# Solvability of a Quadratic Integral Equation of Fredholm Type with Supremum in Hölder Spaces

Authors: J. Caballero Mena; R. Nalepa; K. Sadarangani
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

Using the classical Schauder fixed point theorem, we prove the existence of solutions of a quadratic integral equation of Fredholm type with supremum in the space of functions satisfying the Hölder condition.


---

## Body

## 1. Introduction

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(1)
x
(
t
)
=
1
+
t
x
(
t
)
∫
0
1
‍
Φ
(
s
)
t
+
s
x
(
s
)
d
s
,
where Φ is a continuous function defined on the interval [
0,1
] (see [1–4]).Equations of this type have been studied by several authors [5–11].The aim of this paper is to investigate the existence of solutions of the following quadratic integral equation of Fredholm type with supremum:(2)
x
(
t
)
=
p
(
t
)
+
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
,
t
∈
[
0,1
]
.Differential and integral equations with supremum are adequate models of real-world problems, in which the present state depends significantly on this maximum value on a past time interval [12]. Equations of such kind have been studied in some papers appearing in the literature (see [13–20]).Our solutions are placed in the space of functions satisfying the Hölder condition. A sufficient condition for the relative compactness in these spaces and the classical Schauder fixed point theorem are the main tools used in our study.


## 2. Preliminaries

Our starting point in this section is to introduce the space of functions satisfying the Hölder condition and some properties in this space. These properties appear in [21].Let[
a
,
b
] be a closed interval in ℝ; by C
[
a
,
b
] we denote the space of the continuous functions on [
a
,
b
] equipped with the norm ∥
x
∥
∞
=
sup
⁡
{
|
x
(
t
)
|
:
t
∈
[
a
,
b
]
}.For fixed0
<
α
⩽
1, by H
α
[
a
,
b
], we will denote the space of the real functions x defined on [
a
,
b
] and satisfying the Hölder condition, that is, those functions x for which there exists a constant H
x
α such that
(3)
|
x
(
t
)
-
x
(
s
)
|
⩽
H
x
α
|
t
-
s
|
α
,
for all t
,
s
∈
[
a
,
b
].It is easily seen thatH
α
[
a
,
b
] forms a linear subspace of C
[
a
,
b
].In what follows, forx
∈
H
α
[
a
,
b
], by H
x
α, we will denote the least possible constant for which inequality (3) is satisfied. More “precisely,” we put
(4)
H
x
α
=
sup
⁡
{
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
α
:
t
,
s
∈
[
a
,
b
]
,
t
≠
s
}
.
The space H
α
[
a
,
b
] with 0
<
α
⩽
1 can be normed under the following norm:
(5)
∥
x
∥
α
=
|
x
(
a
)
|
+
sup
⁡
{
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
α
:
t
,
s
∈
[
a
,
b
]
,
t
≠
s
}
,
for x
∈
H
α
[
a
,
b
].It is proved in [21] that (
H
α
[
a
,
b
]
,
∥
·
∥
α
) with 0
<
α
⩽
1 is a Banach space.Now, we recollect some results about the spacesH
α
[
a
,
b
] with 0
<
α
⩽
1 which appear in [21].Lemma 1.
Forx
∈
H
α
[
a
,
b
] with 0
<
α
⩽
1 the following inequality is satisfied:
(6)
∥
x
∥
∞
⩽
max
⁡
(
1
,
(
b
-
a
)
α
)
∥
x
∥
α
.Lemma 2.
For0
<
α
<
γ
⩽
1, one has
(7)
H
γ
[
a
,
b
]
⊂
H
α
[
a
,
b
]
⊂
C
[
a
,
b
]
.Moreover, for x
∈
H
γ
[
a
,
b
] the following inequality holds:
(8)
∥
x
∥
α
⩽
max
⁡
(
1
,
(
b
-
a
)
γ
-
α
)
∥
x
∥
γ
.The following sufficient condition for relative compactness in the spacesH
α
[
a
,
b
] with 0
<
α
⩽
1 appears in Example 6 of [21].Theorem 3.
Suppose that0
<
α
<
β
⩽
