# Eigenvalues of a Class of Singular Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

Authors: Daliang Zhao; Yansheng Liu
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

This paper is devoted to investigating the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces:μ
x
′′
+
f
(
t
,
x
)
=
0
,
t
∈
(
0,1
)
,
t
≠
t
i;
Δ
x
|
t
=
t
i
=
α
i
x
(
t
i
-
0
)
,
i
=
1,2
,
…
,
k
;
a
x
(
0
)
-
b
x
′
(
0
)
=
θ
;
c
x
(
1
)
+
d
x
′
(
1
)
=
θ
, where θ denotes the zero element of Banach space, Δ
x
|
t
=
t
i
=
x
(
t
i
+
0
)
-
x
(
t
i
-
0
), α
i
>
-
1, a
,
b
,
c
,
d
∈
R
+
,
γ
=
a
c
+
a
d
+
b
c
>
0, μ is a parameter, and f
(
t
,
x
) may be singular at t
=
0
,
1 and x
=
θ. The arguments are mainly based upon the theory of fixed point index, measure of noncompactness, and the special cone, which is constructed to overcome the singularity.


---

## Body

## 1. Introduction

Consider the following eigenvalue problems of singular boundary value problem (SBVP) with impulse in Banach spaceE:
(1)
μ
x
′′
+
f
(
t
,
x
)
=
0
,
t
∈
(
0,1
)
,
t
≠
t
i
;
Δ
x
|
t
=
t
i
=
α
i
x
(
t
i
-
0
)
,
i
=
1,2
,
…
,
k
;
a
x
(
0
)
-
b
x
′
(
0
)
=
θ
;
c
x
(
1
)
+
d
x
′
(
1
)
=
θ
,
where θ denotes the zero element of Banach space E, Δ
x
|
t
=
t
i
=
x
(
t
i
+
0
)
-
x
(
t
i
-
0
)
,
0
=
t
0
<
t
1
<
t
2
<
⋯
<
t
k
<
t
k
+
1
=
1, α
i
>
-
1, a
,
b
,
c
,
d
∈
R
+
,
γ
=
a
c
+
a
d
+
b
c
>
0, μ is a parameter, f
(
t
,
x
) may be singular at t
=
0,1 and x
=
θ.During the last few decades, the existence of positive solution of nonlinear singular boundary value problems has gained considerable popularity (see [1–12] and references therein). In recent years, there were also a lot of papers which dealt with eigenvalue problems (see [13–27]). Some of them considered singular case (see, for instance, [13–16, 18, 19], etc.).On the other hand, as we know, the theory of impulsive differential equations has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations such as physics, chemical technology, population dynamics, biotechnology, and economics. It has emerged as an important area of investigation in recent years (see [28–36] and references therein).To the best of our knowledge, there is no paper studying the eigenvalue problems of the impulsive singular boundary value problem in Banach spaces. The main purpose of this paper is to fill this gap. By using the theory of fixed point index, measure of noncompactness, and the special cone which is constructed to overcome the singularity, we investigate the existence of eigenvalues of (1).The main features of the present paper are as follows. By virtue of a special transformation, we first convert (1) into another solvable form such that the associated operator can be used to overcome the influence of impulse and parameter μ. Then a special cone is constructed to deal with the singularity of (1).This paper is organized as follows. In Section2, we provide some basic definitions, preliminaries facts, and lemmas. Meanwhile, some transformations are introduced to convert (1) into another solvable form. In Section 3, the main results are presented and proved. Finally, an example is worked out to demonstrate the application of the main result.


## 2. Preliminaries and Conversion of (1)

LetP be a normal solid cone of real Banach space E. Without loss of generality, suppose the normal constant is 1. Let P
* denote the dual cone of P, J
=
[
0,1
], and P
r
=
{
x
∈
P
∶
∥
x
∥
<
r
}, P
¯
r
=
{
x
∈
P
∶
∥
x
∥
≤
r
}
(
r
>
0
). Denoted by C
[
J
,
E
] the Banach space of all continuous functions x
∶
J
→
E with norm ∥
x
∥
c
=
max
⁡
t
∈
J
∥
x
(
t
)
∥.We defineP
C
[
J
,
E
]
=
{
x
∶
x is a map from J into E such that x
(
t
) is continuous at t
≠
t
i and left continuous at t
=
t
i and its right limit at t
=
t
i
(denoted by x
(
t
i
+
)) exists for i
=
1,2
,
…
,
m
}, and P
C
1
[
J
,
E
]
=
{
x
∈
P
C
[
J
,
E
]
∶
x
′
(
t
) is continuous at t
≠
t
i and left continuous at t
=
t
i and the right limit at t
=
t
i
(denoted by x
′
(
t
i
+
)) exists for i
=
1,2
,
…
,
m
}.LetY
=
{
y
∈
C
1
[
J
,
E
]
∶
a
y
(
0
)
-
b
y
′
(
0
)
=
θ
,
c
y
(
1
)
+
d
y
′
(
1
)
=
θ
}, and X
=
{
x
∈
P
C
1
[
J
,
E
]
∶
Δ
x
|
t
=
t
i
=
α
i
x
(
t
i
-
0
)
,
Δ
x
′
|
t
=
t
i
=
α
i
x
′
(
t
i
-
0
)
,
a
x
(
0
)
-
b
x
′
(
0
)
=
θ
,
c
x
(
1
)
+
d
x
′
(
1
)
=
θ
}. It is well known that Y is a Banach space with the norm ∥
y
∥
1=max
⁡
{
max
⁡
t
∈
J
∥
y
(
t
)
∥, max
⁡
t
∈
J
∥
y
′
(
t
)
∥
}.Evidently,P
C
[
J
,
E
] and P
C
1
[
J
,
E
] are Banach spaces with norm ∥
x
∥
P
C
=
sup
⁡
t
∈
J
∥
x
(
t
)
∥ and ∥
x
∥
P
C
1
=
max
⁡
{
sup
⁡
t
∈
J
∥
x
(
t
)
∥, sup
⁡
t
∈
J
∥
x
′
(
t
)
∥
}, respectively. Furthermore, X is a closed subspace of P
C
1
[
J
,
E
].For eachy
∈
Y, let x
(
t
)
=
∏
0
<
t
i
<
t
(
1
+
α
i
)
y
(
t
); it is easy to see x
∈
X. Conversely, for each x
∈
X, let y
(
t
)
=
x
(
t
)
/
∏
0
<
t
i
<
t
(
1
+
α
i
); then y
∈
Y.We convert (1) into another solvable form first.Lemma 1.
