A QUADRATIC INTEGRAL EQUATION IN THE SPACE OF FUNCTIONS WITH TEMPERED MODULI OF CONTINUITY†

Journal of Applied Mathematics & Informatics.
2015.
May,
33(3_4):
351-363

- Received : July 25, 2014
- Accepted : November 04, 2014
- Published : May 30, 2015

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In this paper, we investigate existence of solutions to a class of quadratic integral equation of Fredholm type in the space of functions with tempered moduli of continuity. Two numerical examples are given to illustrate our results.
AMS Mathematics Subject Classification : 26A33, 26A51, 26D15.
in
C_{ω,g}
[
a
,
b
] (see Section 2), where the functions
f
and
k
will be defined in the later.
By using a sufficient condition for the relative compactness in the space of functions with tempered moduli of continuity (see Theorem 2.5) and the classical Schauder fixed point theorem, we derive new existence result (see Theorem 3.5). Finally, two numerical examples are given to illustrate our results.
Definition 2.1
(see Section 2
[21]
). A function
ω
: ℝ
_{+}
→ ℝ
_{+}
is said to be a modulus of continuity if
ω
(0) = 0,
ω
(
ϵ
) > 0 for
ϵ
> 0, and
ω
is nondecreasing on ℝ.
Let
C
[
a
,
b
] be the space of continuous functions on [
a
,
b
] equipped with ║
x
║
_{∞}
= sup{|
x
(
t
)| :
t
∈ [
a
,
b
]} for
x
∈
C
[
a
,
b
]. We denote
C_{ω,g}
[
a
,
b
] be the set of all real functions defined on [
a
,
b
] such that their growths are tempered by the modulus of continuity
ω
with respect to a function
g
. That is, there exists a constant
such that
for all
t
,
s
∈ [
a
,
b
] where
g
: [
a
,
b
] → ℝ is a monotonic function.
Without loss of generality, we suppose that the above
g
be a increasing function and
g
(
t
) −
g
(
s
) ≥ 0 for
t
≥
s
in the this paper. Obviously,
C_{ω,g}
[
a
,
b
] is a linear subspace of
C
[
a
,
b
].
For
x
∈
C_{ω,g}
[
a
,
b
], we denote
be the least possible constant for which inequality (2) is satisfied. More precisely, we set
Next, the space
C_{ω,g}
[
a
,
b
] can be equipped with the norm
for
x
∈
C_{ω,g}
[
a
,
b
]. Then (
C_{ω,g}
[
a
,
b
], ║۰║
_{ω,g}
) is a Banach space.
Inspired by the properties of the space of Hölder functions in
[21
, see (41), (45)], we give the following sharp results.
Lemma 2.2.
For any x
∈
C_{ω,g}
[
a
,
b
]
, the following inequality is satisfied
Proof
. For any
x
∈
C_{ω,g}
[
a
,
b
] and
t
∈ [
a
,
b
] we obtain
Lemma 2.3.
Suppose that ω
_{2}
(
g
(
t
) −
g
(
s
)) ≤
Gω
_{1}
(
g
(
t
) −
g
(
s
))
for t , s
∈ [
a
,
b
]
where G
> 0.
Then we have
Moreover, for any x
∈
C
_{ω2,g}
[
a
,
b
]
the following inequality holds
Proof
. For any
x
∈
C
_{ω2,g}
[
a
,
b
], we obtain
This shows that
x
∈
C
_{ω1,g}
[
a
,
b
] and hence we infer that inclusions hold. Further,
Remark 2.1.
In particular, if
then the above imbedding relations also hold and for any
x
∈
C
_{ω2,g}
[
a
,
b
], we have ║
x
║
_{ω1,g}
≤ max{1,
M
}║
x
║
_{ω2,g}
= ║
x
║
_{ω2,g}
, where
M
is a arbitrarily small positive number.
Theorem 2.4
(see Theorem 5
[21]
).
Assume that ω
_{1}
,
ω
_{2}
are moduli of continuity being continuous at zero and such that
.
Further, assume that
(
X
,
d
)
is a compact metric space. Then, if A is a bounded subset of the space C
_{ω2,g}
(
X
)
then A is relatively compact in the space C
_{ω1,g}
(
X
).
Theorem 2.5.
Suppose that
.
Denote
.
Then
is compact in the space C
_{ω1,g}
[
a
,
b
].
Proof
. By Theorem 2.4, since
is a bounded subset in
C
_{ω2,g}
[
a
,
b
], it is a relatively compact subset of
C
_{ω1,g}
[
a
,
b
]. Suppose that
and
with
x
∈
C
_{ω1,g}
[
a
,
b
]. This means that for
ε
> 0 we can find
n
_{0}
∈ ℕ such that
for any
n
≥
n
_{0}
, or, equivalently
for any
n
≥
n
_{0}
.
This implies that
x_{n}
(
a
) →
x
(
a
).
Moreover, if in (3) we put
s
=
a
, then we get
for any
n
≥
n
_{0}
.
The last inequality implies that
for any
n
≥
n
_{0}
and for any
t
∈ [
a
,
b
].
Therefore, for any
n
≥
n
_{0}
and any
t
∈ [
a
,
b
] and taking into account (3) and (4), we have
Consequently,
Next, we will prove that
.
In fact, since
, we have that
for any
t
,
s
∈ [
a
,
b
] with
t
>
s
, and, accordingly,
for any
t
,
s
∈ [
a
,
b
].
Letting in the above inequality with
n
→ ∞ and taking into account (5), we deduce that
for any
t
,
s
∈ [
a
,
b
].
Hence we get
