Generalization of
ρ
Attractive Elements in Modular Function
Spaces
MOHAMMAD AMRO
1
, ABDALLA TALLAFHA
1
, WASFI SHATANAWI
2,3
1
Department of Mathematics, University of Jordan, Amman,
JORDAN
2
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh,
SAUDI ARABIA
3
Department of Mathematics, Hashemite University, Zarqa,
JORDAN
Abstract: In this paper, we introduce two new classes of mappings called ραand ραnon-
spreading mappings to broaden the idea of attractive elements in modular function spaces (MFS). In the
MFS that are put up, we also demonstrate several approximation results and existence results. Illustration
examples are provided to clarify the results.
Keywords:- Attractive points, modular spaces, non-spreading mappings,
attractive elements
,
non-spreading mappings,
non-spreading mappings.
Received: October 2, 2022. Revised: December 6, 2022. Accepted: January 5, 2023. Published: February 2, 2023.
1 Introduction
In this section, firstly we will introduce a
literature review of the most relevant work done
in
attractive elements in MFS. Afterward, we
will provide the theoretical background listing the
most related topics to our work. Lastly, we provide
in detail information on our mathematical
definitions and theorems applied.
Literature review
In [20] authors developed the following notion of
attractive points of nonlinear mapping in Hilbert
spaces:
Let
be a nonempty subset of a Hilbert space
and
. Then the set of attractive points
is given by,
They provided evidence for the idea that there are
attractive points in a Hilbert space for the so-
called hybrid mappings. With the exception and
closedness, they continued to demonstrate a weak
Mann-type convergence theorem.
Research on attractive points gained momentum
as a result of the hypothesis provided by [3].
Different mapping classes were combined. Non-
spreading mappings are considered to be a new
class of mappings proposed by, [18].
A mapping
is said to be non-spreading
mapping if for any
. Then,
.
Using the Hausdorff metric, [19], developed the
category of
non-spreading multivalued
mappings based on generalized non-spreading
mappings. In CAT(0) spaces, [1],
investigated the
convergence theorems and attractive points for
normally generalized hybrid mappings (a non-
linear generalization of a Hilbert space known as
"Hadamard spaces").
The strong and weak convergence theorem of the
Ishikawa iteration for an
generalized
hybrid mapping in a uniformly convex Banach
space was confirmed by [4], in 2015 as well. For
normally generalized hybrid mappings in CAT(0)
spaces, [2]
,
developed an attractive point theorem.
Recently, many mathematicians have developed
an interest in fixed point theory in MFS. The first
proposed the concept of MFS was introduced in,
[7], and it was furthermore generalized in, [8].
Continuing in the same direction, [9], [10],
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worked on fixed point theory in the field of MFS
where he proved Banach contraction principle in
that space. An introduction of some fixed points
for generalized contraction mappings in MFS was
carried out, [11].
The proof of results of approximating fixed points
in MFS was proposed for the first time by [12],
after that,
a
multivalued
quasi nonexpansive
mappings in MFS was handled, [13]. Cyclic
Kannan maps in MFS were investigated by [14],
where sufficient conditions for the existence and
uniqueness of fixed points were given. Detailed
discussions on modular spaces are also provided
in, [5], [15], [16].
In 2021, [6], introduced the notion of
attractive elements in MFS, they also
established a class of mappings called
non-spreading mappings and verified the
existence results and some approximation results
in the setup of MFS.
The efforts mentioned above encourage us to
broaden the idea of attractive elements in the
context of MFS. This paper's main goal is to
define classes of
and
non-
spreading mappings. This will enable us to
demonstrate both the existence and
approximation results for attractive elements in
MFS, various numerical examples will be used to
support our findings.
Theoretical background
Now we will review some fundamental concepts
and definitions related to our topic, before
introducing the definitions, it is worth mentioning
that we will use the symbols as defined in Table
1:
Table 1. symbols.