Ifx
∈
X is a solution of impulsive SBVP (1), then y
(
t
)
=
x
(
t
)
/
∏
0
<
t
i
<
t
(
1
+
α
i
) satisfies SBVP:
(2)
μ
y
′′
+
1
∏
0
<
t
i
<
t
(
1
+
α
i
)
f
(
t
,
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
y
(
t
)
)
=
0
,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
t
∈
(
0,1
)
;
a
y
(
0
)
-
b
y
′
(
0
)
=
θ
;
c
y
(
1
)
+
d
y
′
(
1
)
=
θ
.
Conversely, if y
∈
Y is a solution of SBVP (2), then x
(
t
)
=
∏
0
<
t
i
<
t
(
1
+
α
i
)
y
(
t
) is a solution of impulsive SBVP (1).Lemma 2.
y
∈
Y is a solution of SBVP (2) if and only if y is a solution of the integral equation:
(3)
μ
y
(
t
)
=
∫
J
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
×
f
(
t
,
∏
0
<
t
i
<
s
‍
(
1
+
α
i
)
y
(
s
)
)
d
s
,
where
(4)
G
(
t
,
s
)
=
{
γ
-
1
(
a
t
+
b
)
(
c
(
1
-
s
)
+
d
)
,
0
≤
t
≤
s
≤
1
,
γ
-
1
(
a
s
+
b
)
(
c
(
1
-
t
)
+
d
)
,
0
≤
s
≤
t
≤
1
.By Lemmas1 and 2, we can obtain the following.Lemma 3.
x
(
t
)
∈
P
C
[
J
,
E
]
⋂
‍
X is a solution of impulsive SBVP (1) if and only if x
∈
P
C
[
J
,
E
]
⋂
‍
X is a solution of the integral equation:
(5)
μ
x
(
t
)
=
∫
J
‍
G
*
(
t
,
s
)
f
(
s
,
x
(
s
)
)
d
s
,
where
(6)
G
*
(
t
,
s
)
=
∏
0
<
t
i
<
t
(
1
+
α
i
)
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
.Letα
(
·
) and α
p
c
(
·
) donated the Kuratowski noncompactness measure of bounded sets in E and P
C
[
J
,
E
], respectively. If nontrivial function x
∈
P
C
[
J
,
E
] satisfies problem (5) for some μ
≠
0, then μ is called an eigenvalue and x is called an eigenfunction of problem (1) corresponding to the eigenvalue μ.It is well known that the following conclusions hold.Lemma 4 (see [17]).
LetP be a cone in Banach space E, P
r
=
{
x
∈
P
∶
∥
x
∥
<
r
}
(
r
>
0
), and P
¯
r
=
{
x
∈
P
∶
∥
x
∥
≤
r
}. Let operator A
:
P
¯
r
→
P be a strict set contraction. If ∥
A
x
∥
≥
∥
x
∥ and A
x
≠
x for x
∈
∂
P
r, then i
(
A
,
P
r
,
P
)
=
0.Lemma 5 (see [17]).
H
⊂
C
[
J
,
E
] is relatively compact if and only if H is equicontinuous and for any t
∈
J
,
H
(
t
) is a relatively compact set in E.


## 3. Main Results

For convenience, let(7)
σ
*
=
:
min
⁡
1
≤
j
≤
k
∏
1
≤
i
≤
j
‍
(
1
+
α
i
)
,
σ
*
=
:
max
⁡
1
≤
j
≤
k
∏
1
≤
i
≤
j
‍
(
1
+
α
i
)
,
Ω
(
t
)
=
ϱ
(
t
)
∏
0
<
t
i
<
t
(
1
+
α
i
)
σ
*
,
ϱ
(
t
)
=
min
⁡
{
b
+
a
t
b
+
a
,
d
+
c
(
1
-
t
)
d
+
c
}
,
∀
t
∈
(
0,1
)
.For the forthcoming analysis, we list the following assumptions:(
H
1
)
f
∈
C
[
(
0,1
)
×
P
∖
{
θ
}
,
P
],
(8)
∥
f
(
t
,
x
)
∥
≤
g
(
t
)
∥
l
(
x
)
∥
,
t
∈
(
0,1
)
,
x
∈
P
∖
{
θ
}
,
where g
:
(
0,1
)
→
(
0
,
+
∞
)
,
l
∈
C
[
P
∖
{
θ
}
,
P
] satisfy
(9)
∫
0
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
R
]
d
s
<
+
∞
,
∀
R
>
r
>
0
,
where l
[
r
,
R
]
=
sup
⁡
x
∈
P
¯
R
∖
P
r
∥
l
(
x
)
∥
<
+
∞.
(
H
2
)
There exists a constantL
≥
0 such that
(10)
α
(
f
(
t
,
D
)
)
≤
L
α
(
D
)
,
∀
t
∈
(
0,1
)
,
D
⊂
P
¯
R
∖
P
r
,
∀
R
>
r
>
0
,
L
<
σ
*
2
M
0
σ
*
,
M
0
=
max
⁡
s
∈
J
⁡
γ
-
1
(
a
s
+
b
)
(
c
(
1
-
s
)
+
d
)
>
0
.
(
H
3
)
There existu
0
∈
P
∖
{
θ
}
,
[
a
0
,
b
0
]
⊂
(
0,1
) such that
(11)
f
(
t
,
x
)
≥
h
(
∥
x
∥
)
u
0
,
t
∈
J
0
=
[
a
0
,
b
0
]
,
2
σ
*
∥
u
0
∥
σ
*
max
⁡
t
∈
J
∫
a
0
b
0
‍
G
(
t
,
s
)
Ω
(
s
)
d
s
mm
<
liminf
⁡
t
→
0
+
h
(
t
)
t
≤
+
∞
,
where h
∈
C
[
R
+
,
R
+
].
(
H
4
)
There existϕ
∈
P
* and [
a
1
,
b
1
]
⊂
(
0,1
) such that ϕ
(
x
)
>
0 for x
>
θ and
(12)
liminf
⁡
∥
x
∥
→
∞
,
x
∈
P
ϕ
(
f
(
t
,
x
)
)
ϕ
(
x
)
=
ξ
(
t
)
uniformly for t
∈
J
1
=
[
a
1
,
b
1
], and ∫
J
1
‍
G
(
t
,
s
)
ξ
(
s
)
d
s
>
σ
*
/
σ
*.
