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Abstract
The aim of the present paper is to introduce a new family of measures of noncompactness on the Fréchet space LPloc(ℝN) (1 ≤ p < ∞). Further, we prove a fixed point theorem on the family of measures of noncompactness in LPloc(ℝN). As an application, we investigate the existence of entire solutions for some classes of nonlinear functional integral equations of convolution type associated with the new family of measures of noncompactness. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.
PUBLIC INTEREST STATEMENT
Metric fixed point theory is a powerful tool for solving several 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 and differential equations. In this paper, we introduce a new family of measures of noncompactness in the Frechet space lploc(ℝN) and by applying this family of measures of noncompactness, we discuss the existence of solutions for some classes of nonlinear functional integral equations.
1. Introduction and preliminaries
The concept of a measure of noncompactness (MNC) plays a signification role in the nonlinear functional analysis. This notion was initiated by Kuratowski in 1930 (Kuratowski, Citation1930). In 1955 G. Darbo, using the concept of a measure of noncompactness, proved a theorem guaranteeing the existence of fixed points of the so-called condensing operators (Darbo, Citation1955). In recent years, a lot of authors such as Aghajani, Allahyari, and Mursaleen (Citation2014), Aghajani, Banaś, and Jalilian (Citation2011), Aghajani, O’Regan, and Shole Haghighi (Citation2015), Arab and Mursaleen (Citation2018), Banaś and O’Regan (Citation2008), Das, Hazarika, Arab, & Mursaleen (Citation2017); Maleknejad, Torabi, and Mollapourasl (Citation2011) and CitationOlszowy (Citation2010, 2012) studied the existence of solutions of integral equations in one or two variables on some spaces. Nonlinear functional integral equations of convolution type play important roles in applied problems, especially numerous branches of mathematical physics such as neutron transportation, radiation, and gas kinetic theory (see e.g., Zabrejko et al., Citation1968 and the references therein). Equations of this type have been considered in many previous studies Askhabov and Mukhtarove (Citation1987) and Jingqi (Citation1985), which showed that these equations have solutions in some function spaces. As an example, Khosravi et al. defined a new measure of noncompactness on the Banach space
and studied the existence of entire solutions for a class of nonlinear functional integral equations of convolution type in Khosravi, Allahyari, and Shole Haghighi (Citation2015).
On the other hand, a locally integrable function is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions. Regarding the above subject, Olszowy introduced a new family of measures of noncompactness on the space
consisting of all real functions locally integrable on
, equipped with a suitable topology. Then, she studied the existence of solutions of a nonlinear Volterra integral equation in the space
(cf. Olszowy, Citation2014).
In this paper, we define the new family of measures of noncompactness in Fréchet spaces
. In addition, we study the existence of solutions for some classes of nonlinear functional integral equations of convolution type (1.1)
by using an extension of Darbo’s fixed point theorem associated with this new family of measures of noncompactness.
In the sequel, we introduce some notations, definitions and preliminary facts which will be needed further on.
Denoted by the set of real numbers,
and put
. Let
be a real Banach space with zero element
. For a nonempty subset
of
, the symbols
and
will denote the closure and closed convex hull of
, respectively. Moreover, let
indicate the family of nonempty and bounded subsets of
and
indicate the family of all nonempty and relatively compact subsets of
Let
denote the space of Lebesgue integrable functions on
with the standard norm
We say that a function belongs to
if
for every compact set
. In other words,
if only if
for all
Also if
, then
. Recall that
is the countable union of compact
-
, i.e.,
with
where
Moreover, any compact subset of
is contained in some
and so the topology on
is given by the countable family of seminorms
for each
. Then,
becomes a Fréchet space with respect to the metric
A sequence is convergent to
in
if and only if for each
,
is convergent to
in
.
A nonempty subset is said to be bounded if
for all .
Further, let denote the family of nonempty and bounded subsets of
and
the family of all nonempty and relatively compact subsets of
In what follow, we recall the well-known Darbo’s fixed point theorem.
Theorem 1.1. (Darbo Banaś & Goebel, Citation1980) Let be a nonempty, bounded, closed and convex subset of a Banach space
and let
be a continuous mapping. Assume that a constant
exists such that
for any nonempty subset of
, where
is a measure of noncompactness defined in
. Then
has a fixed point in the set
.
