# Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations withp-Laplacian Operator

Authors: Ilkay Yaslan Karaca; Fatma Tokmak
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper studies the existence of solutions for a nonlinear boundary value problem of impulsive fractional
differential equations withp-Laplacian operator. Our results are based on some standard fixed point theorems.
Examples are given to show the applicability of our results.


---

## Body

## 1. Introduction

Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in the modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, and fitting of experimental data. Fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes. The interest in the study of fractional-order differential equations lies in the fact that fractional-order models are found to be more accurate than the classical integer-order models; that is, there are more degrees of freedom in the fractional-order models. In consequence, the subject of fractional differential equations is gaining more and more attention. For some recent development on the topic, see [1–11] and the references therein.Since thep-Laplacian operator and fractional calculus arise from many applied fields such as turbulent filtration in porous media, blood flow problems, rheology, modeling of viscoplasticity, and material science, it is worth studying the fractional p-Laplacian differential equations. The research of boundary value problems for p-Laplacian equations of fractional order has just begun in recent years; see [12–16].The theory of impulsive differential equations is adequate mathematical models for description of evolution processes characterized by the combination of a continuous and jumps change of their state. Impulsive differential equations have become an important area of research in recent years of the needs of modern technology, engineering, economic and physics. Moreover, impulsive differential equations are richer in applications compared to the corresponding theory of ordinary differential equations. Many of the mathematical problems encountered in the study of impulsive differential equations cannot be treated with the usual techniques within the standard framework of ordinary differential equations. For the introduction of the basic theory of impulsive equations, see [17–22] and the references therein. It is worthwhile mentioning that impulsive differential equations of fractional order have not been much studied and many aspects of these equations are yet to be explored. The recent results on impulsive fractional differential equations can be found in [23–30].The investigation on fractionalp-Laplacian impulsive differential equations has not been appreciated well enough. As far as we know, there is only one paper that has considered nonlinear fractional impulsive differential equation with p-Laplacian operator; see [31].Motivated by the above mentioned work onp-Laplacian and impulsive boundary value problems of fractional order, in this study, we consider the following problem:
(1)
(
ϕ
p
(
C
D
0
+
α
u
(
t
)
)
)
′
=
f
(
t
,
u
(
t
)
)
,
1
<
α
≤
2
,
t
∈
J
′
,
Δ
u
(
t
k
)
=
I
k
(
u
(
t
k
)
)
,
Δ
u
′
(
t
k
)
=
J
k
(
u
′
(
t
k
)
)
,
k
=
1,2
,
…
,
m
,
a
u
(
0
)
-
b
u
′
(
0
)
=
0
,
c
u
(
1
)
+
d
u
′
(
1
)
=
0
,
where C
D
0
+
α is the Caputo fractional derivative, ϕ
p is a p-Laplacian operator, ϕ
p
(
s
)
=
|
s
|
p
-
2
s, p
>
1, ϕ
p is invertible, and (
ϕ
p
)
-
1
=
ϕ
q where (
1
/
p
)
+
(
1
/
q
)
=
1, f
∈
𝒞
(
[
0,1
]
×
ℝ
,
ℝ
), I
k
,
J
k
∈
𝒞
(
ℝ
,
ℝ
), a
,
b
,
c
,
d
∈
[
0
,
∞
) with a
c
+
a
d
+
b
c
>
0, J
=
[
0,1
], 0
=
t
0
<
t
1
<
⋯
<
t
k
<
⋯
<
t
m
<
t
m
+
1
=
1, J
′
=
J
∖
{
t
1
,
t
2
,
…
,
t
m
}, Δ
u
(
t
k
)
=
u
(
t
k
+
)
-
u
(
t
k
-
), where u
(
t
k
+
) and u
(
t
k
-
) denote the right and the left limits of u
(
t
) at t
=
t
k
(
k
=
1,2
,
…
,
m
), respectively. Δ
u
′
(
t
k
) have a similar meaning for u
′
(
t
).This paper is organized as follows. In Section2, we provide some necessary definitions and preliminary lemmas which are key tools for our main results. We give and prove our main results in Section 3. Finally, in Section 4, we give some examples to demonstrate our main results.