(
H
5
)
There existϕ
∈
P
* and [
a
1
,
b
1
]
⊂
(
0,1
) such that ϕ
(
x
)
>
0 for x
>
θ and
(13)
liminf
⁡
∥
x
∥
→
0
,
x
∈
P
ϕ
(
f
(
t
,
x
)
)
ϕ
(
x
)
=
ξ
(
t
)
uniformly for t
∈
J
1
=
[
a
1
,
b
1
], and ∫
J
1
‍
G
(
t
,
s
)
ξ
(
s
)
d
s
>
σ
*
/
σ
*.Define(14)
Q
=
{
x
∈
P
C
[
J
,
P
]
∶
x
(
t
)
≥
Ω
(
t
)
x
(
s
)
,
∀
t
,
s
∈
J
}
.
It is easy to check that Q is a cone in space P
C
[
J
,
E
] and Q
⊂
P
C
[
J
,
P
]. Let Q
r
=
{
x
∈
Q
∶
∥
x
∥
<
r
}. As in [2], we can prove that
(15)
G
(
t
,
τ
)
≥
ϱ
(
t
)
G
(
s
,
τ
)
,
∀
t
,
s
,
τ
∈
[
0,1
]
,
where G
(
t
,
s
) is defined in Lemma 2. So for all t
,
s
,
τ
∈
[
0,1
], we have
(16)
G
*
(
t
,
s
)
≥
ϱ
(
t
)
∏
0
<
t
i
<
t
(
1
+
α
i
)
∏
0
<
t
i
<
τ
(
1
+
α
i
)
·
∏
0
<
t
i
<
τ
(
1
+
α
i
)
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
τ
,
s
)
≥
Ω
(
t
)
G
*
(
τ
,
s
)
.To solve eigenvalues of impulsive SBVP (1), we first consider operator A associated with (5) and defined by
(17)
(
A
x
)
(
t
)
=
∫
J
‍
G
*
(
t
,
s
)
f
(
s
,
x
(
s
)
)
d
s
,
∀
x
∈
Q
∖
{
θ
}
.
For ∀
x
∈
Q
∖
{
θ
}, from the definition of Q, we have
(18)
Ω
(
t
)
∥
x
∥
p
c
≤
∥
x
(
t
)
∥
≤
∥
x
∥
p
c
,
t
∈
J
.
By (
H
1
), we know
(19)
∥
(
A
x
)
(
t
)
∥
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
‍
G
(
s
,
s
)
∥
f
(
s
,
x
(
s
)
)
∥
d
s
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
m
×
G
(
s
,
s
)
g
(
s
)
∥
l
(
x
(
s
)
)
∥
d
s
≤
σ
*
σ
*
∫
0
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
∥
x
∥
p
c
,
∥
x
∥
p
c
]
d
s
<
+
∞
.
So the operator A is well defined in Q
∖
{
θ
}.For the sake of overcoming the singularity, choosee
∈
int
⁡
P with ∥
e
∥
=
1 and consider the approximate problem of (17):
(20)
(
A
m
x
)
(
t
)
=
∫
0
1
‍
G
*
(
t
,
s
)
f
(
s
,
x
(
s
)
+
e
m
)
d
s
.
For any x
∈
Q
,
t
,
τ
∈
J,
(21)
(
A
m
x
)
(
t
)
=
∫
0
1
‍
G
*
(
t
,
s
)
f
(
s
,
x
(
s
)
+
e
m
)
d
s
≥
Ω
(
t
)
∫
0
1
‍
G
*
(
τ
,
s
)
f
(
s
,
x
(
s
)
+
e
m
)
d
s
=
Ω
(
t
)
(
A
m
x
)
(
τ
)
.
Hence A
m
Q
⊂
Q.Lemma 6.
Let conditions(
H
1
) and (
H
2
) be satisfied; then for any r
>
0, the operator A
m is a strict set contraction from Q into Q.Proof.
ObviouslyA
m
Q
⊂
Q. Now we prove that A
m is continuous. Let ∥
x
n
-
x
∥
p
c
→
0 as n
→
∞
(
x
n
,
x
∈
Q
¯
r
); we have
(22)
∥
(
A
m
x
n
)
(
t
)
-
(
A
m
x
)
(
t
)
∥
≤
∏
0
<
t
i
<
t
(
1
+
α
i
)
∫
0
1
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
n
m
m
m
m
m
m
m
m
m
×
∥
f
(
s
,
x
n
(
s
)
+
e
m
)
m
m
m
m
m
m
m
m
m
m
m
m
m
-
f
(
s
,
x
(
s
)
+
e
m
)
∥
d
s
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
s
,
s
)
g
(
s
)
m
m
m
m
m
m
m
m
m
m
n
×
(
∥
l
(
x
n
(
s
)
+
e
m
)
∥
m
m
m
m
m
m
m
m
m
m
m
m
i
+
∥
l
(
x
(
s
)
+
e
m
)
∥
)
≤
2
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
s
,
s
)
g
(
s
)
m
m
m
m
m
m
m
n
m
m
m
m
×
l
[
Ω
(
s
)
r
,
r
+
1
]
d
s
≤
2
σ
*
σ
*
∫
0
1
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
]
d
s
.
So (
H
1
) and the dominated convergence theorem imply that
(23)
lim
⁡
n
→
∞
(
A
m
x
n
)
(
t
)
=
(
A
m
x
)
(
t
)
.
We now show that
(24)
lim
⁡
n
→
∞
∥
A
m
x
n
-
A
m
x
∥
p
c
=
0
.
In fact, if (24) is not true, then there exist a positive number ε
0 and a sequence {
x
n
i
}
⊂
{
x
n
} such that
(25)
∥
A
m
x
n
i
-
A
m
x
∥
p
c
≥
ε
0
(
i
=
1,2
,
…
)
.
Since {
A
m
x
n
} is relatively compact, there is a subsequence of {
A
m
x
n
i
} which converges to some x
0
∈
Q. Without loss of generality, we may assume that {
A
m
x
n
i
} itself converges to x
0; that is,
(26)
lim
⁡
i
→
∞
∥
A
m
x
n
i
-
x
0
∥
p
c
=
0
.
