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Abstract
In this paper, we extend Darbo’s fixed point theorem in Banach space via the concept of the class of operators O(f;.) and obtain a tripled fixed point theorem. The main tool applied in our investigation is the technique of measures of noncompactness. Finally, as an application of our obtained results, we analyze the existence of solutions for a system of integral equations.
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Metric fixed point theory is a powerful tool for solving many problems in various parts of mathematics and its applications. 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 extend the Darbo’s fixed point theorem and analyze the existence of solutions for a system of integral equations.
1. Introduction and preliminaries
The concept of a measure of noncompactness (MNC) was introduced by Kuratowski (1930). These measures are very useful tools in the wide area of nonlinear functional analysis. Darbo’s fixed point theorem (Darbo, Citation1955) which ensures the existence of fixed point is an important application of this measure, since it generalizes both the classical Schauder fixed point and Banach contraction principle. Thereafter, many papers have been studied in the MNC and it’s application, see for example (Aghajani et al., Citation2014; Aghajani, Banás, & Jalilian, Citation2011; Aghajani, Banás, & Sabzali, Citation2013; Banaei, Citation2018; Banaei, & Ghaemi, Citation2019; Banaei, Ghaemi, & Saadati, 2017; Habib, & Benaicha, Citation2018; Ilkhan, & Kara; Mondal, 2019) . The aim of this paper is to generalize the Darbo’s fixed point theorem via the concept of the class of operators and extend the results of the theorems in, (Samadi, & Ghaemi, Citation2014). In this work, first, some definitions and results are recalled. In the second section, main results and an extension of Darbo fixed point theorem are presented. finally, we apply our extension to study the existence of solutions for the system
where .
Now, we introduce some notations and definitions which are used throughout this paper. Let be the set of real numbers,
and
be a real Banach space with the zero element 0. We write
to denote the closed ball centered at
with radius
. If
be a nonempty subset of
then the symbols
and
stand for the closure and closed convex hull of
, respectively. Moreover,
is the family of nonempty bounded subset of
and
denote its subfamily consisting of all relatively compact sets.
Definition 1.1 (Banas & Goebel, Citation1980). A mapping is said to be a measure of noncompactness in
if it satisfies the following conditions:
The family
is nonempty and
.
.
.
.
for
.
If
such that
and
, then
is nonempty.
The following concept of was given by Altun and Turkoglu (Altun & Turkoglu, Citation2007).
Let be the class of all functions
and let
be the class of all operators
satisfying the following conditions:
for
and
.
for
.
.
for some
.
Definition 1.2. (Banaei, Ghaemi, & Saadati, Citation2017). A triple of a mapping
is called a triple fixed pointed if
The following theorems are basic for our main results.
Theorem 1.1. (Schauder (Akmerov, Kamenski, Potapov, Rodkina, & Sadovskii, Citation1992)). Let be a nonempty, bounded, closed and convex subset of a Banach space
. Then every compact and continuous map
has at least one fixed point.
Theorem 1.2 .(Darbo 1955). 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 MNC defined in
. Then
has at least a fixed point in
.
Theorem 1.3. (Aghajani, Citation2014). Let are measures of noncompactness in
respectively. Suppose
is a convex function and
if and only if
= 0 for
Then, the function
=
defines a measure of noncompactness in
, where
denote the natural projection of
into
for
.
Remark 1.1. Aghajani et al. (Citation2013) illustrated the Theorem 1.3 by the following example. Let be as follows:
They showed that
defines an MNC in the space where
,
are the natural projection of
into
.
Samadi and Ghaemi (Samadi, Citation2014) generalized Darbo’s fixed point theorem as fallow.
Theorem 1.4. Let be a nonempty, bounded, closed and convex subset of a Banach space
. Assume
be a continuous operator such that satisfying
for all nonempty subset of
, where
is an arbitrary MNC in
,
:
is nondecreasing function such that
if and only if
=0 and
is a continuous function such that
for all
. Then F has a fixed point in U.
The Darbo’s fixed point theorem is followed if (identity map) and
in the Theorem 1.4.
Now we recall one of the theorems in this paper which extends and generalizes Darbo’s fixed point theorem by using the concept of .
Theorem 1.5. Let be a nonempty, bounded, closed and convex subset of a Banach space
and
be a continuous operator such that
for of
,
and
:
is a nondecreasing function such that
if and only if
=0. Let
is a continuous function such that
for each
, where
is an arbitrary MNC. Then
has at least one fixed point in
.
Proof. (Banaei, Ghaemi, & Saadati, Citation2017).
Remark 1.2. The Theorem 1.4 is followed if and
in Theorem 1.5.
The following Corollary is immediate of Theorem 1.5.