Symbol
Meaning
nonempty set
nontrivial
algebra of subsets of
a nontrivial
ring of subsets of
linear space of all simple functions
with supports from
space of all extended measurable
functions
Note that:
1)
is closed with respect to forming of
countable intersections, and finite unions
and differences.
i.e., suppose that
for any
and
. Assume that there
exists an increasing sequence of sets
such that
.
2)
A measurable function
such that there exists a sequence
Definition 1.1, [3].
Let
be an even, convex, and
nontrivial function. We say that
is a regular
convex function pseudomodular if:
a)
;
b)
is monotone, i.e.,
for
any
where
c)
is orthogonally subadditive, i.e.,
such that
denotes the
characteristic function of the set
.
d)
has Fatou property, that is,
where
e)
is order continuous in
, i.e.,
, and
Definition 1.2, [2].
A set
is
null if
A property holds
almost everywhere (
) if the set
is
null.
We identify any pair of measurable sets whose
symmetric difference is
null as well as any
pair of measurable functions differing only on a
null set. For this, we define
.} where each
is an equivalence class of functions equal
, rather than an individual function.
Definition 1.3, [2].
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Let
be a regular convex function
pseudomodular. Then, we say that
is a regular
convex function modular if
.
The class of all nonzero regular convex function
modular defined on
is denoted by
.
Definition 1.4, [7]
.
Let
be a convex function modular. Then the
modular function space
is defined as:
.
Generally, the modular
is not subadditive so it
is not like a norm.
Therefore, the modular space
can be fitted with
an
norm defined by:
.
If
is a convex modular. Then,
.
defines a norm on the modular space
, and is
called the Luxemburg norm.
Definition 1.5, [10]
.
Let
be a modular space. Then:
a)
The sequence
is said to be
convergent to
if
;
b)
The sequence
is said to be
Cauchy if
;
c)
We say that
is
complete if and
only if any
Cauchy sequence in
convergent.
Definition 1.6, [7].
A subset
of
is called:
a)
closed if the
limit of a
-
convergent sequence of
always belongs
to
;
b)
compact if every sequence in
has a
convergent subsequence in
;
c)
bounded if
;
d)
The
distance between
is
defined as:
.
The nomenclature defined for
is similar to
metric spaces but
does not satisfy triangle
inequality. Hence, if a sequence in
is
convergent it does not imply
Cauchy. This
is only true if and only if
satisfies
condition.
Definition 1.7, [17]
.
The modular function
is said
to satisfy
th
e
condition if
as
approaches
, whenever
as
approaches
.
The modular
satisfies some uniform convexity
type properties.
Definition 1.8, [16]
.
Let
:
a)
For
. Define,
Let
,
and
if
. We say
that
satisfies
if for every
,
.
a)
Note that for every
,
for every
small
enough.
We say that
satisfies
if for
every
,
, there exists
depending only upon
and
such that
for
any
.
b)
We say that
satisfies
if for
every
, there exists
depending upon
and
such that
for
any
. Note that
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. If
satisfies
condition, then
is equivalent to
.
Definition 1.9, [6].
We will say that
is uniformly continuous if for
every
and
, there exists
such that:
A sequence
is called bounded
away from
if there exists
such that
for every
. Similarly,
is
called bounded away from
if there exists
such that
for every
. The
following lemma helps study the convergence of
fixed points as well as attractive elements in the
MFS.
Lemma 1.1, [6]
.
Let
satisfy
and let
be bounded away from
. If
such
that:
then
The following theorem is necessary because MFS
do not satisfy the triangle inequality.
Theorem 1.1, [6]
.
Let
satisfy
condition. Let
and
be two sequences in
. Then:
and
Definition 1.10, [6]
.
Let
be convex and
bounded. A
function
is called a
type if
there exists a sequence
of elements of
such that for any
,
Now the following lemma establishes an
important minimizing sequence property of
uniformly convex MFS which is used to prove the
existence of fixed points.
Lemma 1.2, [17]
.