By virtue of (23) and (26), we have x
0
=
A
m
x, and so (26) contradicts with (25). Hence (24) holds, and the continuity of A
m is proved.
By(
H
1
), it is easy to see that A
m is bounded from Q into Q.
Now we will prove that the operatorA
m is a strict set contraction from Q into Q. Let
(27)
(
A
m
(
n
)
x
)
(
t
)
=
∫
1
/
n
1
-
(
1
/
n
)
‍
G
*
(
t
,
s
)
f
(
s
,
x
(
s
)
+
e
m
)
d
s
.
By (
H
1
), as n
→
∞, we have
(28)
∥
(
A
m
x
)
(
t
)
-
(
A
m
(
n
)
x
)
(
t
)
∥
≤
∫
0
1
/
n
‍
G
*
(
t
,
s
)
∥
f
(
s
,
x
(
s
)
+
e
m
)
∥
d
s
mm
+
∫
1
-
(
1
/
n
)
1
‍
G
*
(
t
,
s
)
∥
f
(
s
,
x
(
s
)
+
e
m
)
∥
d
s
⟶
0
.
So for any bounded set U
⊂
Q, we have
(29)
lim
⁡
n
→
∞
(
A
m
(
n
)
x
)
(
t
)
=
(
A
m
x
)
(
t
)
,
∀
x
∈
U
,
t
∈
J
.
Hence
(30)
d
h
(
(
A
m
(
n
)
U
)
(
t
)
,
(
A
m
U
)
(
t
)
)
⟶
0
,
n
⟶
∞
,
where d
h
(
·
,
·
) denotes the Hausdorff metric, which implies
(31)
α
(
(
A
m
(
n
)
U
)
(
t
)
)
⟶
α
(
(
A
m
U
)
(
t
)
)
,
n
⟶
∞
,
t
∈
J
.
For any t
∈
J, by (
H
2
), we have
(32)
α
(
(
A
m
(
n
)
U
)
(
t
)
)
=
α
(
{
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
m
m
m
m
m
n
m
×
∫
1
/
n
1
-
(
1
/
n
)
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
m
m
m
m
m
m
m
m
m
m
×
f
(
s
,
x
(
s
)
+
e
m
)
d
s
,
x
∈
U
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
}
)
≤
(
1
-
1
2
n
)
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
m
m
i
×
α
(
c
o
¯
{
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
m
m
l
l
l
m
n
m
×
f
(
s
,
x
(
s
)
+
e
m
)
∶
s
∈
[
1
n
,
1
-
1
n
]
,
x
∈
U
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
)
≤
∏
0
<
t
i
<
t
(
1
+
α
i
)
‍
m
m
m
×
α
(
{
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
s
,
s
)
m
m
m
m
l
l
l
l
m
i
×
f
(
s
,
x
(
s
)
+
e
m
)
∶
s
∈
[
1
n
,
1
-
1
n
]
,
x
∈
U
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
)
≤
σ
*
σ
*
M
0
α
(
f
(
s
,
U
(
I
n
)
)
)
,
I
n
=
[
1
n
,
1
-
1
n
]
,
s
∈
I
n
,
≤
L
M
0
σ
*
σ
*
α
(
U
(
I
n
)
)
≤
2
L
M
0
σ
*
σ
*
α
p
c
(
U
)
<
α
p
c
(
U
)
.
It is easy to show that α
p
c
(
A
m
(
U
)
)
=
sup
⁡
t
∈
J
α
(
(
A
m
(
U
)
)
(
t
)
). Therefore, we can have α
p
c
(
A
m
U
)
≤
2
L
M
0
(
σ
*
/
σ
*
)
α
p
c
(
U
)
<
α
p
c
(
U
). Consequently, the operator A
m is a strict set contraction from Q into Q. The proof is thus completed.Theorem 7.
Let conditions(
H
1
)
,
(
H
2
), and (
H
3
) be satisfied; then there exists a positive number δ such that for any given r
∈
(
0
,
δ
), the impulsive SBVP (1) has a nontrivial eigenfunction x
r
∈
P
C
[
J
,
P
] with ∥
x
r
∥
p
c
=
r corresponding to eigenvalue μ
r
≥
1.Proof.
By Lemma6, the operator A
m is a strict set contraction from Q
¯
r into Q. Observing (
H
3
), if lim
⁡
inf
⁡
t
→
0
+
(
h
(
t
)
/
t
)
≠
+
∞, we can choose a v
0 with
(33)
1
2
liminf
⁡
t
→
0
+
h
(
t
)
t
≤
v
0
<
liminf
⁡
t
→
0
+
h
(
t
)
t
,
and there is a δ
>
0 such that
(34)
h
(
t
)
≥
v
0
t
,
∀
0
<
t
<
δ
.
Choose r
∈
(
0
,
δ
) and m sufficiently large such that (
1
/
m
)
<
δ
-
r; we have
(35)
(
A
m
x
)
(
t
)
=
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
i
×
G
(
t
,
s
)
f
(
s
,
x
(
s
)
+
e
m
)
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
a
0
b
0
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
m
m
m
m
m
m
m
m
i
×
h
(
∥
x
(
s
)
+
e
m
∥
)
u
0
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
a
0
b
0
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
i
×
G
(
t
,
s
)
v
0
∥
x
(
s
)
∥
u
0
d
s
≥
σ
*
σ
*
∫
a
0
b
0
‍
G
(
t
,
s
)
v
0
Ω
(
s
)
∥
x
∥
P
C
u
0
d
s
.
So
(36)
∥
A
m
x
∥
P
C
≥
σ
*
σ
*
v
0
∥
u
0
∥
max
⁡
t
∈
J
∫
a
0
b
0
‍
G
(
t
,
s
)
Ω
(
s
)
d
s
∥
x
∥
P
C
>
∥
x
∥
p
c
.
By virtue of Lemma 4, we have i
(
A
m
,
Q
r
,
Q
)
=
0. Since i
(
θ
,
Q
r
,
Q
)
=
1, it follows from the homotopy invariance of fixed point index for strict set contraction that there exist x
m
∈
∂
Q
¯
r and λ
m
∈
(
0,1
) such that
(37)
x
m
=
λ
m
A
m
x
m
,
i
.
e
.
A
m
x
m
=
λ
m
-
1
x
m
,
where m
≥
m
0
=
[
1
/
(
δ
-
r
)
]
+
1.