1 and that A is a bounded subset in H
β
[
a
,
b
]
(this means that there exists a constant M
>
0 such that |
x
(
t
)
-
x
(
s
)
|
⩽
M
|
t
-
s
|
β for any x
∈
A and for any t
,
s
∈
[
a
,
b
]
). Then A is a relatively compact subset of H
α
[
a
,
b
].Remark 4.
Suppose that0
<
α
<
β
⩽
1 and by B
r
β we denote the ball centered at θ and radius r in the space H
β
[
a
,
b
]; that is, B
r
β
=
{
x
∈
H
β
[
a
,
b
]
:
∥
x
∥
β
⩽
r
}. Then B
r
β is compact in the space H
α
[
a
,
b
].Proof.
In fact, by Theorem3, since B
r
β is a bounded subset in H
β
[
a
,
b
], B
r
β is a relatively compact subset of H
α
[
a
,
b
].
Suppose that(
x
n
)
⊂
B
r
β and x
n
→
∥
·
∥
α
x with x
∈
H
α
[
a
,
b
]. This means that for ε
>
0 we can find n
0
∈
ℕ such that
(9)
∥
x
n
-
x
∥
α
⩽
ε
for
any
n
⩾
n
0
,
or, equivalently
(10)
|
x
n
(
a
)
-
x
(
a
)
|
+
sup
⁡
{
(
|
t
-
s
|
α
)
-
1
:
t
,
s
∈
[
a
,
b
]
,
t
≠
s
(
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
s
)
-
x
(
s
)
)
|
)
}
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
s
)
-
x
(
s
)
)
|
|
t
-
s
|
α
:
000000000
t
,
s
∈
[
a
,
b
]
,
t
≠
s
(
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
s
)
)
|
)
(
|
t
-
s
|
α
)
-
1
[
a
,
b
]
(
|
x
n
(
x
n
x
(
s
)
)
|
)
}
(
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
s
)
-
x
(
s
)
)
|
)
|
t
-
s
|
α
}
⩽
ε
for
any
n
⩾
n
0
.
This implies that x
n
(
a
)
→
x
(
a
).
Moreover, if in (10) we put s
=
a, then we get
(11)
sup
⁡
{
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
a
)
-
x
(
a
)
)
|
|
t
-
a
|
α
:
t
∈
[
a
,
b
]
,
t
≠
a
}
<
ε
00000000000000000000000000000000000000
for
any
n
⩾
n
0
.
The last inequality implies that
(12)
|
x
n
(
t
)
-
x
(
t
)
-
(
x
n
(
a
)
-
x
(
a
)
)
|
<
ε
|
t
-
a
|
α
⩽
ε
(
b
-
a
)
α
00000000000000000
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 (10) and (12), we have
(13)
|
x
n
(
t
)
-
x
(
t
)
|
⩽
|
(
x
n
(
t
)
-
x
(
t
)
)
-
(
x
n
(
a
)
-
x
(
a
)
)
|
+
|
x
n
(
a
)
-
x
(
a
)
|
<
ε
(
b
-
a
)
α
+
ε
=
ε
(
1
+
(
b
-
a
)
α
)
.
Consequently,
(14)
∥
x
n
-
x
∥
∞
⟶
0
.
Next, we will prove thatx
∈
B
r
β.
In fact, since(
x
n
)
⊂
B
r
β
⊂
H
β
[
a
,
b
], we have that
(15)
|
x
n
(
t
)
-
x
n
(
s
)
|
|
t
-
s
|
β
⩽
r
for
any
t
,
s
∈
[
a
,
b
]
with
t
≠
s
,
and, accordingly,
(16)
|
x
n
(
t
)
-
x
n
(
s
)
|
⩽
r
|
t
-
s
|
β
for
any
t
,
s
∈
[
a
,
b
]
.
Letting n
→
∞ in the above inequality and taking into account (14), we deduce that
(17)
|
x
(
t
)
-
x
(
s
)
|
⩽
r
|
t
-
s
|
β
for
any
t
,
s
∈
[
a
,
b
]
.
Hence, we get
(18)
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
β
⩽
r
for
any
t
,
s
∈
[
a
,
b
]
,
and this means that x
∈
B
r
β.
This proves thatB
r
β is a closed subset of H
α
[
a
,
b
]. Thus, B
r
β is a compact subset of H
α
[
a
,
b
].
This finishes the proof.Next, we recall some results appearing in [18].In what follows, we consider[
a
,
b
]
=
[
0,1
].Lemma 5.
Letr
:
[
0,1
]
→
[
0,1
] be a continuous and nondecreasing function and x
∈
C
[
0,1
]. Let G
x be the function defined by
(19)
(
G
x
)
(
t
)
=
max
⁡
τ
∈
[
0
,
r
(
t
)
]
|
x
(
τ
)
|
for
t
∈
[
0,1
]
.
Then G
x
∈
C
[
0,1
].Under the above assumption we have the following.Lemma 6.
Let(
x
n
) be a sequence in C
[
0,1
] and x
∈
C
[
0,1
] such that x
n
→
x in C
[
0,1
]. Then
(20)
∥
G
x
n
-
G
x
∥
∞
⩽
∥
x
n
-
x
∥
∞
.Finally, we recall Schauder’s fixed point theorem.Theorem 7.
LetQ be a nonempty, convex, and compact subset of a Banach space (
X
,
∥
·
∥
) and let T
:
Q
→
Q be a continuity mapping. Then T has at least one fixed point in Q.