## 2. Preliminaries

In this section, we present auxiliary lemmas which will be used later.LetJ
0
=
[
0
,
t
1
], J
1
=
(
t
1
,
t
2
]
,
…
,
J
m
-
1
=
(
t
m
-
1
,
t
m
], J
m
=
(
t
m
,
T
], and we introduce the following spaces.P
C
(
J
,
ℝ
)
=
{
u
:
J
→
ℝ
∣
u
∈
𝒞
(
J
k
)
,
k
=
0,1
,
…
,
m
and
u
(
t
k
+
)
exist
,
k
=
1,2
,
…
,
m
} with the norm ∥
u
∥
=
sup
⁡
t
∈
J
|
u
(
t
)
|, and P
C
1
(
J
,
ℝ
)
=
{
u
:
J
→
ℝ
∣
u
∈
𝒞
1
(
J
k
)
,
k
=
0,1
,
…
,
m
,
and
u
(
t
k
+
)
,
u
′
(
t
k
+
)
exist
,
k
=
1,2
,
…
,
m
} with the norm ∥
u
∥
P
C
1
=
max
⁡
{
∥
u
∥
,
∥
u
′
∥
}. Obviously, P
C
(
J
,
ℝ
) and P
C
1
(
J
,
ℝ
) are Banach spaces.Definition 1.
For a continuous functionf
:
[
0
,
∞
)
→
R, the Caputo derivative of fractional order α is defined as
(2)
C
D
0
+
α
f
(
t
)
=
1
Γ
(
n
-
α
)
∫
0
t
‍
(
t
-
s
)
n
-
α
-
1
f
(
n
)
(
s
)
d
s
,
n
=
[
α
]
+
1
,
where [
α
] denotes the integer part of real number α.Definition 2.
The Riemann-Liouville fractional integral of orderα is defined as
(3)
I
0
+
α
f
(
t
)
=
1
Γ
(
α
)
∫
0
t
‍
(
t
-
s
)
α
-
1
f
(
s
)
d
s
,
α
>
0
,
provided the integral exists.Lemma 3 (see [7]).
Letα
>
0; then the differential equation
(4)
C
D
0
+
α
f
(
t
)
=
0
has solution f
(
t
)
=
c
0
+
c
1
t
+
c
2
t
2
+
⋯
+
c
n
-
1
t
n
-
1, c
i
∈
ℝ, i
=
0,1
,
2
,
…
,
n
-
1, n
=
[
α
]
+
1.Lemma 4 (see [7]).
Letα
>
0, f
∈
L
(
0,1
); then
(5)
C
D
0
+
α
I
0
+
α
f
(
t
)
=
f
(
t
)
.Definition 5.
A functionu
∈
P
C
1
(
J
,
ℝ
) with its Caputo derivative of order α existing on J is a solution of (1) if it satisfies (1).
We need the following known results to prove the existence of solutions for (1).Theorem 6 (see [32]).
LetE be a Banach space. Assume that Ω is an open bounded subset of E with θ
∈
Ω and let T
:
Ω
-
→
E be a completely continuous operator such that
(6)
∥
T
u
∥
≤
∥
u
∥
,
∀
u
∈
∂
Ω
.
Then T has a fixed point in Ω
-.Theorem 7 (see [32]).
LetE be a Banach space. Assume that T
:
E
→
E is a completely continuous operator and the set V
=
{
u
∈
E
∣
u
=
μ
T
u
,
0
<
μ
<
1
} is bounded. Then T has a fixed point in E.Lemma 8.