LetB
=
:
{
x
m
∶
m
≥
m
0
}. Obviously, B is uniformly bounded. We shall show that B is equicontinuous. By virtue of boundary condition, we need to consider only the following eight cases.
Case   1.
b
=
d
=
0, the boundary value condition is x
(
0
)
=
x
(
1
)
=
0. So we need only to show that {
x
m
(
t
)
} uniformly converge to θ with respect to m
≥
m
0 as t
→
0
+
,
t
→
1
-
0, and B is equicontinuous on any closed subinterval of (
0,1
).
By(
H
1
), let
(38)
η
=
:
∫
0
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
<
+
∞
.
Using the absolute continuity of integration, for any ε
>
0, there exists a δ
1
>
0, such that
(39)
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
<
ε
as |
t
1
-
t
2
|
<
δ
1
,
t
1
,
t
2
∈
J. Since
(40)
x
m
(
t
)
=
λ
m
∫
0
1
‍
G
*
(
t
,
s
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
≤
λ
m
σ
*
σ
*
[
(
1
-
t
)
∫
0
t
‍
s
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
mmmmm
+
t
∫
t
1
‍
(
1
-
s
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
]
,
we have
(41)
∥
x
m
(
t
)
∥
≤
σ
*
σ
*
{
(
1
-
t
)
∫
0
t
‍
s
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
m
m
m
m
m
+
t
∫
t
1
‍
(
1
-
s
)
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
}
≤
σ
*
σ
*
{
(
1
-
t
)
∫
0
t
‍
s
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
m
m
m
m
m
+
t
∫
t
1
‍
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
}
.
To prove that {
x
m
(
t
)
} uniformly converge to θ with respect to m
≥
m
0 as t
→
0
+, we need only to show
(42)
lim
⁡
t
→
0
+
(
1
-
t
)
∫
0
t
‍
s
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
=
0
,
(43)
lim
⁡
t
→
0
+
t
∫
t
1
‍
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
=
0
.
Notice that
(44)
(
1
-
t
)
∫
0
t
‍
s
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
∫
0
t
‍
(
1
-
s
)
s
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
=
∫
0
t
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
.
This together with (39) implies that (42) holds. For ε
,
δ
1 in (39), choose
(45)
δ
2
=
min
⁡
{
δ
1
,
ε
δ
1
η
}
.
Then for t
∈
(
0
,
δ
2
), (39) implies that
(46)
t
∫
t
1
‍
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
t
∫
t
δ
1
‍
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
∫
δ
1
1
‍
t
s
s
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
∫
t
δ
1
‍
s
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
t
δ
1
∫
δ
1
1
‍
s
(
1
-
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
=
∫
t
δ
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
t
δ
1
∫
δ
1
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
ε
+
t
δ
1
∫
0
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
2
ε
.
Hence (43) holds. Very similarly, we can obtain that {
x
m
(
t
)
} uniformly converge to θ with respect to m
≥
m
0 as t
→
1
-
0. Now, we show that B is equicontinuous on [
ζ
,
1
-
ζ
] for any [
ζ
,
1
-
ζ
]
⊂
(
0,1
)
,
ζ
∈
(
0
,
1
/
2
). Notice that
(47)
x
m
(
t
2
)
-
x
m
(
t
1
)
=
λ
m
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
mim
×
[
(
1
-
t
2
)
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
mmnm
+
t
2
∫
t
2
1
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
mmnm
-
(
1
-
t
1
)
∫
0
t
1
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
mnmm
-
t
1
∫
t
1
1
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
]
=
λ
m
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
m
i
m
×
[
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
(
t
2
-
t
1
)
m
m
m
i
m
i
×
(
-
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
m
m
mmmnmi
+
∫
t
2
1
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
i
m
m
i
i
m
m
i
×
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
)
m
m
m
i
i
m
+
(
1
-
t
1
)
∫
t
1
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
i
m
i
×
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
m
m
m
i
i
m
-
t
1
∫
t
1
t
2
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
i
m
m
i
×
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
]
.
So for anyt
1
,
t
2
∈
[
ζ
,
1
-
ζ
]
,
t
2
>
t
1
,
m
≥
m
0, we have
(48)
∥
x
m
(
t
2
)
-
x
m
(
t
1
)
∥
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
m
m
m
×
{
(
t
2
-
t
1
)
[
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
i
m
m
×
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
m
m
m
m
m
m
m
m
l
m
m
m
+
∫
t
2
1
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
l
i
m
m
m
×
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
]
mmnmm
+
(
1
-
t
1
)
∫
t
1
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmmmmmm
m
m
n
m
m
×
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
mmnmm
+
t
1
∫
t
1
t
2
‍
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
n
m
m
m
m
m
m
m
×
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
(
t
2
-
t
1
)
[
∫
0
t
2
‍
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
i
i
m
m
i
×
{
(
t
2
-
t
1
)
(
1
1
-
t
2
∫
0
t
2
‍
(
1
-
s
)
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmm
m
m
m
m
m
m
m
m
m
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mmmmmmmmmm
+
1
t
2
∫
t
2
1
‍
s
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
1
1
-
t
2
∫
0
t
2
‍
(
1
-
s
)
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
)
m
m
i
mm
+
1
-
t
1
1
-
t
2
∫
t
1
t
2
‍
s
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
i
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mmmim
+
∫
t
1
t
2
‍
s
(
1
-
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
i
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
(
t
2
-
t
1
)
(
1
1
-
t
2
∫
0
t
2
‍
(
1
-
s
)
s
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
≤
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
m
i
m
n
i
×
{
(
t
2
-
t
1
)
(
1
1
-
t
2
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmi
m
m
m
m
m
m
m
m
i
m
m
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mmmmmmmmmm
+
1
t
2
∫
t
2
1
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmmmimmmmmm
mmm
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
1
1
-
t
2
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
)
mimmmmi
+
1
-
t
1
1
-
t
2
∫
t
1
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
i
m
m
m
m
m
i
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mimmmmi
+
∫
t
1
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
i
m
m
m
m
m
i
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
(
t
2
-
t
1
)
(
1
1
-
t
2
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
≤
∏
0
<
t
i
<
t
(
1
+
α
i
)
‍
m
m
m
i
×
{
(
t
2
-
t
1
)
(
1
ζ
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
mmm
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
m
m
m
m
m
m
m
m
m
l
l
m
+
1
ζ
∫
t
2
1
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
i
m
l
l
m
m
m
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
1
ζ
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
)
mlllmmi
+
(
1
-
ζ
ζ
+
1
)
∫
t
1
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmlllmmmmmmmmmmi
×
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
(
t
2
-
t
1
)
(
1
1
-
t
2
∫
0
t
2
‍
G
(
s
,
s
)
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
}
≤
σ
*
σ
*
{
(
t
2
-
t
1
)
2
η
ζ
mmimmi
+
1
ζ
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
(
t
2
-
t
1
)
2
η
ζ
+
1
ζ
∫
t
1
t
2
‍
G
(
s
,
s
)
}
.