## 3. Main Result

In this section we will investigate the solvability of the integral equation (2) in the Hölder spaces.We will formulate the following assumptions:(i)
p
∈
H
β
[
0,1
] (0
<
β
⩽
1);
(ii)
k
:
[
0,1
]
×
[
0,1
]
→
ℝ is a continuous function such that it satisfies the Hölder condition with exponent β with respect to the first variable; that is, there exists a constant k
β
>
0 such that
(21)
|
k
(
t
,
ϱ
)
-
k
(
s
,
ϱ
)
|
⩽
k
β
|
t
-
s
|
β
,
for any t
,
s
,
ϱ
∈
[
0,1
];
(iii)
r
:
[
0,1
]
→
[
0,1
] is a continuous and nondecreasing function;
(iv)
the following inequality is satisfied:(22)
∥
p
∥
β
(
2
K
+
k
β
)
<
1
4
,
where the constant K is defined by
(23)
K
=
sup
⁡
{
∫
0
1
‍
|
k
(
t
,
τ
)
|
d
τ
:
t
∈
[
0,1
]
}
and whose existence is guaranteed by virtue of (ii).Theorem 8.
Under assumptions(
i
)
–
(
i
v
), (2) has at least one solution belonging to the space H
α
[
0,1
], where α is arbitrarily fixed number satisfying 0
<
α
<
β.Proof.
Let us consider the operatorF defined on the space H
β
[
0,1
] by
(24)
(
F
x
)
(
t
)
=
p
(
t
)
+
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
,
for t
∈
[
0,1
].
First, we will prove thatF transforms the space H
β
[
0,1
] into itself.
In fact, we take an arbitrary functionx
∈
H
β
[
0,1
] and t
,
s
∈
[
0,1
] such that t
≠
s. Then, in view of assumptions (i) and (ii), we get
(25)
|
(
F
x
)
(
t
)
-
(
F
x
)
(
s
)
|
|
t
-
s
|
β
=
|
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
p
(
t
)
+
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
p
(
s
)
-
x
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
×
(
|
t
-
s
|
β
)
-
1
⩽
|
p
(
t
)
-
p
(
s
)
|
|
t
-
s
|
β
+
|
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
x
(
s
)
×
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
×
(
|
t
-
s
|
β
)
-
1
+
|
x
(
s
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
x
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
×
(
|
t
-
s
|
β
)
-
1
⩽
|
p
(
t
)
-
p
(
s
)
|
|
t
-
s
|
β
+
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
β
·
∫
0
1
‍
|
k
(
t
,
τ
)
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
+
|
x
(
s
)
|
∫
0
1
‍
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
t
-
s
|
β
⩽
|
p
(
t
)
-
p
(
s
)
|
|
t
-
s
|
β
+
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
β
∥
x
∥
∞
×
∫
0
1
‍
|
k
(
t
,
τ
)
|
d
τ
+
∥
x
∥
∞
∥
x
∥
∞
∫
0
1
‍
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
d
τ
|
t
-
s
|
β
⩽
|
p
(
t
)
-
p
(
s
)
|
|
t
-
s
|
β
+
K
∥
x
∥
∞
×
|
x
(
t
)
-
x
(
s
)
|
|
t
-
s
|
β
+
∥
x
∥
∞
2
∫
0
1
‍
k
β
|
t
-
s
|
β
d
τ
|
t
-
s
|
β
⩽
H
p
β
+
K
∥
x
∥
∞
H
x
β
+
k
β
∥
x
∥
∞
2
.
By Lemma 1, ∥
x
∥
∞
⩽
∥
x
∥
β, and, since H
x
β
⩽
∥
x
∥