For a giveny
∈
𝒞
[
0,1
], a function u is a solution of the impulsive boundary value problem of fractional order
(7)
(
ϕ
p
(
C
D
0
+
α
u
(
t
)
)
)
′
=
y
(
t
)
,
1
<
α
≤
2
,
t
∈
J
′
,
Δ
u
(
t
k
)
=
I
k
(
u
(
t
k
)
)
,
Δ
u
′
(
t
k
)
=
J
k
(
u
′
(
t
k
)
)
,
k
=
1,2
,
…
,
m
,
a
u
(
0
)
-
b
u
′
(
0
)
=
0
,
c
u
(
1
)
+
d
u
′
(
1
)
=
0
,
if and only if u is a solution of the impulsive fractional integral equation
(8)
u
(
t
)
=
{
∫
0
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
a
t
+
b
a
c
+
a
d
+
b
c
×
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
𝒜
,
t
∈
J
0
;
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
a
t
+
b
a
c
+
a
d
+
b
c
×
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
∑
i
=
1
k
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
‍
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
I
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
-
1
‍
(
t
k
-
t
i
)
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
J
i
(
u
(
t
i
)
)
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
J
i
(
u
(
t
i
)
)
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
-
𝒜
,
t
∈
J
k
,
k

=

1,2
,
…
,
m
,
where
(9)
𝒜
=
a
t
+
b
a
c
+
a
d
+
b
c
×
{
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
+
c
I
i
(
u
(
t
i
)
)
+
d
J
i
(
u
(
t
i
)
)
∫
0
s
‍
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
iii
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
w
w
i
+
J
i
(
u
(
t
i
)
)
∫
0
s
‍
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
i
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
w
w
i
+
J
i
(
u
(
t
i
)
)
∫
0
s
‍
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
}
.Proof.
Letu be a solution of (7). Then, for t
∈
J
0, there exist constants c
0
,
c
1
∈
ℝ such that
(10)
u
(
t
)
=
I
0
+
α
ϕ
q
(
∫
0
t
‍
y
(
s
)
d
s
)
-
c
0
-
c
1
t
=
1
Γ
(
α
)
∫
0
t
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
0
-
c
1
t
,
(11)
u
′
(
t
)
=
1
Γ
(
α
-
1
)
∫
0
t
‍
(
t
-
s
)
α
-
2
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
1
.
Fort
∈
J
1, there exist constants d
0
,
d
1
∈
ℝ such that
(12)
u
(
t
)
=
1
Γ
(
α
)
∫
t
1
t
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
d
0
-
d
1
(
t
-
t
1
)
,
u
′
(
t
)
=
1
Γ
(
α
-
1
)
∫
t
1
t
‍
(
t
-
s
)
α
-
2
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
d
1
.
Then we have
(13)
u
(
t
1
-
)
=
1
Γ
(
α
)
∫
0
t
1
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
0
-
c
1
t
1
,
u
(
t
1
+
)
=
-
d
0
,
u
′
(
t
1
-
)
=
1
Γ
(
α
-
1
)
∫
0
t
1
‍
(
t
1
-
s
)
α
-
2
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
1
,
u
′
(
t
1
+
)
=
-
d
1
.
In view of Δ
u
(
t
1
)
=
u
(
t
1
+
)
-
u
(
t
1
-
)
=
I
1
(
u
(
t
1
)
) and Δ
u
′
(
t
1
)
=
u
′
(
t
1
+
)
-
u
′
(
t
1
-
)
=
J
1
(
u
(
t
1
)
), we have
(14)
-
d
0
=
1
Γ
(
α
)
∫
0
t
1
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
0
-
c
1
t
1
+
I
1
(
u
(
t
1
)
)
,
-
d
1
=
1
Γ
(
α
-
1
)
∫
0
t
1
‍
(
t
1
-
s
)
α
-
2
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
-
c
1
+
J
1
(
u
(
t
1
)
)
.
Consequently,(15)
u
(
t
)
=
1
Γ
(
α
)
∫
t
1
t
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
1
Γ
(
α
)
∫
0
t
1
‍
(
t
-
s
)
α
-
1
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
(
t
-
t
1
)
Γ
(
α
-
1
)
∫
0
t
1
‍
(
t
1
-
s
)
α
-
2
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
I
1
(
u
(
t
1
)
)
+
(
t
-
t
1
)
J
1
(
u
(
t
1
)
)
-
c
0
-
c
1
t
,
t
∈
J
1
.