for any
t
,
s
∈ [
a
,
b
], and this means that
. This proves that
is a closed subset of
C
_{ω1,g}
[
a
,
b
]. Thus,
is a compact subset of
C
_{ω1,g}
[
a
,
b
]. This finishes the proof. □
C_{ω,g}
[
a
,
b
]. We will use the following assumptions:
(
H
_{1}
)
f
: [
a
,
b
] × ℝ → ℝ is a continuous function and there exists a positive number
k
_{1}
such that
and set
k
= |
f
(
a
,
a
)|. Meanwhile, for any
t
,
s
∈ [
a
,
b
] and
t
>
s
, there exists a positive constant
k
_{2}
such that the inequality
(
H
_{2}
)
k
: [
a
,
b
] × [
a
,
b
] → ℝ is a continuous function satisfies the tempered by the modulus of continuity with respect to the first variable, that is, there exists a constant
K_{ω}
_{2}
such that
for any
t
,
s
,
τ
∈ [
a
,
b
].
(
H
_{3}
) The following inequality is satisfied
where
.
Consider the operator
F
defined on
C
_{ω2,g}
[
a
,
b
] by
Lemma 3.1.
The operator F maps
C
_{ω2,g}
[
a
,
b
]
into itself
.
Proof
. In fact, we take
x
∈
C
_{ω2,g}
[
a
,
b
] and
t
,
s
∈ [
a
,
b
] with
t
>
s
. Then, by assumptions (
H
_{1}
)-(
H
_{3}
), we obtain
By Lemma 2.2, since ║
x
║
_{∞}
≤ max{1,
ω
_{2}
(
g
(
b
) −
g
(
a
))}║
x
║
_{ω2,g}
and, as
, we infer that
This proves that the operator
F
maps
C
_{ω2,g}
[
a
,
b
] into itself. □
Lemma 3.2.
Let
where
r
_{0}
> 0
satisfy-ing the inequality (6). Then
.
Proof
. For any
, one has
Consequently, from above it follows that
F
transforms the ball
into itself, for any
r
_{0}
∈ [
r
_{1}
,
r
_{2}
]; i.e.,
, where
r
_{1}
≤
r
_{0}
≤
r
_{2}
. □
Lemma 3.3.
is a compact subset in C
_{ω1,g}
[
a
,
b
].
Proof
. According to Theorem 2.5, we can know
is a compact subset in
C
_{ω1,g}
[
a
,
b
]. □
Lemma 3.4.
The operator F is continuous on
, where we consider the norm
║ㆍ║
_{ω1,g}
in
.
Proof
. To do this, we fix
and
ε
> 0. Suppose that
and ║
x
−
y
║
_{ω1,g}
≤
δ
, where
δ
is a positive number such that
where
ρ
= max{
ρ
_{1}
,
ρ
_{2}
},
ρ
_{1}
,
ρ
_{2}
is defined below. Then, for any
t
,
s
∈ [
a
,
b
] with
t
>
s
, we have
Define
Since ║
y
║
_{ω1,g}
≤ ║
y
║
_{ω2,g}
(see Remark 2.1) and
, from the above inequality we infer that
On the other hand,
where
which yields that
By (7) and (8), we have
This proves the operator
F
is continuous at the point
for the norm ║ㆍ║
_{ω1,g}
. □
Theorem 3.5.
Under assumptions
(
H
_{1}
)-(
H
_{3}
)
, the equation (1) has at least one solution in the space C
_{ω1,g}
[
a
,
b
].
Proof
. According to Lemma 3.1, Lemma 3.2, Lemma 3.3 and Lemma 3.4, the operator
F
is continuous at the point
for the norm ║·║
_{ω1,g}
. Since
is compact in
C
_{ω2,g}
[
a
,
b
], applying the classical Schauder fixed point theorem we obtain the desired result. □
Example 4.1.
Let us consider the quadratic integral equation
Set
arctan
x
(
t
) and
for
t
,
τ
∈ [1,
e
]. It is easy to see that
which implies
K_{ω}
_{2}
= 1 and
So we can choose
Moreover,
.
On the other hand,
so we can get
,
k
= |
f
(1, 1)| = 0 and
so
.
In what follows, the condition (
H
_{3}
) reduce to the inequality
Obviously, there exist a positive number satisfying these conditions. For example, one can choose
r
= 0.1.
Finally, applying Theorem 3.5, we conclude that the quadratic integral equation has at least one solution in the space
C
_{|ㆍ|α,ln}
.[1,
e
] and displayed in
Fig.1
.
The solution of the equation (9).
Example 4.2.
Consider another quadratic integral equation
Set
. Obviously,
which gives
Then we choose
Moreover,
.
On the other hand,
we can get
and
so derive
.
In what follows, the condition (
H
_{3}
) reduce to the inequality
The condition reduce to
r
< 0.1676. Obviously, there exist a positive number satisfying these conditions. For example, one can choose
r
= 0.16.
Finally, applying Theorem 3.5, we conclude that the quadratic integral equation has at least one solution in the space
C
_{|·|α}
,.[0, 1] and displayed in
Fig.2
.
The solution of the equation (10).
Shan Peng
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China.
e-mail: spengmath@126.com
JinRong Wang
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China.
Department of Mathematics, Guizhou Normal College, Guiyang, Guizhou 550018, P.R. China.
e-mail: sci.jrwang@gzu.edu.cn
Fulai Chen
Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, P.R. China.
e-mail: cflmath@163.com