β, from the last inequality it follows that
(26)
|
(
F
x
)
(
t
)
-
(
F
x
)
(
s
)
|
|
t
-
s
|
β
⩽
H
p
β
+
K
∥
x
∥
β
2
+
k
β
∥
x
∥
β
2
=
H
p
β
+
(
K
+
k
β
)
∥
x
∥
β
2
.
Therefore,
(27)
∥
F
x
∥
β
=
|
(
F
x
)
(
0
)
|
+
sup
⁡
{
|
(
F
x
)
(
t
)
-
(
F
x
)
(
s
)
|
|
t
-
s
|
β
:
t
,
s
∈
[
0,1
]
,
t
≠
s
}
⩽
|
(
F
x
)
(
0
)
|
+
H
p
β
+
(
K
+
k
β
)
∥
x
∥
β
2
⩽
|
p
(
0
)
|
+
|
x
(
0
)
|
∫
0
1
‍
|
k
(
0
,
τ
)
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
+
H
p
β
+
(
K
+
k
β
)
∥
x
∥
β
2
⩽
∥
p
∥
β
+
∥
x
∥
∞
·
∥
x
∥
∞
∫
0
1
‍
|
k
(
0
,
τ
)
|
d
τ
+
(
K
+
k
β
)
∥
x
∥
β
2
⩽
∥
p
∥
β
+
K
∥
x
∥
∞
2
+
(
K
+
k
β
)
∥
x
∥
β
2
⩽
∥
p
∥
β
+
K
∥
x
∥
β
2
+
(
K
+
k
β
)
∥
x
∥
β
2
=
∥
p
∥
β
+
(
2
K
+
k
β
)
∥
x
∥
β
2
<
∞
,
where we have used the fact that ∥
p
∥
β
=
|
p
(
0
)
|
+
H
p
β.
This shows that the operatorF transforms H
β
[
0,1
] into itself.
On the other hand, the inequality(28)
∥
p
∥
β
+
(
2
K
+
k
β
)
r
2
⩽
r
is satisfied by the number
(29)
r
1
=
1
-
1
-
4
∥
p
∥
β
(
2
K
+
k
β
)
2
(
2
K
+
k
β
)
,
r
2
=
1
+
1
-
4
∥
p
∥
β
(
2
K
+
k
β
)
2
(
2
K
+
k
β
)
,
which are positive by virtue of (iv), and, consequently, from (27) we infer that F transforms the ball B
r
0
β
=
{
x
∈
H
β
[
0,1
]
:
∥
x
∥
β
⩽
r
0
} into itself, for any r
0
∈
[
r
1
,
r
2
].
Therefore,F
:
B
r
0
β
→
B
r
0
β, where r
1
⩽
r
0
⩽
r
2.
By Theorem3 and Remark 4, B
r
0
β is a compact subset in the space H
α
[
0,1
] for any 0
<
α
<
β
⩽
1.
Next, we will prove that the operatorF is continuous on B
r
0
β, where in B
r
0
β we consider the induced norm by ∥
·
∥
α, where 0
<
α
<
β
⩽
1.
To this end fixx
∈
B
r
0
β and ε
>
0. Suppose that y
∈
B
r
0
β and ∥
x
-
y
∥
α
⩽
δ, where δ
⩽
ε
/
2
(
2
K
+
3
k
β
)
r
0.
Then, for anyt
,
s
∈
[
0,1
] with t
≠
s, we have
(30)
|
[
(
F
x
)
(
t
)
-
(
F
y
)
(
t
)
]
-
[
(
F
x
)
(
s
)
-
(
F
y
)
(
s
)
]
|
|
t
-
s
|
α
=
|
[
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
]
-
[
x
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
]
|
×
(
|
t
-
s
|
α
)
-
1
=
|
[
x
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
]
+
[
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
]
-
[
x
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
]
-
[
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
]
|
×
(
|
t
-
s
|
α
)
-
1
=
|
[
x
(
t
)
-
y
(
t
)
]
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
+
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
-
[
x
(
s
)
-
y
(
s
)
]
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
[
x
(
t
)
-
y
(
t
)
]
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
·
1
|
t
-
s
|
α
⩽
{
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
|
[
x
(
t
)
-
y
(
t
)
]
-
[
x
(
s
)
-
y
(
s
)
]
|
·
|
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