By a similar process, we can get(16)
u
(
t
)
=
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
∑
i
=
1
k
‍
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
w
w
w
w
w
w
i
×
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
I
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
-
1
‍
(
t
k
-
t
i
)
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
a
w
×
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
-
c
0
-
c
1
t
,
w
w
w
w
w
w
w
w
w
a
t
∈
J
k
,
k
=
1,2
,
…
,
m
.
By conditionsa
u
(
0
)
-
b
u
′
(
0
)
=
0 and c
u
(
1
)
-
d
u
′
(
1
)
=
0, we have
(17)
c
0
=
b
a
c
+
a
d
+
b
c
×
{
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
i
+
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
i
i
w
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
i
i
w
+
c
I
i
(
u
(
t
i
)
)
+
d
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
‍
]
i
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
i
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
i
i
w
i
i
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
i
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
i
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
i
i
i
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
}
,
c
1
=
a
a
c
+
a
d
+
b
c
×
{
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
+
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
w
+
c
I
i
(
u
(
t
i
)
)
+
d
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
)
]
w
w
w
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
w
w
i
w
w
×
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
w
w
w
w
w
w
w
w
w
i
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
w
w
w
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
×
(
∫
0
s
‍
y
(
r
)
d
r
)
d
s
+
J
i
(
u
(
t
i
)
)
∫
t
m
1
‍
(
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
]
}
.
Substituting the value of c
i
(
i
=
0,1
) in (10) and (16), we can get (8). Conversely, assume that u is a solution of the impulsive fractional integral equation (8); then by a direct computation, it follows that the solution given by (8) satisfies (7). This completes the proof.


## 3. Main Results

For the sake of convenience, we set(18)
λ
1
(
t
)
=
a
t
+
b
a
c
+
a
d
+
b
c
,
λ
1
′
(
t
)
=
λ
2
=
a
a
c
+
a
d
+
b
c
.
Define the operator T
:
P
C
(
J
,
ℝ
)
→
P
C
(
J
,
ℝ
) as
(19)
T
u
(
t
)
=
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
-
λ
1
(
t
)
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
ϕ
q
w
w
w
w
w
w
w
i
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
+
∑
i
=
1
k
‍
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
w
w
w
w
w
w
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
+
I
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
-
1
‍
(
t
k
-
t
i
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
-
λ
1
(
t
)
{
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
w
w
i
i
+
c
I
i
(
u
(
t
i
)
)
+
d
J
i
(
u
(
t
i
)
)
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
wwwwwwi
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
wwwww
w
w
i
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
wwwwwi
w
w
w
w
i
w
w
w
w
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
wwwwwi
w
w
w
w
i
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
]
wwwwiwi
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
wwwwwi
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
wwwwwi
w
w
w
w
i
w
w
w
w
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
wwwwwi
w
w
w
w
i
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
]
}
.
Then the problem (1) has a solution if and only if the operator T has a fixed point.Lemma 9.
The operatorT
:
P
C
(
J
,
ℝ
)
→
P
C
(
J
,
ℝ
) defined by (19) is completely continuous.Proof.
