
∥
)
d
s
,
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
×
(
C
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
C
1
K
T
r
+
C
2
)
+
M
∫
0
t
m
f
(
s
)
d
s
Ω
(
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
r
)
=
r
0
.
For t
r
∈
(
t
1
,
t
2
], we have
(56)
r
<
∥
Q
u
r
(
t
r
)
∥
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
(
C
1
∥
u
r
t
r
+
z
t
r
∥
𝒫
+
C
2
)
+
M
(
L
1
∥
u
r
t
r
+
z
t
r
∥
𝒫
+
L
2
)
+
M
L
g
(
L
1
∥
u
r
t
r
+
z
t
r
∥
𝒫
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
Ω
(
∥
u
r
s
+
z
s
∥
)
d
s
,
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
×
(
C
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
C
1
K
T
r
+
C
2
)
+
M
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
M
L
g
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
d
s
Ω
(
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
r
)
,
=
r
1
.
For t
r
∈
(
t
n
,
T
], we have
(57)
r
<
∥
Q
u
r
(
t
r
)
∥
≤
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
(
C
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
C
1
K
T
r
+
C
2
)
+
n
M
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
n
M
L
g
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
d
s
Ω
(
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
r
)
,
=
r
n
,
which implies that r
<
max
⁡
{
r
0
,
r
1
,
…
,
r
n
}. Therefore, we conclude that
(58)
r
<
M
∥
g
(
t
,
φ
)
∥
+
∥
A
-
β
∥
×
(
C
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
C
1
K
T
r
+
C
2
)
+
n
M
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
n
M
L
g
(
L
1
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
L
1
K
T
r
+
L
2
)
+
M
∫
0
t
m
f
(
s
)
d
s
Ω
(
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
r
)
.
We divide both the sides of (58) by r and letting r
→
∞, we obtain that
(59)
1
<
K
T
[
∥
A
-
β
∥
C
1
+
n
M
L
1
+
n
M
L
g
L
1
]
+
M
∫
0
T
‍
m
f
(
s
)
d
s
×
lim
⁡
r
→
∞
sup
⁡
Ω
(
(
K
T
M
H
+
N
T
)
∥
φ
∥
𝒫
+
K
T
r
)
r
,
<
K
T
[
∥
A
-
β
∥
C
1
+
n
M
L
1
(
1
+
L
g
)
]
+
∫
0
T
m
f
(
s
)
d
s
lim
⁡
τ
→
∞
sup
⁡
Ω
(
τ
)
τ
.
This contradicts the inequality (54). Hence, there exists a positive constant r
>
0 such that Q
(
B
r
)
⊂
B
r. Moreover, by uniform continuity of the map t
↦
S
q
(
t
) on (
0
,
T
], we have that set Q
(
B
r
) is equicontinuous. As the proof of Theorem 17, we infer that system (1) has a mild solution.