+
|
x
(
s
)
-
y
(
s
)
|
·
|
∫
0
1
(
k
(
t
,
τ
)
-
k
(
s
,
τ
)
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
|
+
|
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
00000000000000000000
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
-
y
(
s
)
×
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
|
}
·
1
|
t
-
s
|
α
⩽
|
[
x
(
t
)
-
y
(
t
)
]
-
[
x
(
s
)
-
y
(
s
)
]
|
|
t
-
s
|
α
·
∥
x
∥
∞
∫
0
1
‍
|
k
(
t
,
τ
)
|
d
τ
+
[
|
[
x
(
s
)
-
y
(
s
)
]
-
[
x
(
0
)
-
y
(
0
)
]
|
+
|
x
(
0
)
-
y
(
0
)
|
]
·
∥
x
∥
∞
×
∫
0
1
‍
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
|
t
-
s
|
α
d
τ
+
|
y
(
t
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
·
1
|
t
-
s
|
α
+
|
y
(
s
)
∫
0
1
‍
k
(
t
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
-
y
(
s
)
∫
0
1
‍
k
(
s
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
·
1
|
t
-
s
|
α
⩽
∥
x
-
y
∥
α
∥
x
∥
∞
K
+
sup
⁡
t
,
s
∈
[
0,1
]
|
[
x
(
t
)
-
y
(
t
)
]
-
[
x
(
s
)
-
y
(
s
)
]
|
·
∥
x
∥
∞
·
∫
0
1
‍
k
β
|
t
-
s
|
β
|
t
-
s
|
α
d
τ
+
|
x
(
0
)
-
y
(
0
)
|
·
∥
x
∥
∞
×
∫
0
1
‍
k
β
|
t
-
s
|
β
|
t
-
s
|
α
d
τ
+
|
y
(
t
)
-
y
(
s
)
|
|
t
-
s
|
α
×
∫
0
1
‍
|
k
(
t
,
τ
)
|
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
|
d
τ
+
|
y
(
s
)
|
∫
0
1
‍
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
|
t
-
s
|
α
×
|
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
|
d
τ
⩽
K
∥
x
∥
∞
∥
x
-
y
∥
α
+
sup
⁡
t
,
s
∈
[
0,1
]
{
|
[
x
(
t
)
-
y
(
t
)
]
-
[
x
(
s
)
-
y
(
s
)
]
|
|
t
-
s
|
α
·
|
t
-
s
|
α
}
·
∥
x
∥
∞
k
β
|
t
-
s
|
β
-
α
+
k
β
∥
x
∥
∞
|
t
-
s
|
β
-
α
|
x
(
0
)
-
y
(
0
)
|
+
H
y
α
K
∥
x
-
y
∥
∞
+
∥
y
∥
∞
∥
x
-
y
∥
∞
∫
0
1
‍
k
β
|
t
-
s
|
β
|
t
-
s
|
α
d
τ
⩽
K
∥
x
∥
β
∥
x
-
y
∥
α
+
k
β
∥
x
∥
β
∥
x
-
y
∥
α
+
k
β
∥
x
∥
β
∥
x
-
y
∥
α
+
H
y
α
K
∥
x
-
y
∥
α
+
k
β
∥
y
∥
β
∥
x
-
y
∥
α
⩽
(
K
r
0
+
3
k
β
r
0
)
∥
x
-
y
∥
α
+
K
∥
y
∥
α
∥
x
-
y
∥
α
⩽
(
K
r
0
+
3
k
β
r
0
)
×
∥
x
-
y
∥
α
+
K
∥
y
∥
β
∥
x
-
y
∥
α
⩽
(
2
K
r
0
+
3
k
β
r
0
)
∥
x
-
y
∥
α
⩽
(
2
K
r
0
+
3
k
β
r
0
)
δ
<
ε
2
.
On the other hand, we have
(31)
|
(
F
x
)
(
0
)
-
(
F
y
)
(
0
)
|
=
|
x
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
y
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
⩽
|
x
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
}
d
τ
-
x
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
+
|
x
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
-
y
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
⩽
|
x
(
0
)
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
x
(
η
)
|
-
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
+
|
[
x
(
0
)
-
y
(
0
)
]
∫
0
1
‍
k
(
0
,
τ
)
{
max
⁡
η
∈
[
0
,
r
(
τ
)
]
|
y
(
η
)
|
}
d
τ
|
⩽
∥