It is obvious thatT is continuous in view of the continuity of f, I
k, and J
k. Let Ω
⊂
P
C
(
J
,
ℝ
) be bounded. Then, there exist positive constants M
i
>
0
(
i
=
1,2
,
3
) such that |
f
(
t
,
u
)
|
≤
ϕ
p
(
M
1
), |
I
k
(
u
)
|
≤
M
2 and |
J
k
(
u
)
|
≤
M
3, ∀
u
∈
Ω. Thus, ∀
u
∈
Ω, and we have
(20)
|
T
u
(
t
)
|
≤
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
λ
1
(
t
)
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
i
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
∑
i
=
1
k
‍
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
w
w
w
w
i
w
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
+
|
I
i
(
u
(
t
i
)
)
|
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
-
1
‍
(
t
k
-
t
i
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
i
w
w
u
w
+
|
J
i
(
u
(
t
i
)
)
|
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
λ
1
(
t
)
{
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
w
i
i
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
i
w
w
+
c
|
I
i
(
u
(
t
i
)
)
|
+
d
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
]
wwwwww
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
w
w
w
i
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
i
w
w
w
×
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
w
w
a
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
w
w
w
w
w
w
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
i
w
w
w
w
×
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
w
a
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
}
≤
M
1
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
d
s
+
M
1
λ
1
(
t
)
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
d
s
+
∑
i
=
1
m
‍
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
d
s
+
M
2
]
+
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
+
λ
1
(
t
)
{
∑
i
=
1
m
‍
[
M
1
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
w
w
w
w
w
w
w
w
w
w
i
w
w
w
i
w
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
d
s
w
i
w
w
w
w
i
w
i
w
w
+
c
M
2
+
d
M
3
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
]
w
w
w
w
w
i
w
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
w
w
i
w
i
w
×
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
w
w
w
w
w
i
w
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
w
w
i
w
×
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
}
≤
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
M
1
Γ
(
α
+
1
)
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
d
λ
1
(
1
)
(
1
+
m
)
]
M
1
Γ
(
α
)
+
(
1
+
c
λ
1
(
1
)
)
m
M
2
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
m
d
λ
1
(
1
)
]
M
3
.
Since t
∈
[
0,1
], therefore there exists a positive constant M, such that ∥
T
u
∥
≤
M, which implies that the operator T is uniformly bounded.
On the other hand, for anyt
∈
J
k, 0
≤
k
≤
m, we have
(21)
|
(
T
u
)
′
(
t
)
|
≤
∫
t
k
t
‍
(
t
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
λ
2
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
wwwwww
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
∑
i
=
1
k
‍
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
w
w
w
w
w
i
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
λ
2
{
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
i
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
i
w
+
c
|
I
i
(
u
(
t
i
)
)
|
+
d
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
)
]
w
w
w
i
w
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
w
i
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
w
w
w
w
w
w
w
w
i
w
w
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
w
w
w
i
w
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
i
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
w
w
w
w
w
w
w
w
w
w
i
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
i
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
}
≤
M
1
∫
t
k
t
‍
(
t
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
1
λ
2
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
d
s
+
∑
i
=
1
m
‍
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
+
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
+
λ
2
{
∑
i
=
1
m
‍
[
M
1
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
w
w
w
w
w
w
w
w
w
w
i
i
w
w
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
d
s
w
w
w
w
w
w
w
+
c
M
2
+
d
M
3
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
w
w
w
w
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
w
w
×
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
w
w
w
w
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
w
×
[
M
1
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
d
s
+
M
3
]
}
≤
c
λ
2
(
1
+
m
)
M
1
Γ
(
α
+
1
)
+
[
(
1
+
m
)
(
1
+
d
λ
2
)
+
c
(
2
m
-
1
)
]
M
1
Γ
(
α
)
+
m
c
M
2
+
[
m
(
1
+
d
)
+
c
(
2
m
-
1
)
]
M
3
:
=
M
-
.
Hence, for t
1
,
t
2
∈
J
k, t
1
<
t
2, 0
≤
k
≤
m, we have
(22)
|
(
T
u
)
(
t
2
)
-
(
T
u
)
(
t
1
)
|
≤
∫
t
1
t
2
‍
|
(
T
u
)
′
(
s
)
|
d
s
≤
M
-
(
t
2
-
t
1
)
,
which implies that T is equicontinuous on all J
k, k
=
0,1
,
2
,
…
,
m. Thus, by the Arzela-Ascoli theorem, the operator T
:
P
C
(
J
,
ℝ
)
→
P
C
(
J
,
ℝ
) is completely continuous.Theorem 10.