## 4. Example

In this section, we consider an example to illustrate the application of the theory. Here we take the spaceC
0
×
L
2
(
h
,
X
) as the phase space 𝒫 (see, [7]).Now we study the following fractional differential equation with infinite delay:(60)
d
q
d
t
q
[
u
(
t
,
x
)
+
∫
-
∞
t
∫
0
π
b
(
t
-
s
,
μ
,
x
)
u
(
s
,
μ
)
d
μ
d
s
]
=
∂
2
d
x
2
[
u
(
t
,
x
)
+
∫
-
∞
t
∫
0
π
b
(
t
-
s
,
μ
,
x
)
u
(
s
,
μ
)
d
μ
d
s
]
+
ℐ
t
1
-
q
∫
-
∞
t
f
(
t
,
t
-
s
,
x
,
u
(
s
,
x
)
)
d
s
,
t
∈
[
0
,
T
]
,
x
∈
[
0
,
π
]
,
u
(
t
,
0
)
=
u
(
t
,
π
)
=
0
,
0
≤
t
≤
T
,
u
(
τ
,
x
)
=
φ
(
τ
,
x
)
,
0
≥
τ
,
x
∈
[
0
,
π
]
,
Δ
u
(
t
k
,
x
)
=
∫
-
∞
t
k
a
k
(
t
k
-
s
)
u
(
s
,
x
)
d
s
,
k
=
1,2
,
…
,
n
,
where φ
∈
C
0
×
L
2
(
h
;
X
) and 0
<
t
1
<
⋯
<
t
n
<
t
n
+
1
=
T are prefixed numbers. Let X
=
L
2
[
0
,
π
] and consider the operator A
:
D
(
A
)
⊂
X
→
X as A
v
=
v
′′ with the domain
(61)
D
(
A
)
=
{
5
9
v
∈
X
:
v
,
v
′
are
absolutely
continuous
and
v
′′
∈
X
,
v
(
0
)
=
0
=
v
(
π
)
5
9
}
.
Then,
(62)
A
=
∑
n
=
1
∞
n
2
(
v
,
v
n
)
v
n
,
v
∈
D
(
A
)
,
Here v
n
(
x
)
=
(
2
/
π
)
sin
(
n
x
), n
∈
ℕ is the orthogonal set of eigenvectors of A. It is clear that A is the infinitesimal generator of an analytic semigroup {
S
(
t
)
}
t
≥
0 in X which is given by
(63)
S
(
t
)
v
=
∑
n
=
1
∞
e
-
n
2
t
(
v
,
v
n
)
v
n
,
∀
v
∈
X
,
t
>
0
.
By Theorem 3.1 in [27], we get that A is the infinitesimal generator of solution operator {
S
q
(
t
)
}
t
≥
0 and there exists a positive constant M such that ∥
S
q
(
t
)
∥
ℒ
(
X
)
≤
M for t
∈
[
0
,
T
]. The functions b
,
f
,
a
k satisfy the following conditions:(1)
the functionsb
(
s
,
μ
,
x
), (
∂
b
/
∂
x
)
(
s
,
μ
,
x
) are measurable such that b
(
s
,
μ
,
π
)
=
b
(
s
,
μ
,
0
)
=
0 and
(64)
L
g
=
max
⁡
{
(
∫
0
π
‍
∫
-
∞
0
‍
∫
0
π
‍
1
h
(
s
)
(
∂
i
b
(
s
,
μ
,
x
)
∂
x
i
)
2
d
μ
d
s
d
x
)
1
/
2
:
i
=
0,1
(
∫
0
π
‍
∫
-
∞
0
‍
∫
0
π
‍
1
h
(
s
)
(
∂
i
b
(
s
,
μ
,
x
)
∂
x
i
)
2
d
μ
d
s
d
x
)
1
/
2
}
<
∞
;
(2)
f
:
ℝ
4
→
ℝ is continuous and there exist a η
:
C
(
ℝ
2
,
ℝ
) such that
(65)
∥
f
(
t
1
,
t
2
,
x
,
u
)
∥
≤
η
(
t
1
,
t
2
)
∥
u
∥
,
(
t
1
,
t
2
,
x
,
u
)
∈
ℝ
4
;
(3)
the functionsa
k
:
[
0
,
∞
)
→
ℝ, k
=
1,2
,
…
,
n are continuous and
(66)
L
I
=
(
∫
-
∞
0
(
a
k
(
s
)
)
2
h
(
s
)
d
s
)
1
/
2
<
∞
.The impulsive system (60) might be reformulated as the abstract impulsive Cauchy problem (1) where
(67)
g
(
t
,
ϕ
)
(
x
)
=
∫
-
∞
0
‍
∫
0
π
b
(
s
,
y
,
x
)
ϕ
(
s
,
y
)
d
y
d
s
,
F
(
t
,
ϕ
)
(
x
)
=
∫
-
∞
0
f
(
t
,
s
,
x
,
ϕ
(
s
,
x
)
)
d
s
,
I
k
(
ϕ
)
(
x
)
=
∫
-
∞
0
a
k
(
s
)
ϕ
(
s
,
x
)
d
s
,
k
=
1
,
…
,
n
.
It may be verified that g, F satisfy the assumptions (Hf), (Hg), and (HI); that is, g
(
t
,
·
) and I
k, k
=
1
,
…
,
n are bounded linear operators and the range of g
(
·
) is contained in X
1
/
2, ∥
A
1
/
2
g
(
t
,
·
)
∥
≤
L
g, ∥
I
k
∥
≤
L
I, k
=
1
,
…
,
n, and ∥
F
(
t
,
ϕ
)
∥
≤
m
F
(
t
)
∥
ϕ
∥
𝒫 for every t
∈
[
0
,
T
], where m
F
(
t
) is defined as m
F
(
t
)
=
(
∫
-
∞
0
η
(
t
,
s
)
2
/
h
(
s
)
d
s
)
1
/
2. Applying Theorem 17, we obtain that problem (60) has a mild solution.



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*Source: 780636-2014-05-04.xml*