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A QUADRATIC INTEGRAL EQUATION IN THE SPACE OF FUNCTIONS WITH TEMPERED MODULI OF CONTINUITY†
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
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : July 25, 2014
  • Accepted : November 04, 2014
  • Published : May 30, 2015
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SHAN PENG
JINRONG WANG
FULAI CHEN

Abstract
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.
Keywords
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
PPT Slide
Lager Image
in Cω,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.
2. Preliminaries
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 Cω,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
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
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, Cω,g [ a , b ] is a linear subspace of C [ a , b ].
For x Cω,g [ a , b ], we denote
PPT Slide
Lager Image
be the least possible constant for which inequality (2) is satisfied. More precisely, we set
PPT Slide
Lager Image
Next, the space Cω,g [ a , b ] can be equipped with the norm
PPT Slide
Lager Image
for x Cω,g [ a , b ]. Then ( Cω,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 Cω,g [ a , b ] , the following inequality is satisfied
PPT Slide
Lager Image
Proof . For any x Cω,g [ a , b ] and t ∈ [ a , b ] we obtain
PPT Slide
Lager Image
Lemma 2.3. Suppose that ω 2 ( g ( t ) − g ( s )) ≤ 1 ( g ( t ) − g ( s )) for t, s ∈ [ a , b ] where G > 0. Then we have
PPT Slide
Lager Image
Moreover, for any x C ω2,g [ a , b ] the following inequality holds
PPT Slide
Lager Image
Proof . For any x C ω2,g [ a , b ], we obtain
PPT Slide
Lager Image
This shows that x C ω1,g [ a , b ] and hence we infer that inclusions hold. Further,
PPT Slide
Lager Image
Remark 2.1. In particular, if
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. 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
PPT Slide
Lager Image
. Denote
PPT Slide
Lager Image
. Then
PPT Slide
Lager Image
is compact in the space C ω1,g [ a , b ].
Proof . By Theorem 2.4, since
PPT Slide
Lager Image
is a bounded subset in C ω2,g [ a , b ], it is a relatively compact subset of C ω1,g [ a , b ]. Suppose that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
with x C ω1,g [ a , b ]. This means that for ε > 0 we can find n 0 ∈ ℕ such that
PPT Slide
Lager Image
for any n n 0 , or, equivalently
PPT Slide
Lager Image
for any n n 0 .
This implies that xn ( a ) → x ( a ).
Moreover, if in (3) we put s = a , then we get
PPT Slide
Lager Image
for any n n 0 .
The last inequality implies that
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Consequently,
PPT Slide
Lager Image
Next, we will prove that
PPT Slide
Lager Image
.
In fact, since
PPT Slide
Lager Image
, we have that
PPT Slide
Lager Image
for any t , s ∈ [ a , b ] with t > s , and, accordingly,
PPT Slide
Lager Image
for any t , s ∈ [ a , b ].
Letting in the above inequality with n → ∞ and taking into account (5), we deduce that
PPT Slide
Lager Image
for any t , s ∈ [ a , b ].
Hence we get
PPT Slide
Lager Image
for any t , s ∈ [ a , b ], and this means that
PPT Slide
Lager Image
. This proves that
PPT Slide
Lager Image
is a closed subset of C ω1,g [ a , b ]. Thus,
PPT Slide
Lager Image
is a compact subset of C ω1,g [ a , b ]. This finishes the proof. □
3. Main results
In this section, we will study the solvability of the equation (1) in Cω,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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
( 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 Kω 2 such that
PPT Slide
Lager Image
for any t , s , τ ∈ [ a , b ].
( H 3 ) The following inequality is satisfied
PPT Slide
Lager Image
where
PPT Slide
Lager Image
.
Consider the operator F defined on C ω2,g [ a , b ] by
PPT Slide
Lager Image
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
PPT Slide
Lager Image
By Lemma 2.2, since ║ x ≤ max{1, ω 2 ( g ( b ) − g ( a ))}║ x ω2,g and, as
PPT Slide
Lager Image
, we infer that
PPT Slide
Lager Image
This proves that the operator F maps C ω2,g [ a , b ] into itself. □
Lemma 3.2. Let
PPT Slide
Lager Image
where r 0 > 0 satisfy-ing the inequality (6). Then
PPT Slide
Lager Image
.
Proof . For any
PPT Slide
Lager Image
, one has
PPT Slide
Lager Image
Consequently, from above it follows that F transforms the ball
PPT Slide
Lager Image
into itself, for any r 0 ∈ [ r 1 , r 2 ]; i.e.,
PPT Slide
Lager Image
, where r 1 r 0 r 2 . □
Lemma 3.3.
PPT Slide
Lager Image
is a compact subset in C ω1,g [ a , b ].
Proof . According to Theorem 2.5, we can know
PPT Slide
Lager Image
is a compact subset in C ω1,g [ a , b ]. □
Lemma 3.4. The operator F is continuous on
PPT Slide
Lager Image
, where we consider the norm ║ㆍ║ ω1,g in
PPT Slide
Lager Image
.
Proof . To do this, we fix
PPT Slide
Lager Image
and ε > 0. Suppose that
PPT Slide
Lager Image
and ║ x y ω1,g δ , where δ is a positive number such that
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Define
PPT Slide
Lager Image
Since ║ y ω1,g ≤ ║ y ω2,g (see Remark 2.1) and
PPT Slide
Lager Image
, from the above inequality we infer that
PPT Slide
Lager Image
On the other hand,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
which yields that
PPT Slide
Lager Image
By (7) and (8), we have
PPT Slide
Lager Image
This proves the operator F is continuous at the point
PPT Slide
Lager Image
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
PPT Slide
Lager Image
for the norm ║·║ ω1,g . Since
PPT Slide
Lager Image
is compact in C ω2,g [ a , b ], applying the classical Schauder fixed point theorem we obtain the desired result. □
4. Examples
Now we make two examples illustrating the main results in the above section.
Example 4.1. Let us consider the quadratic integral equation
PPT Slide
Lager Image
Set
PPT Slide
Lager Image
arctan x ( t ) and
PPT Slide
Lager Image
for t , τ ∈ [1, e ]. It is easy to see that
PPT Slide
Lager Image
which implies Kω 2 = 1 and
PPT Slide
Lager Image
So we can choose
PPT Slide
Lager Image
Moreover,
PPT Slide
Lager Image
.
On the other hand,
PPT Slide
Lager Image
so we can get
PPT Slide
Lager Image
, k = | f (1, 1)| = 0 and
PPT Slide
Lager Image
so
PPT Slide
Lager Image
.
In what follows, the condition ( H 3 ) reduce to the inequality
PPT Slide
Lager Image
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 .
PPT Slide
Lager Image
The solution of the equation (9).
Example 4.2. Consider another quadratic integral equation
PPT Slide
Lager Image
Set
PPT Slide
Lager Image
. Obviously,
PPT Slide
Lager Image
which gives
PPT Slide
Lager Image
Then we choose
PPT Slide
Lager Image
Moreover,
PPT Slide
Lager Image
.
On the other hand,
PPT Slide
Lager Image
we can get
PPT Slide
Lager Image
and
PPT Slide
Lager Image
so derive
PPT Slide
Lager Image
.
In what follows, the condition ( H 3 ) reduce to the inequality
PPT Slide
Lager Image
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 .
PPT Slide
Lager Image
The solution of the equation (10).
BIO
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
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