This together with (39) implies that B is equicontinuous on J.
Case   2.
b
=
0
,
d
≠
0
,
c
≠
0, the boundary value condition is x
(
0
)
=
0
,
c
x
(
1
)
+
d
x
′
(
1
)
=
0. We need to show that {
x
m
(
t
)
} uniformly converge to θ with respect to m
≥
m
0 as t
→
0
+. This can be obtained by the similar way in Case 1, so we omit it. Next, we show B is equicontinuous on [
ι
,
1
] for any ι
∈
(
0,1
). In fact, for any t
1
,
t
2
∈
[
ι
,
1
]
,
t
2
>
t
1
,
m
≥
m
0, we have
(49)
∥
x
m
(
t
2
)
-
x
m
(
t
1
)
∥
≤
∫
0
t
1
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
∫
t
1
t
2
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
∫
t
2
1
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
≤
(
t
2
-
t
1
)
c
c
(
1
-
s
)
+
d
σ
*
σ
*
mm
×
∫
0
t
1
‍
s
(
c
(
1
-
s
)
+
d
)
c
+
d
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
(
t
2
-
t
1
)
1
s
σ
*
σ
*
mm
×
∫
t
2
1
‍
s
(
c
(
1
-
s
)
+
d
)
c
+
d
g
(
s
)
∥
l
(
x
m
(
s
)
+
e
m
)
∥
d
s
≤
(
t
2
-
t
1
)
c
d
σ
*
σ
*
∫
0
t
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
(
t
2
-
t
1
)
1
ι
σ
*
σ
*
∫
t
2
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
(
t
2
-
t
1
)
η
σ
*
σ
*
(
c
d
+
1
ι
)
m
m
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
.
This together with (39) implies that B is equicontinuous on J.
Case   3.
b
=
0
,
d
≠
0
,
c
=
0, the boundary value condition is x
(
0
)
=
0
,
x
′
(
1
)
=
0. The equicontinuity of B on J of this case can be obtained by the same way of Case 2, so we omit it.
Case   4.
d
=
0
,
b
≠
0
,
a
≠
0, the boundary value condition is a
x
(
0
)
-
b
x
′
(
0
)
=
0
,
x
(
1
)
=
0. Similarly, by Case 1, we can get that {
x
m
(
t
)
} uniformly converge to θ with respect to m
≥
m
0 as t
→
1
-. For any ς
∈
(
0,1
), we need to show that B is equicontinuous on [
0
,
ς
]. In fact, for any t
1
,
t
2
∈
[
0
,
ς
]
,
t
2
>
t
1
,
m
≥
m
0, we have
(50)
∥
x
m
(
t
2
)
-
x
m
(
t
1
)
∥
≤
∫
0
t
1
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
∫
t
1
t
2
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
mm
+
∫
t
2
1
‍
|
G
*
(
t
2
,
s
)
-
G
*
(
t
1
,
s
)
|
∥
f
(
s
,
x
m
(
s
)
+
e
m
)
∥
d
s
≤
(
t
2
-
t
1
)
1
1
-
s
σ
*
σ
*
∫
0
t
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
(
t
2
-
t
1
)
a
a
s
+
b
σ
*
σ
*
∫
t
2
1
‍
G
(
s
,
s
)
g
(
s
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
×
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
(
t
2
-
t
1
)
1
1
-
ς
σ
*
σ
*
∫
0
t
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
mm
+
(
t
2
-
t
1
)
b
a
σ
*
σ
*
∫
t
2
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
≤
(
t
2
-
t
1
)
η
σ
*
σ
*
(
1
1
-
ς
+
b
a
)
mm
+
2
σ
*
σ
*
∫
t
1
t
2
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
r
+
1
m
0
]
d
s
.
This together with (39) implies that B is equicontinuous on J.
Case   5.
d
=
0
,
b
≠
0
,
a
=
0, the boundary value condition is x
′
(
0
)
=
0
,
x
(
1
)
=
0.
Case   6.
a
=
0
,
b
d
≠
0
,
c
≠
0, the boundary value condition is x
′
(
0
)
=
0
,
c
x
(
1
)
+
d
x
′
(
1
)
=
0.
Case   7.
a
≠
0
,
b
d
≠
0
,
c
=
0, the boundary value condition is a
x
(
0
)
-
b
x
′
(
0
)
=
0
,
x
′
(
1
)
=
0.
Case   8.
b
d
≠
0
,
a
c
≠
0, the boundary value condition is a
x
(
0
)
-
b
x
′
(
0
)
=
0
,
c
x
(
1
)
+
d
x
′
(
1
)
=
0.
By the very similar methods in Cases 2 and 4, we can prove thatB is equicontinuous on J in Cases 5–8. Hence, the proofs are omitted. Next we show that B is relatively compact. It is easy to see that
(51)
α
(
B
(
t
)
)
≤
α
(
{
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
m
m
m
m
m
m
m
m
m
i
×
f
(
s
,
x
m
(
s
)
+
e
m
)
d
s
∶
m
≥
m
0
∏
0
<
t
i
<
t
‍
}
)
m
m
m
m
m
m
m
m
m
m
m
≤
2
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
G
(
t
,
s
)
m
m
m
m
m
m
m
m
m
m
×
α
(
{
f
(
s
,
x
m
(
s
)
+
e
m
)
:
m
≥
m
0
}
)
d
s
≤
2
σ
*
σ
*
L
∫
0
1
‍
G
(
t
,
s
)
α
(
B
(
s
)
)
d
s
.
