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Abstract
In this paper, by applying the behavior of measure of noncompactness in L∞(n), we study the existence of solutions of an infinite system of integral equations in the space L∞(
n). Finally, an example is included to show the usefulness of the outcome.
PUBLIC INTEREST STATEMENT
Metric fixed point theory is a powerful tool for solving many problems in various parts of mathematics. In particular, the technique of measure of noncompactness is a very useful tool for studying the existing solutions of integral equations. In this paper, we study the existence of solutions of an infinite system of integral equations in the space {L^\infty }({{\mathbb R}^n}).
1. Introduction
The concept measure of noncompactness (M.N.C) was introduced by the essential article of Kuratowski (Kuratowski, Citation1930). (Darbo, Citation1955) applied this measure to generalize both the Banach contraction principle and the Schauder fixed point theorem. Measures of noncompactness are very suitable tools which are widely applied for proving solvability of nonlinear differential and integral equations in Banach spaces (Aghajani, Allahyari, & Mursaleen, Citation2014; Aghajani & Shole Haghighi, Citation2014; Banaei & Ghaemi, Citation2017; Banaei, Ghaemi, & Saadati, Citation2017; Banás, Jleli, Mursaleen, & Samet, Citation2017; Hazarika, Srivastava, Arab, & Rabbani, Citation2018; Srivastava, Das, Hazarika, & Mohiuddine, Citation2018). Also, many authors extended Dorbo fixed point theorem and applied it to investigate the solvability of integral equations in two variables (Aghajani & Shole Haghighi, Citation2014; Arab, Allahyari, & Shole Haghighi, Citation2014; Das, Hazarika, Arab, & Mursaleen, Citation2017; Srivastava, Das, Hazarika, & Mohiuddine, Citation2019). Srivastava et al. (Srivastava et al., Citation2019) studied existence of solution for non-linear functional integral equations of two variables in Banach Algebra. Moreover, they investigated existence of solutions of infinite systems of differential equations of general order with boundary conditions in the spaces and
via the measure of noncompactness in (Srivastava et al., Citation2018). (Das et al., Citation2017) studied solvability of infinite system of integral equations in the sequence spaces
and
. (Arab et al., Citation2014) investigated the existence of solutions of infinite systems of integral equations in the Fréchet space. On the other hand Allahyari (Allahyari, Citation2017) introduced the construction of (M.N.C) in
. In this paper, by applying (M.N.C) in
(Allahyari, Citation2017), we give a fixed point theorem and study the existence of solutions of infinite systems of nonlinear functional integral equations of Urysohn type in two variables.
where ,
,
and (i ∈
). The importance of the space
is that the functions in this space do not need to be continuous. The structure of this paper is as follows. In Section 2, some definitions and concepts are recalled. Sections
is devoted to prove a fixed point theorem. In section 4, as an application for the obtained results, we present an existence theorem. Finally, in section 5 an example is given to illustrate the effectiveness of our results.
2. Preliminaries
Here, we recall some facts which will be used in our main results. Throughout this article, let , denote the set of real numbers,
and put
countable cartesian product of
with itself. Let
be a real Banach space. Moreover,
denotes the closed ball centered at
with radius
. The symbol
stands for the ball
. For
, a nonempty subset of
, we denote by
and
the closure and the closed convex hull of
, respectively. Furthermore, let us denote by
the family of nonempty bounded subsets of
and by
its subfamily consisting of all relatively compact subsets of
.
Definition 2.1 ((Banás & Goebel, Citation1980)). A mapping is said to be a measure of noncompactness in
if it satisfies the following conditions:
1° The family is nonempty and
.
2°
3°
4°
5° , for
.
6° If is a sequence of closed sets from
such that
, for
and if
then
.
The following theorems are basic for our main results.
Theorem 2.2 ((Darbo, Citation1955)). Let be a nonempty, bounded, closed and convex subset of a Banach space
and
be a continuous mapping. Assume that there exists a constant
such that
for any nonempty subset
of
,where
is a
defined in
. Then
has at least a fixed point in
.
Theorem 2.3 ((Aghajani et al., Citation2014)). Suppose are measures of noncompactness in Banach spaces
respectively. Moreover assume that the function
is convex and
if and only if
for
. Then
defines a measure of noncompactness in where
denotes the natural projection of X into
, for
.
Theorem 2.4 (Tychonoff fixed point theorem (Banás et al., Citation2017)). Let be a Hausdorff locally convex linear topological space,
a convex subset of
and
a continuous mapping such that
3. Main results
In this section, we recall the definition of a measure of noncompactness in which has been presented in (Allahyari, Citation2017). First, we recall the compact subsets of
.
Theorem 3.1. Let be a bounded set in
. Then
is relatively compact if the following conditions are satisfied:
(i) uniformly with respect to
for any
, where
.