Theorem 1.2. (Tychonoff fixed point theorem Agarwal, Meehan, & O’Regan, Citation2001) Let be a Hausdorff locally convex linear topological space,
a nonempty convex subset of
and
a continuous mapping such that
with compact. Then
has at least one fixed point.
2. Main results
In this section, we will introduce a family of measures of noncompactness in the space . Before that, we characterize the construction of compact subsets of
. First, we quote a useful theorem in (Brezis, Citation2011).
Theorem 2.1. (Brezis, Citation2011, Theorem 4.26) (Kolmogorov-M. Riesz-Fréchet). Let be a bounded set in
with
. Assume that
i.e.,
such that
with
. Then the closure of
in
is compact for any measurable set
with finite measure.
Here
,
and
denotes the restrictions to
of the functions in
.
Lemma 2.2 Let be a set in
.
is totally bounded in
if and only if
is totally bounded in
for each
.
Proof. First, assume that is totally bounded in
. We prove
is totally bounded in
for all
. For this, fixed
and
. Without loss of generality we may assume that
. Since
is totally bounded so there exists
-cover
for
. Take
for
and let
. Then there exists
such that
So, and since
so we have
. Thus,
is an
-cover for
.
Next, assume that is totally bounded for each
. We prove that
is totally bounded in
. For this, fixed
. We may assume that
. Take
such that
. Also, suppose that
is an
-cover for
. So, there exists
such that
is an
-cover for
. Let
. Thus, there exists
such that
for all . Now, for
we have
Also, since so, for
we have
Hence, we deduce
Thus is an
-cover for
and so
is totally bounded. □
Theorem 2.3. Let and
. Then
is totally bounded if, and only if, the following hold:
(i) For every there is some
so that
for all
(ii) For every and
there is some
so that
for all and
Proof. Assume that is totally bounded. For any
, there exists a finite
-cover
for
. Choose
. Let
then there exists
such that
we have
So
and
It implies the boundedness of , thus condition
holds. To establish condition
, let
and
be given. Let
be an
-cover of
. If
then
for some
. Take
It yields that
Now, since the space is dense in
thus there exists a continuous function
such that
Also, there exists such that for all
such that
we have
Therefore, applying (2.1), (2.2) and (2.3) we can write
thus condition holds.
Conversely, by using Lemma 2.2, it is enough to show that is totally bounded in
for every
. Now, since
satisfies the condition (ii) so by Theorem 2.1 the closure of
is compact in
. This completes the proof. □
Now, we are ready to describe a family of measures of noncompactness in the space .
Definition 2.4. A family of mappings
, is said to be a family of measures of noncompactness in
if it fulfils the following conditions:
The family
for
is nonempty and
.
for
.
for
.
for
.
for
and
.
If
is a sequence of closed chains of
such that
for
and if
for each
, then the intersection set
is nonempty.
We say that a family of measures of noncompactness is regular, if it additionally satisfies the following conditions:
for
.
for
.
for
and
.
for
.
Theorem 2.5. Suppose and
is a bounded subset of the space
For
, and
let
where . Then
,
defines a regular family of measures of noncompactness on
Proof. Assume that , then
for each
and therefore
for all
. Thus, for each
,
exists such that
. It implies that
for all
and
with
This means that,
On the other hand, we have
for all . According to Theorem 2.3, we infer that the closure of
in
is compact and
Thus
holds.
is obvious by the definition of
.
Now, we prove . Suppose that
and
. Therefore, a sequence
in
exists such that
converges to
in
From the definition of
we have
for any ,
and
with
By letting
, we obtain
Therefore,
Consequently, , from
we infer that
The properties and
are simple consequences of the following inequality
To prove , assume that
,
,
for
and
For any
, take an
In the first step, we claim that
is a compact set in
To establish this claim, we need to check conditions
and
of Theorem 2.3. Let
be fixed and take any
. Since
, then
exists such that
Hence, we can find sufficiently small so that
Thus, for all and
with
, we can write
The set is compact, hence
and
exist such that
for ,
and
with
It enforces that
for all ,
and
Thus,
for and for all
We know that
Then, all the hypotheses of Theorem 2.3 are satisfied and so is compact. Therefore, a subsequence
and
exist such that
converges to
. Since
,
and
for all
, we yield
that finishes the proof of .
The properties -
are obvious. Now, we check that condition
holds. Take
. Thus, the closure of
in
is compact. By Theorem 2.3, for any
and
,
exists such that
for all , and
with
Then for all we have
Therefore,
It in turn implies that
for all . Taking into account
, we obtain
The following theorem is a version of Darbo’s fixed point theorem in .