Hence, we have
(52)
α
p
c
(
B
)
≤
2
σ
*
σ
*
L
∫
0
1
‍
G
(
t
,
s
)
d
s
α
p
c
(
B
)
≤
2
σ
*
σ
*
L
∫
0
1
‍
G
(
s
,
s
)
d
s
α
p
c
(
B
)
≤
2
L
M
0
σ
*
σ
*
α
p
c
(
B
)
.
Observing L
<
σ
*
/
2
M
0
σ
*, so α
p
c
(
B
)
=
0, which means B is relatively compact.
It follows from Lemma5 that there is a convergent subsequence of {
x
m
}, and without loss of generality, we may assume that {
x
m
} itself converges to some x
r
∈
∂
Q
¯
r, and lim
⁡
m
→
∞
λ
m
=
λ
r
∈
[
0,1
]. Hence (40) and the dominated convergence theorem imply that
(53)
x
r
(
t
)
=
λ
r
∫
0
1
‍
G
*
(
t
,
s
)
f
(
s
,
x
r
(
s
)
)
d
s
=
λ
r
A
x
r
(
t
)
,
t
∈
J
.
Obviously, λ
r
>
0. Consequently, A
x
r
=
λ
r
-
1
x
r
=
μ
r
x
r, where μ
r
=
λ
r
-
1
≥
1.
The caseliminf
⁡
t
→
0
+
(
h
(
t
)
/
t
)
=
+
∞ can be proved similarly, so it is omitted. Then the theorem is proved.Theorem 8.
Let conditions(
H
1
)
,
(
H
2
), and (
H
4
) be satisfied; then there exists a positive number M such that for any given r
>
M
/
min
⁡
t
∈
J
Ω
(
t
), the impulsive SBVP (1) has a nontrivial eigenfunction x
r
∈
P
C
[
J
,
P
] with ∥
x
r
∥
P
C
=
r corresponding to eigenvalue μ
r
≥
1.Proof.
By(
H
4
), there exist ε
0
>
0 and M
>
0 such that
(54)
∫
a
1
b
1
‍
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
0
)
d
s
>
σ
*
σ
*
,
and ϕ
(
f
(
t
,
x
)
)
≥
(
ξ
(
t
)
-
ε
0
)
ϕ
(
x
) for t
∈
J
1=[
a
1
,
b
1
]
,
x
∈
P with ∥
x
∥
>
M.
Chooser
>
M
/
min
⁡
t
∈
J
Ω
(
t
). If there exists x
0
∈
∂
Q
¯ such that x
0
≥
A
m
x
0, then
(55)
∥
x
0
(
s
)
∥
≥
Ω
(
s
)
∥
x
0
∥
p
c
=
Ω
(
s
)
r
>
M
,
s
∈
J
.
Hence, for t
∈
J
1, we have
(56)
ϕ
(
x
0
(
t
)
)
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmmmmmmmmm
×
‍
G
(
t
,
s
)
ϕ
(
f
(
s
,
x
0
(
s
)
+
e
m
)
)
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
J
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmmmmmmmm
×
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
0
)
ϕ
(
x
0
(
s
)
+
e
m
)
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
J
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
mmmmmmm
mmmm
×
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
0
)
ϕ
(
x
0
(
s
)
)
d
s
≥
σ
*
σ
*
∫
J
1
‍
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
0
)
ϕ
(
x
0
(
s
)
)
d
s
.
Obviously,
(57)
inf
⁡
t
∈
J
1
∥
x
0
(
t
)
∥
≥
Ω
(
t
)
∥
x
0
∥
P
C
=
Ω
(
t
)
r
>
0
.
The continuity of x
0
(
t
) implies that ϕ
(
x
0
(
t
)
) is continuous on J
1. Hence, this together with (57) and (
H
4
) implies that min
⁡
t
∈
J
1
ϕ
(
x
0
(
t
)
)
>
0. Observing (54) and (56), we can obtain a contradiction. Applying Lemma 4, we know that the fixed point index i
(
A
m
,
Q
r
,
Q
)
=
0. Since i
(
θ
,
Q
r
,
Q
)
=
1, it follows from the homotopy invariance of fixed point index for strict set contraction that there exist x
m
∈
∂
Q
¯
r and 0
<
λ
m
<
1 such that
(58)
x
m
=
λ
m
A
m
x
m
,
i
.
e
.
A
m
x
m
=
λ
m
-
1
x
m
=
μ
m
x
m
,
where μ
m
=
λ
m
-
1
>
1. The rest of the proof is completely similar to that of Theorem 7. So it is omitted.Theorem 9.
Let conditions(
H
1
)
,
(
H
2
), and (
H
5
) be satisfied; then there exists a positive number δ such that for any given r
∈
(
0
,
δ
), the impulsive SBVP (1) has a nontrivial eigenfunction x
r
∈
P
C
[
J
,
P
] with ∥
x
r
∥
P
C
=
r corresponding to eigenvalue μ
r
≥
1.Proof.
By(
H
5
), there exist ε
1
>
0 and δ
>
0 such that
(59)
∫
J
1
‍
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
1
)
d
s
>
1
,
and ϕ
(
f
(
t
,
x
)
)
≥
(
ξ
(
t
)
-
ε
1
)
ϕ
(
x
) for t
∈
J
1
=
[
a
1
,
b
1
]
,
x
∈
P with ∥
x
∥
<
δ. Choose r
∈
(
0
,
δ
) and m such that (
1
/
m
)
<
δ
-
r. If there exists x
0
∈
∂
Q
¯
r such that x
0
≥
A
m
x
0, then for t
∈
J
1, we have
(60)
ϕ
(
x
0
(
t
)
)
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
0
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
m
×
G
(
t
,
s
)
ϕ
(
f
(
s
,
x
0
(
s
)
+
e
m
)
)
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
J
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
m
×
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
1
)
ϕ
(
x
0
(
s
)
+
e
m
)
d
s
≥
∏
0
<
t
i
<
t
‍
(
1
+
α
i
)
∫
J
1
‍
1
∏
0
<
t
i
<
s
(
1
+
α
i
)
m
m
m
m
m
m
m
m
m
m
m
×
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
1
)
ϕ
(
x
0
(
s
)
)
d
s
≥
σ
*
σ
*
∫
J
1
‍
G
(
t
,
s
)
(
ξ
(
s
)
-
ε
1
)
ϕ
(
x
0
)
d
s
.
Obviously,
(61)
inf
⁡
t
∈
J
1
∥
x
0
(
t
)
∥
≥
Ω
(
t
)
∥
x
0
∥
P
C
=
Ω
(
t
)
r
>
0
.