Letlim
⁡
u
→
0
f
(
t
,
u
)
/
ϕ
p
(
u
)
=
0, lim
⁡
u
→
0
I
k
(
u
)
/
u
=
0, and lim
⁡
u
→
0
J
k
(
u
)
/
u
=
0, and then the problem (1) has at least one solution.Proof.
Sincelim
⁡
u
→
0
f
(
t
,
u
)
/
ϕ
p
(
u
)
=
0, lim
⁡
u
→
0
I
k
(
u
)
/
u
=
0, and lim
⁡
u
→
0
J
k
(
u
)
/
u
=
0, therefore there exists a constant r
>
0 such that |
f
(
t
,
u
)
|
≤
ϕ
p
(
δ
1
)
|
ϕ
p
(
u
)
|, |
I
k
(
u
)
|
≤
δ
2
|
u
|, and |
J
k
(
u
)
|
≤
δ
3
|
u
| for 0
<
|
u
|
<
r, where δ
i
>
0
(
i
=
1,2
,
3
) satisfy the inequality
(23)
{
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
δ
1
Γ
(
α
+
1
)
ia
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
d
λ
1
(
1
)
(
1
+
m
)
]
δ
1
Γ
(
α
)
ia
+
(
1
+
c
λ
1
(
1
)
)
m
δ
2
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
a
w
w
w
w
w
w
w
w
w
w
i
w
w
+
m
d
λ
1
(
1
)
(
1
+
c
λ
1
(
1
)
)
]
δ
3
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
δ
1
Γ
(
α
+
1
)
}
≤
1
.
Let us setΩ
=
{
u
∈
P
C
(
J
,
ℝ
)
∣
∥
u
∥
<
r
} and take u
∈
P
C
(
J
,
ℝ
) such that ∥
u
∥
=
r; that is, u
∈
∂
Ω. Then, by the process used to obtain (20), we have
(24)
|
T
u
(
t
)
|
≤
{
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
δ
1
Γ
(
α
+
1
)
aa
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
d
λ
1
(
1
)
(
1
+
m
)
]
δ
1
Γ
(
α
)
aa
+
(
1
+
c
λ
1
(
1
)
)
m
δ
2
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
a
w
w
w
i
i
w
w
w
w
w
w
w
w
w
+
m
d
λ
1
(
1
)
(
1
+
c
λ
1
(
1
)
)
]
δ
3
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
δ
1
Γ
(
α
+
1
)
}
∥
u
∥
.
Thus, it follows that ∥
T
u
∥
≤
∥
u
∥, u
∈
∂
Ω. Therefore, by Theorem 6, the operator T has at least one fixed point, which in turn implies that the problem (1) has at least one solution u
∈
Ω
-.Theorem 11.
Assume that there exist positive constantsM
i
(
i
=
1,2
,
3
) such that
(25)
|
f
(
t
,
u
)
|
≤
ϕ
p
(
M
1
)
,
|
I
k
(
u
)
|
≤
M
2
,
|
J
k
(
u
)
|
≤
M
3
,
f
o
r
t
∈
J
,
u
∈
ℝ
a
n
d
k
=
1,2
,
…
,
m
.
Then the problem (1) has at least one solution.Proof.