x
∥
∞
K
∥
x
-
y
∥
∞
+
∥
x
-
y
∥
α
∥
y
∥
∞
K
⩽
∥
x
∥
β
K
∥
x
-
y
∥
α
+
∥
x
-
y
∥
α
∥
y
∥
β
K
⩽
2
K
r
0
∥
x
-
y
∥
α
<
2
K
r
0
δ
<
ε
2
.
From (30) and (31), we infer that
(32)
∥
F
x
-
F
y
∥
α
=
|
(
F
x
)
(
0
)
-
(
F
y
)
(
0
)
|
+
sup
⁡
{
|
[
(
F
x
)
(
t
)
-
(
F
y
)
(
t
)
]
-
[
(
F
x
)
(
s
)
-
(
F
y
)
(
s
)
]
|
|
t
-
s
|
α
:
t
,
s
∈
[
0,1
]
,
t
≠
s
|
[
(
F
x
)
(
t
)
-
(
F
y
)
(
t
)
]
-
[
(
F
x
)
(
s
)
-
(
F
y
)
(
s
)
]
|
|
t
-
s
|
α
}
<
ε
2
+
ε
2
=
ε
.
This shows that the operator F is continuous at the point x
∈
B
r
0
β, where x is arbitrary.
Finally, applying Theorem7, we deduce that desired result.
Finally, we illustrate our results by presenting an example.
Example
9. Let us consider the following quadratic integral equation:
(33)
x
(
t
)
=
q
t
+
r
+
x
(
t
)
×
∫
0
1
‍
m
t
2
+
τ
3
{
max
⁡
η
∈
[
0
,
τ
/
(
τ
+
1
)
]
|
x
(
η
)
|
}
d
τ
,
where t
∈
[
0,1
] and q
,
r, and m are positive constants.
Notice that (33) is a particular case of (2), where p
(
t
)
=
q
t
+
r
,
k
(
t
,
τ
)
=
m
t
2
+
τ
3 and r
(
τ
)
=
τ
/
(
τ
+
1
).
It is easily seen that(34)
|
p
(
t
)
-
p
(
s
)
|
⩽
q
|
t
-
s
|
1
/
2
,
for
any
t
,
s
∈
[
0,1
]
,
and, consequently, p
(
t
)
=
q
t
+
r
∈
H
1
/
2
[
0,1
].
Moreover, using the inequality proved in [11], we have
(35)
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
=
|
m
t
2
+
τ
3
-
m
s
2
+
τ
3
|
⩽
m
3
|
t
-
s
|
2
/
3
,
for any t
,
s
,
τ
∈
[
0,1
]. Therefore, assumption (ii) of Theorem 8 is satisfied, since for any t
,
s
,
τ
∈
[
0,1
]
(36)
|
k
(
t
,
τ
)
-
k
(
s
,
τ
)
|
⩽
m
3
|
t
-
s
|
2
/
3
=
m
3
|
t
-
s
|
1
/
2
|
t
-
s
|
1
/
6
⩽
m
3
|
t
-
s
|
1
/
2
,
where k
β
=
m
3.
Moreover, we have that(37)
∥
p
∥
1
/
2
=
|
p
(
0
)
|
+
sup
⁡
{
|
p
(
t
)
-
p
(
s
)
|
|
t
-
s
|
1
/
2
:
t
,
s
∈
[
0,1
]
,
t
≠
s
}
=
r
+
q
.
Since r
(
τ
)
=
τ
/
(
τ
+
1
), the function r is continuous and increasing on [
0,1
] and 0
⩽
r
(
τ
)
⩽
1
/
2 for all τ
∈
[
0,1
]. Thus, assumption (iii) of Theorem 8 is satisfied.
In our case, the constantK is given by
(38)
K
=
sup
⁡
{
∫
0
1
‍
|
k
(
t
,
τ
)
|
d
τ
:
t
∈
[
0,1
]
}
=
sup
⁡
{
∫
0
1
‍
m
t
2
+
τ
3
d
τ
:
t
∈
[
0,1
]
}
=
sup
⁡
{
3
4
[
(
m
t
2
+
1
)
4
3
-
(
m
t
2
)
4
3
]
:
t
∈
[
0,1
]
}
=
3
4
[
(
m
+
1
)
4
3
-
m
4
3
]
.
In our case, the inequality appearing in assumption (iv) of Theorem 8 takes the form
(39)
∥
p
∥
1
/
2
(
2
K
+
k
β
)
=
(
r
+
q
)
(
3
2
[
(
m
+
1
)
4
3
-
m
4
3
]
+
m
3
)
<
1
4
.
It is easily seen that the above inequality is satisfied when, for example, q
=
r
=
1
/
216 and m
=
1
/
128.
Therefore, using Theorem8, we infer that (33) for q
=
r
=
1
/
216 and m
=
1
/
128 has at least one solution in the space H
α
[
0,1
] with 0
<
α
<
1
/
2.



---

*Source: 856183-2014-04-29.xml*