The continuity of x
0
(
t
) implies that ϕ
(
x
0
(
t
)
) is continuous on J
1. Hence, this together with (61) and (
H
5
) implies that min
⁡
t
∈
J
1
ϕ
(
x
0
(
t
)
)
>
0. Observing (59) and (60), then we can obtain a contradiction. Applying Lemma 4, we know that the fixed point index i
(
A
m
,
Q
r
,
Q
)
=
0. Since i
(
θ
,
Q
r
,
Q
)
=
1, it follows from the homotopy invariance of fixed point index for strict set contraction that there exist x
m
∈
∂
Q
¯
r and 0
<
λ
m
<
1 such that
(62)
x
m
=
λ
m
A
m
x
m
,
i
.
e
.
A
m
x
m
=
λ
m
-
1
x
m
=
μ
m
x
m
,
where μ
m
=
λ
m
-
1
>
1. The rest of the proof is completely similar to that of Theorem 7. So it is omitted.Example 10.
Consider the following impulsive SBVP:(63)
μ
x
n
′′
+
π
t
(
1
-
t
)
(
1
3
n
(
t
x
n
+
e
2
+
x
n
+
1
)
+
arctan
t
n
∥
x
∥
)
=
0
,
m
m
m
m
m
m
m
i
m
m
m
m
m
m
m
l
m
l
m
m
i
t
∈
(
0,1
)
,
t
≠
1
5
;
Δ
x
n
|
t
=
1
/
5
=
2
5
x
n
,
2
x
n
(
0
)
-
x
n
′
(
0
)
=
θ
;
x
n
(
1
)
=
θ
,
(
n
=
1,2
,
3
,
…
)
.Conclusion. There exists a positive number δ such that for any given r
∈
(
0
,
δ
), impulsive SBVP (63) has a nontrivial, nonnegative eigenfunction x
r
n
(
n
=
1,2
,
3
,
…
) satisfying sup
⁡
n
x
r
n
(
t
)
<
∞ for 0
≤
t
≤
1 and max
⁡
t
∈
[
0,1
]
sup
⁡
n
x
r
n
(
t
)
=
r corresponding to μ
r
≥
1.Proof.
LetE
=
l
∞
=
{
x
=
(
x
1
,
…
,
x
n
,
…
)
∶
sup
⁡
n
|
x
n
|
<
∞
} with norm ∥
x
∥
=
sup
⁡
n
|
x
n
| and P
=
{
x
=
(
x
1
,
…
,
x
n
,
…
)
∈
l
∞
∶
x
n
≥
0
,
n
=
1,2
,
3
,
…
}. Then, P is a normal solid cone of E with normal constant 1, and system (63) can be regarded as an impulsive SBVP in E
=
l
∞ of the form (1), where x
(
t
)
=
(
x
1
(
t
)
,
x
2
(
t
)
,
x
3
(
t
)
,
…
)
,
f
(
t
,
x
)
=
(
f
1
,
f
2
,
f
3
,
…
), and
(64)
f
n
(
t
,
x
)
=
π
t
(
1
-
t
)
(
1
3
n
(
t
x
n
+
e
2
+
x
n
+
1
)
+
arctan
t
n
∥
x
∥
)
.
It is easy to see that f
(
t
,
x
) is singular at t
=
0,1 and x
=
θ.
Letg
(
t
)
=
π
/
t
(
1
-
t
) and l
(
x
)
=
(
l
1
(
x
)
,
l
2
(
x
)
,
l
3
(
x
)
,
…
), where
(65)
l
n
(
x
)
=
1
3
n
(
x
n
+
e
2
+
x
n
+
1
)
+
π
4
n
∥
x
∥
.
So we have
(66)
∥
f
(
t
,
x
)
∥
≤
g
(
t
)
∥
l
(
x
)
∥
,
t
∈
(
0,1
)
,
x
∈
P
∖
{
θ
}
,
l
[
r
,
R
]
=
sup
⁡
x
∈
P
¯
R
∖
P
r
∥
l
(
x
)
∥
<
R
+
e
2
+
R
+
π
4
r
<
+
∞
,
llllllllllllllllllllllllllllllllllllllllllllllllllllll
∀
R
>
r
>
0
.
By virtue of ∫
0
1
‍
(
π
/
t
(
1
-
t
)
)
d
t
=
π
2, G
(
s
,
s
)
=
(
1
/
3
)
(
1
+
2
s
)
(
1
-
s
), and
(67)
M
0
=
max
⁡
s
∈
J
G
(
s
,
s
)
=
3
8
,
Ω
(
s
)
=
min
⁡
{
1
+
2
s
3
,
1
-
s
}
≥
1
3
(
1
+
2
s
)
(
1
-
s
)
,
we can obtain
(68)
∫
0
1
‍
G
(
s
,
s
)
g
(
s
)
l
[
Ω
(
s
)
r
,
R
]
d
s
mm
≤
3
π
2
8
(
R
+
e
2
+
R
)
+
π
3
4
r
<
+
∞
.
So condition (
H
1
) is satisfied. On the other hand, by the diagonal method of choosing subsequence, we can see that L
=
0 in condition (
H
2
); that is, condition (
H
2
) holds.
We now show that condition(
H
3
) is satisfied for
(69)
J
0
=
[
1
2
,
9
10
]
,
h
(
t
)
=
1
t
,
u
0
=
(
2
π
arctan
1
2
,
2
2
π
arctan
1
2
,
2
3
π
arctan
1
2
,
…
)
.
In fact,
(70)
f
n
(
t
,
x
)
≥
π
t
(
1
-
t
)
arctan
t
n
∥
x
∥
≥
2
n
π
arctan
1
2
·
1
∥
x
∥
,
mmmmmmmmmmmmmmmmmm
∀
t
∈
[
1
2
,
9
10
]
,
which means that condition (
H
3
) is satisfied for
(71)
u
0
=
(
2
π
arctan
⁡
1
2
,
2
2
π
arctan
1
2
,
2
3
π
arctan
1
2
,
…
)
,
J
0
=
[
1
2
,
9
10
]
,
h
(
t
)
=
1
t
;
that is, condition (
H
3
) is satisfied.
Hence, our conclusion follows from Theorem7.



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*Source: 720494-2014-02-23.xml*