1. Introduction

Fractional integral and differential equations play increasingly important roles in the modeling of real world problems. Some problems in physics, mechanics and other fields can be described with the help of all kinds of fractional differential and integral equations. For more recent development on Riemann-Liouville, Caputo and Hadamard fractional calculus, the reader can refer to the monographs
[1
,
2
,
3
,
4
,
5
,
6]
.
Quadratic integral equations arise naturally in applications of real world problems. For example, problems in the theory of radiative transfer in the theory of neutron transport and in the kinetic theory of gases lead to the quadratic equation
[7
,
8
,
9
,
10]
. There are many interesting existence results for all kinds of quadratic integral equations, one can refer to
[11
,
12
,
13
,
14
,
15
,
16
,
17]
. Our group extend to study the existence, local attractivity and stability of solutions to fractional version Urysohn type quadratic integral equations
[18]
and Erdélyi- Kober type quadratic integral equations
[19]
and Hadamard types quadratic integral equations
[20]
in the space of continuous functions.
Very recently, Banaś and Nalepa
[21]
study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity and derive the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Hölder condition. Further, Caballero et al.
[22]
study the solvability of a quadratic integral equation of Fredholm type in Hölder spaces.
The aim of the paper is to investigate the existence of solutions of the following integral equation of Fredholm type
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2. Preliminaries

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3. Main results

In this section, we will study the solvability of the equation (1) in
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4. Examples

Now we make two examples illustrating the main results in the above section.
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Citing 'A QUADRATIC INTEGRAL EQUATION IN THE SPACE OF FUNCTIONS WITH TEMPERED MODULI OF CONTINUITY†
'

@article{ E1MCA9_2015_v33n3_4_351}
,title={A QUADRATIC INTEGRAL EQUATION IN THE SPACE OF FUNCTIONS WITH TEMPERED MODULI OF CONTINUITY†}
,volume={3_4}
, url={http://dx.doi.org/10.14317/jami.2015.351}, DOI={10.14317/jami.2015.351}
, number= {3_4}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={PENG, SHAN
and
WANG, JINRONG
and
CHEN, FULAI}
, year={2015}
, month={May}