Theorem 2.6. Let be a nonempty, closed and convex subset of a Fréchet space
and
be a family of measures of noncompactness on
. Let
be a continuous operator such that
where for all
Then
has at least one fixed point in the set
.
Proof. By induction, we define a sequence by letting
and
,
We have
, therefore by continuing this process we obtain
If for some integer
, and for all
, then
is relatively compact. Thus, Theorem 1.2 implies that
has a fixed point. Now, assume
exists such that
for any
. By (2.4) we have
Since , then
is a positive decreasing sequence of real numbers. Thus, there is an
such that
as
We show that
. Suppose, to the contrary that
. Then from (2.5) we have
It enforces that , which is a contradiction. Consequently
, and hence
as
Using this fact and since the sequence
is nested, in view of part
of Definition 2.4, we conclude that the set
is nonempty, closed, convex and
Furthermore, the set
is invariant under the operator
and
. By applying Theorem 1.2, we find that the operator
has a fixed point.
3. Application
To verify the applicability of our results, in the following section, we shall present an existence result for a large class of nonlinear functional integral equations of convolution type on the space . We provide an illustrative example to show the effectiveness and applicability of our results.
Definition 3.1. (Aghajani et al., Citation2015) We say that a function satisfies the Carathéodory conditions if:
The function
is measurable for any
The function
is continuous for almost all
Theorem 3.2. Assume that the following conditions are satisfied.
satisfies the Caratéodory conditions, and a constant
and a function
exist such that
for any and almost all
.
The operator
acts continuously from the space
into itself and an increasing function
exists such that
for any
A function
exists such that the following relation holds for all
and all
.
Then, the nonlinear functional integral equation of convolution type (1.1) has at least one solution in the space .
Before giving the proof, we quote the following result, which is a version of Young’s Theorem (Brezis, Citation2011, Theorem 4.15).
Remark 3.3. Under hypothesis , the linear operator
defined by
is a continuous operator, and moreover, we have
Proof. First, we define the operator by formula
where . By considering the Caratéodory conditions, we infer that
is measurable for any
Now, we prove that
for any
For this purpose, one only needs to prove that
for all
Let us fix . By using condition
we have
for any and for a.e.
Thus, we get
According to Remark 3.3 and we deduce
Hence, and
is well defined. We consider the subset
of
given by
Then, the subset is nonempty, convex, closed and bounded in
, and condition
ensures that
transforms
into itself. Furthermore,
is continuous in
, because
,
and
are continuous for a.e.
In order to finish the proof, we show that satisfies in Theorem 2.6.
Let us take a nonempty and bounded subset of the set
and an arbitrary positive number
. Let
and
with
for arbitrarily small positive number
. We obtain
Therefore, in view of Young’s Theorem we get
By using the above estimate we have
Since is a compact set in
and
is a compact set in
, we have
,
as
Then we obtain
where for each
. By (3.1) and Theorem 2.6 we infer that, the operator
has a fixed point
in
and therefore the nonlinear functional integral equation of convolution type (1.1) has at least one solution in the space
□
Example 3.4. Consider the following functional integral Equation (3.2)
where and
is the Euclidean norm. We study the solvability of integral Equation (3.2) on the space
for
Observe that Equation (3.2) is a special case of the Equation (1.1) when
In this example, hypothesis of Theorem 3.2 holds with
and
. Indeed, we have
It is easy to see that satisfies assumption
. In fact, for each
we can write
for all . Thus, we have
and so
.
Furthermore, satisfies hypothesis
with
. Obviously, the function
is increasing and
. Next, we show that
satisfies assumption
. Indeed, we have
Finally, the inequality from assumption , has the form
so we can take
Consequently, all the conditions of Theorem 3.2 are satisfied, and thus functional integral equation of convolution type (3.2) has at least one solution in the space if
.
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Funding
Notes on contributors
Hojjatollah Amiri Kayvanloo
The authors have studied the existence of solutions for some classes of nonlinear functional integral equations of convolution type by using measure of noncompactness. In particular, they introduced a new family of measures of noncompactness on the Frechet space lploc(ℝN) Also, by this family of measures of noncompactness, they investigated the existence of solutions for some classes of nonlinear functional integral equations. The results of their article extend and improve previously known results.
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