Corollary 1.6. Let be a nonempty, bounded, closed and convex subset of a Banach space
,
and
:
are continuous functions. Suppose that there exists a constant
such that for all
,
where is an arbitrary measure of noncompactness and
. Then
has at least one fixed point in
.
Remark 1.3. The Darbo’s fixed point theorem is followed if ,
and
in Corollary 1.6.
2. Main results
In this section, we state one of the main results in this paper which extends and generalizes Darbo’s fixed point theorem by using the concept of .
(2.1)
Theorem 2.1. Let be a nonempty, bounded, closed and convex subset of a Banach space
and
be a continuous function such that
for any subset of
, where
is an arbitrary MNC,
is nondecreasing function such that
for
and
:
is a continuous function such that
if and only if
=0. Also,
and
for all
. Then
has at least a tripled fixed point.
Proof. From Remark 1.1 we have . Now we define a mapping
by
It is clear that is continuous. We prove that
satisfies all the conditions of Theorem 1.5. Let
be any nonempty subset. By (2.1) and
we obtain
From Theorem 1.5 deduce that has at least a fixed point in
and
has at least a tripled fixed point.
As application of Theorem 2.1, we can get the following theorem.
(2.2)
Theorem 2.2 Let be a nonempty, bounded, closed and convex subset of a Banach space
and
be a continuous function such that
for any subset of
, where
is an arbitrary MNC,
is nondecreasing function such that
for
and
:
is a continuous function such that
if and only if
=0. Also,
and
for all
. Then
has at least a tripled fixed point.
Proof. From Remark 1.1, we know that is a MNC. Now we define a mapping
by
It is clear that satisfies all the conditions of Theorem.
Corollary 2.3. Let be a nonempty, bounded, closed and convex subset of a Banach space
and
be a continuous function such that
for any subset of
, where
is an arbitrary MNC. Also,
is a nondecreasing function such that
for
. Then
has at least a tripled fixed point.
3. Application
In this section as an application of Theorem 2.2, we study the existence of solutions for the system of integral Equations (1.1). Let the Banach space consisting of all real functions defined, bounded and continuous on
equipped with the standard norm
We will use a measure of noncompactness in the space , which is given in (Banás & Goebel, Citation1980). Let
be fix a nonempty bounded subset of
and a positive number
. For
and
we denote
and
and
Moreover, for three fixed numbers the function
on the family
is defined as the following formula
where ,
and
Now consider the following assumptions:
are continuous, nondecreasing and
The function
is continuous and bounded.
is continuous and there exists a positive constant
such that
.
Moreover,
uniformly respect to
The functions
are continuous and there exists an upper semicontinuous and nondecreasing function
with
for each
. Also, there exists two bounded functions
with bound
and a positive constant such that
and
for all and
. We suppose that
has the property
for all . Moreover, we assume that
.
The functions
defined by
and
are bounded on
with
(3.1)
Theorem 3.1 If the assumptions -
are satisfied, then the system of integral Equations (1.1) has at least one solution
.
Define the operator
associated with the system of integral Equations (1.1) by
where,
We divide our proof in three steps.
Step 1: In the first step we prove that transforms the space
into
.
Obviously, that is continuous on
. Now we show that
For arbitrarily fixed
we have
Indeed,
and
By , we have
Therefore, maps the space
into
and
,
where
Step 2: We prove that the map is continuous.
Let us fix arbitrarily and take
such that
Then
By using condition there exist
such that for
Suppose that We have from (3.4) and (3.5)
If , then we obtain
where
By using continuity of on
we have
as
From inequalities (3.4) and (3.7) we deduce that
We conclude that is continuous on
Step 3: Now we prove that for any nonempty set ,
By using assumptions for any
and
where
Now we have
By considering upper semicontinuity of the function , from (3.9) and
we have
Fix arbitrary and
Choose
such that
,
Without loss of generality, suppose that
and
Then, for
we have
and
and
Therefore,
where
Since is an arbitrary element of
the inequality (3.11) implies that
By considering the uniform continuity of the functions ,
and
on
and
and
respectively, we conclude that
and
It is obvious that constant
is finite and
,
as
. Thus, we get
From inequality (3.10), (3.13) and definition measure of noncompactness we have
By applying Corollary 2.3 the desired result is obtained.
Additional information
Funding
Notes on contributors
Shahram Banaei
Shahram Banaei The research field of author is fixed point theory with its applications. He is assistant professor and faculty member in Islamic Azad university. The author has studied Darbo’s fixed point theorem in Banach space. Also, he give an application of their obtained results and analyze the existence of solutions for a system of integral equations by using the technique of measure of noncompactness.
References
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