Assume that
is
. Let
be a
closed
bounded convex nonempty subset
of
. Let
be a
type defined on
. Then any
minimizing sequence in
is
-convergent. Its
-
limit is independent of the minimizing sequence.
The following lemma is a modification of the
above that is used to prove the existence of
attractive elements without the condition of
closedness.
Lemma 1.3, [6].
Assume that
is
. Let
be a
bounded convex nonempty subset of
. Let
be a
type defined on
. Then any minimizing
sequence in
is
convergent in
. Its
limit
is independent of the minimizing sequence.
Definition 1.11, [3]
.
Let
. The growth function of a modular
function
denoted by
is defined as :
Note that if
, then
.
Definition 1.12, [3]
.
Let
be a nonempty subset of a Hilbert space
.
Let
be a mapping.
is said to be a
fixed point of
if
.
The set of all fixed points is denoted by
.
Definition 1.13, [14].
Let
be a nonempty subset of a Hilbert space
.
A mapping
is said to be:
(a)
nonexpansive mapping if
.
(b)
quasi-nonexpansive mapping if
.
Theorem 1.2, [6].
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Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded convex subset of
. Let
be a
non-spreading mapping
with
Then
has a
attractive
point.
Theorem 1.3, [6]
.
Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded,
closed convex
subset of
. Let
be a
non-
spreading mapping with
Then
has a fixed point.
2 Main Results
We start this section by giving the notions of
mappings
and
non-spreading
mappings. Then, we explain the concept of
ρ
attractive elements and prove the existence and
some of the convergent results.
Definition 2.1
Let
. And
. Then:
(1)
is
non-spreading mapping if
for
we have:
(2)
is
non-spreading mapping
if for
we have:
Note that for
we get a
non-spreading mapping which is similar to
non-spreading mapping with
is
quasi-nonexpansive mapping.
In fact, if
is a fixed point of
, then from the
previous definition taking
we get:
Definition 2.2 (
attractive element)
Let
be a convex modular function,
be a
nonempty subset of
, and
be a
mapping. Then a function
is called
attractive element of
if
we have
The set of all
attractive elements of
is
denoted by
Now, before we prove the existence of
attractive element of
, we start with the
following two lemmas.
Lemma 2.1
Let
be uniformly continuous. Let
(nonempty) and
, with
Then
is closed.
Remark:
To prove this, we have to show that for any
such that
, then
.
Proof:
Let
Since
is uniformly continuous and taking
The equation above showcase clearly that an
attractive point is not necessarily a fixed point
from their definitions. In this point, it is worth
mentioning that if the mapping is
quasi-
nonexpansive mapping then the
attractive
elements which are in
must be fixed points of
Lemma 2.2
Let
be uniformly continuous,
(nonempty), and
be an
quasi-
nonexpansive mapping. Then
.
Proof:
.
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Conversely,
-quasi-
nonexpansive mapping, then for
Now, we have to prove the existence of
attractive point for
non-spreading
mapping for
Theorem 2.1
Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded convex subset of
. Let
be a
non-spreading mapping
with
Then
has a
attractive point.
Proof:
Let
. Define the
type,
by
By Lemma 1.3.
a minimizing sequence (say
) of
, s.t.
But
and
is
bounded
(Definition 1.6), we get:
Also
Now, by Definition 2.1 (1), we have:
Letting
we have:
Thus
is also
a minimizing sequence of
.
Now, depending on Lemma 1.3
converges to
some
and for any other minimizing
sequence converges to
, then
.
So, we have to show that
is the
attractive
point of
.
By Definition 2.1 (1) and uniformly continuous of
, we get:
.
So,
Hence
ℎ
is a
attractive point of
.
Consequently, we have to prove the existence of
attractive point for
non-spreading
mapping for
Theorem 2.2
Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded convex subset of
. Let
be a
non-spreading
mapping with
Then
has a
attractive point.
Proof:
Let
. Define the
type,
by
By Lemma 1.3.
a minimizing sequence (say
) of
, s.t.