Let us show that the setV
=
{
u
∈
P
C
(
J
,
ℝ
)
∣
u
=
μ
T
u
,
0
<
μ
<
1
} is bounded. Let u
∈
V, and then u
=
μ
T
u, 0
<
μ
<
1. For any t
∈
J, we have
(26)
u
(
t
)
=
∫
t
k
t
‍
μ
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
-
λ
1
(
t
)
∫
t
m
1
‍
(
μ
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
μ
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
×
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
+
∑
i
=
1
k
‍
μ
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
w
w
w
w
w
i
w
w
×
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
+
I
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
-
1
‍
μ
(
t
k
-
t
i
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
i
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
+
∑
i
=
1
k
‍
μ
(
t
-
t
k
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
i
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
-
λ
1
(
t
)
{
∑
i
=
1
m
‍
μ
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
w
w
i
w
×
ϕ
q
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
i
w
w
w
+
c
I
i
(
u
(
t
i
)
)
+
d
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
(
t
i
-
s
)
α
-
1
Γ
(
α
)
)
]
w
w
w
w
w
w
+
c
∑
i
=
1
m
-
1
‍
μ
(
t
m
-
t
i
)
w
w
w
w
w
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
i
w
w
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
w
i
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
w
w
w
w
w
w
+
c
∑
i
=
1
m
‍
μ
(
1
-
t
m
)
w
w
w
w
w
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
w
w
w
×
(
∫
0
s
‍
f
(
r
,
u
(
r
)
)
d
r
)
d
s
w
w
w
w
w
w
w
w
w
w
w
+
J
i
(
u
(
t
i
)
)
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
]
}
.
Combining (25) and (26) and employing the procedure used to obtain (20), we obtain
(27)
|
u
(
t
)
|
=
μ
|
T
u
(
t
)
|
≤
∫
t
k
t
‍
(
t
-
s
)
α
-
1
Γ
(
α
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
λ
1
(
t
)
∫
t
m
1
‍
(
c
(
1
-
s
)
α
-
1
Γ
(
α
)
+
d
(
1
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
+
∑
i
=
1
k
‍
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
1
Γ
(
α
)
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
+
|
I
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
∑
i
=
1
k
-
1
‍
(
t
k
-
t
i
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
∑
i
=
1
k
‍
(
t
-
t
k
)
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
+
λ
1
(
t
)
{
∑
i
=
1
m
‍
[
∫
t
i
-
1
t
i
‍
(
c
(
t
i
-
s
)
α
-
1
Γ
(
α
)
+
d
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
w
w
w
w
w
w
w
w
w
w
w
×
ϕ
q
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
w
w
+
c
|
I
i
(
u
(
t
i
)
)
|
+
d
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
)
]
w
w
w
w
w
i
a
i
+
c
∑
i
=
1
m
-
1
‍
(
t
m
-
t
i
)
w
w
w
w
w
w
w
w
w
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
w
w
a
×
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
w
w
w
a
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
w
w
w
w
w
i
a
i
+
c
∑
i
=
1
m
‍
(
1
-
t
m
)
w
w
w
w
w
w
w
w
a
×
[
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
ϕ
q
w
w
w
w
w
w
w
w
w
w
w
w
a
×
(
∫
0
s
‍
|
f
(
r
,
u
(
r
)
)
|
d
r
)
d
s
w
w
w
w
w
w
w
a
w
w
w
+
|
J
i
(
u
(
t
i
)
)
|
∫
t
i
-
1
t
i
‍
(
t
i
-
s
)
α
-
2
Γ
(
α
-
1
)
]
}
≤
{
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
M
1
Γ
(
α
+
1
)
ww
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
d
λ
1
(
1
)
(
1
+
m
)
]
M
1
Γ
(
α
)
w
w
+
(
1
+
c
λ
1
(
1
)
)
m
M
2
w
w
+
[
(
2
m
-
1
)
(
1
+
c
λ
1
(
1
)
)
+
m
d
λ
1
(
1
)
]
M
3
(
1
+
m
)
(
1
+
c
λ
1
(
1
)
)
M
1
Γ
(
α
+
1
)
}
:
=
M
,
which implies that ∥
u
∥
≤
M for any t
∈
J. So, the set V is bounded. Thus, by Theorem 7, the operator T has at least one fixed point. Hence the problem (1) has at least one solution.


## 4. Examples

Example 1.