(ii) For there is some
so that for every
We recall the Euclidean norm on the space to be
for . Let
denote the space of all Lebesgue measurable functions on
with the standard norm
Theorem 3.2 ((Allahyari, Citation2017)). Let be a bounded set in
. Then
is relatively compact if and only if
uniformly with respect to ,where
.
Suppose is a bounded subset of the space
. For
and
., let us denote
and
We know that the function is a (M.N.C) in
(Allahyari, Citation2017).
Theorem 3.3. Let
be nonempty, closed and convex subsets of
,
an arbitrary
in
and
. Let
be a continuous operator such that
where and
. Then there exists
( countable cartesian product of
with itself) such that
for all .
Proof. Consider the operator defined by
for all . Also,
is a measure of noncompactness in the space
where
, denote the natural projections of
and
. Now, by induction, we define a sequence
such that
and
,
. Then we have
, and by continuing this process we obtain
If there exists an integer such that
, then
is relatively compact and since
, therefore, Theorem 2.4 implies that
has a fixed point. Thus, there exists
such that
for
. By our assumptions, we have
Since , so
is a positive decreasing sequence of real numbers. Therefore, there is a
such that
as
. On the other hand, from the inequality (5) we have
This shows that . Consequently
. Hence, we derive that
as
. Since the sequence
is nested, from
of Definition 2.1 we infer that the set
is closed and convex subset of the set
. Now, using Theorem 2.4 implies that
has a fixed point.
4. Application
In this section, we present an existence result for the system of integral equations of Urysohn type in two variables in the spaces , where
.
Definition 4.1 ((Banás et al., Citation2017)). A function is said to have the Carathéodory property if
(i) For all the function
is measurable on
.
(ii) For almost all the function
is continuous on
.
We will consider the Equation under the following hypotheses:
(A1) are measurable functions.
(A2)
satisfies the Carathéodory conditions and
. Furthermore, there exists
such that
(A3)
satisfies the Carathéodory conditions,
and there exists nondecreasing function
such that for all
and
with
we have
Moreover, for any
uniformly with respect to such that
(A4) The following equality holds:
(A5) The inequality
has a positive solution .
Theorem 4.2. Under assumptions -
,the Equation
has at least one solution in the space
,where
.
Proof. First we fix arbitrary . Define
by
In view of the Carathéodory conditions, we infer that is measurable for any
.
Now, we show that . To this end, from conditions
-
we have
Therefore, we obtain
Thus, and
is well defined. From (4.9) and using
, we infer the function
maps
into
.
Now we show that is continuous function. Let us fix
and consider
such that
. Then, we have
By applying condition and
we select
,
such that for
the following inequalities holds
From (4.10), we have
Now, for almost all , we have
where
In view of the Carathéodory conditions for on the compact set
, we have
as
. Thus from (4.11), (4.12) and (4.13) we infer that
is a continuous function on
. In order to finish the proof, we show that
satisfies assumptions imposed in Theorem 3.3. Let
be nonempty and bounded subset of
. Suppose that
and
are arbitrary constants. For almost all
and
we have
where
Since is arbitrary element of
in (4.14), we have
In view of the Carathéodory conditions for and
on the compact set
and
, respectively and Corollary 3.2, we infer that
and
as
. Therefore, we have
Now taking and by applying assumption
we obtain
On the other hand, for all and
, we get
Thus, we have
If take in the inequality (4.16), then using
we have
If we consider in the inequality (4.17), then
Moreover, combining (4.15) and (4.18) imply that
Since is a constant, (4.19) shows that
and we get
Taking , we have:
Now by applying Theorem 3.3, there exists that is solution of the system of integral EquationEquation (1.1)
(1.1)
(1.1) and this completes the proof.□
Now, we study the following example to show the usefullnees of the Theorem 4.2.
Example 4.3. Consider the system of integral equations
where the symbol shows the integer part of
and
.
EquationEquationEquation (4.20(4.20)
(4.20)
(4.20)
(4.20) ) is a special case of Equation. (1.1) with
Condition is satisfied. Suppose that
and
. We have
The case can be treated in the same way. Also,
and
satisfies the Carathéodory conditions and
. Thus, hypothesis
holds.
for any number .
Also,
uniformly with respect to such that
. Therefore, condition
is satisfied. Moreover,
which shows that condition holds.
It is easy to check that condition satisfies i.e.,
We take ,which shows the inequality of condition
holds. Therefore, the system of integral EquationEquationEquation (4.20
(4.20)
(4.20)
(4.20)
(4.20) ) has at least one solution.
Additional information
Funding
Notes on contributors
Ayub Samadi
The research fields of authors are fixed point theory with its applications. We are assistant professor and faculty member at Islamic Azad University. The authors have studied Darbo’s fixed point theorem in Banach space. Also, we give an application of obtained results and analyze the existence of solutions integral equations by using the technique of measure of noncompactness.
References
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