For1
<
α
≤
2, consider the following fractional order impulsive boundary value problem:
(28)
(
ϕ
3
(
D
C
0
+
α
u
(
t
)
)
)
′
=
t
arctan
3
u
(
t
)
+
e
2
t
u
5
(
t
)
,
0
<
t
<
1
,
t
≠
t
1
,
0
<
t
1
<
1
,
Δ
u
(
t
1
)
=
sin
(
1
-
cos
⁡
u
(
t
1
)
)
,
Δ
u
′
(
t
1
)
=
e
u
2
(
t
1
)
-
cos
⁡
2
u
(
t
1
)
,
u
(
0
)
-
2
u
′
(
0
)
=
0
,
3
u
(
1
)
+
u
′
(
1
)
=
0
.
Here m
=
1, p
=
3, a
=
d
=
1, b
=
2, c
=
3, f
(
t
,
u
)
=
t
arctan
3
u
(
t
)
+
e
2
t
u
5
(
t
), I
1
(
u
(
t
1
)
)
=
sin
(
1
-
cos
⁡
u
), and J
1
(
u
(
t
1
)
)
=
e
u
2
-
cos
⁡
2
u. Clearly
(29)
lim
⁡
u
→
0
f
(
t
,
u
)
ϕ
p
(
u
)
=
lim
⁡
u
→
0
t
arctan
3
u
(
t
)
+
e
2
t
u
5
(
t
)
u
|
u
|
=
{
lim
⁡
u
→
0
+
t
arctan
3
u
(
t
)
+
e
2
t
u
5
(
t
)
u
2
,
lim
⁡
u
→
0
-
t
arctan
3
u
(
t
)
+
e
2
t
u
5
(
t
)
-
u
2
,
=
{
0
,
0
,
=
0
,
lim
⁡
u
→
0
I
1
(
u
)
u
=
lim
⁡
u
→
0
sin
(
1
-
cos
⁡
u
)
u
=
0
,
lim
⁡
u
→
0
J
1
(
u
)
u
=
lim
⁡
u
→
0
e
u
2
-
cos
⁡
2
u
u
=
lim
⁡
u
→
0
2
u
e
u
2
+
2
cos
⁡
u
sin
u
=
0
.
Furthermore, in this case, δ
i
(
i
=
1,2
) given by (23) satisfy the inequality
(30)
(
19
5
Γ
(
α
+
1
)
+
5
2
Γ
(
α
)
)
δ
1
+
19
10
δ
2
+
11
5
δ
3
≤
1
.
Thus all the assumptions of Theorem 10 hold. Hence, the conclusion of Theorem 10 applies and the impulsive fractional boundary value problem (28) has at least one solution.Example 2.
Consider the following fractional order impulsive boundary value problem:(31)
(
ϕ
4
(
C
D
0
+
α
u
(
t
)
)
)
′
=
e
3
t
cos
⁡
3
u
(
t
)
1
+
u
2
(
t
)
,
0
<
t
<
1
,
t
≠
1
3
,
Δ
u
(
1
3
)
=
2
cos
⁡
2
u
(
1
/
3
)
u
2
(
1
/
3
)
+
5
,
Δ
u
′
(
1
/
3
)
=
e
-
2
u
2
(
1
/
3
)
(
1
+
sin
(
1
+
e
2
u
(
1
/
3
)
)
)
,
2
u
(
0
)
-
u
′
(
0
)
=
0
,
u
(
1
)
+
5
u
′
(
1
)
=
0
.
Here 1
<
α
≤
2, m
=
1, p
=
4, a
=
2, b
=
c
=
1, d
=
5, |
f
(
t
,
u
)
|
=
|
e
3
t
cos
⁡
3
u
/
(
1
+
u
2
)
|
≤
e
3
=
ϕ
4
(
e
), |
I
1
(
u
)
|
=
|
2
cos
⁡
2
u
/
(
u
2
+
5
)
|
≤
2
/
5, and |
J
1
(
u
)
|
=
|
e
-
2
u
2
(
1
+
sin
(
1
+
e
2
u
)
)
|
≤
2.
In this case,M
1
=
e, M
2
=
2
/
5, M
3
=
2, and the conditions of Theorem 11 can readily be verified. Thus, by the conclusion of Theorem 11, the problem (28) has at least one solution.



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*Source: 692703-2014-03-26.xml*