Since
and
is
bounded
(Definition 1.6) we get:
Also
Now, by Definition 2.1 (2), we have:
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Letting
we have:
Since
, we get:
Thus,
is also a
minimizing sequence of
.
Now, depending on Lemma 1.3,
converges
to some
and for any other minimizing
sequence converges to
. Then
.
So, we have to show that
is the
attractive
point of
.
Since
is uniformly continuous, we have
Therefore
is a
attractive point of
.
Note that Theorem 2.1 is a special case of
Theorem 2.2 when (
).
As a special case, if we take
,
then Definition 2.1 (2) implies that
which is Definition 2.1. in, [6], (
non-
spreading mapping).
In our main result if we take
then we obtain the results of Theorem 1.2 and 1.3,
[6].
Corollary 2.1, [6].
Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded convex subset of
. Let
be a
non-spreading mapping
with
Then
has a
attractive
point.
Corollary 2.2, [6]
.
Let
be complete,
is
and
uniformly continuous. Assume that
is a
nonempty
bounded,
closed convex
subset of
. Let
be a
non-
spreading mapping with
Then
has a fixed point.
Theorem 2.3
Let
satisfy
and
condition. Let
is a nonempty convex subset of
and
be a
non-spreading mapping with
. Suppose
is nonempty,
define the sequence
as follows:
(1)
with
. Then
exists
for
and
.
Proof:
Suppose that
, since
is convex, we
get:
(2)
Also, we have:
(3)
Therefore:
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Now because of
is
bounded and
is a nonincreasing sequence, we get that
exists for
.
However, we have to show that
.
Let
(4)
For
, we have
.
So,
(5)
Also,
Which implies that
(6)
And
(7)
Thus,
.
(8)
By (5), (6), (8) and Lemma 1.1, we get:
.
Now we have to prove
.
For
, then
such that:
By the definition of growth function, we have:
Thus,
. (9)
Now,
So, by Theorem 1.1 and (9), we get:
Therefore,
Now,
(10)
By (7) and (10), we get:
Consequently,
(11)
Hence, by (4), (5), (11), and Lemma 1.1, we
obtain:
Definition 2.3
Let
be a nonempty subset of
. A mapping
is said to satisfy condition
if there
exists a nondecreasing function
with
,
such that
where
.
The following example explains a mapping that
satisfies the condition
.
Example 2.1
Let ℎ (the set of real numbers) be the space
modulared as
. Let
ℎ
, define
as
.
Clearly,
is
non-spreading mapping.
Clear that
ℎ
is an attractive point of
if
.
Suppose that
, then:
(12)
Hence, we have
. Because
must satisfy
equation (12)
. Therefore
∞
.
Now define a continuous nondecreasing function
∞
∞
by
. Then we get:
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∞
Thus,
.
Theorem 2.4
Let
satisfy
,
condition, and
is uniformly continuous. Let
be a nonempty
convex subset of
and
be a
non-spreading mapping with
. Assume
and
satisfies the
condition
. Let
be a sequence defined as
follows:
with
. Then
converges to
attractive point of
.
Proof:
It’s clear that
and
. Then by condition
and
Theorem 2.3, we get:
,
.
So,
.
Follow that
, since
.
Now we have to show that
is
-cauchy.
Because of
, let
,
then
such that for
:
and
Then
such that
.
Now for
, by convexity of
and since
is nonincreasing we get:
Hence, by
condition,
is
cauchy
sequence. Since
is complete,
is
converge to some
.
Now let
. Then by convexity
of
and Theorem 2.3 we get:
Moreover, by definition 2.1 (2) and uniform
convexity of
we have the following:
This implies:
Thus,
Therefore,
and
.
Definition 2.4, [18].
Let
be a subset of
. A mapping
is said to be
demicompact if it has the
property that whenever
is
bounded
and the
is
converge, then
subsequence which is
converge.
Theorem 2.5
Let
satisfy
and
condition. In
addition,
is uniformly continuous. Let
be a
nonempty convex subset of
and
be a
non-spreading mapping with
and
demicompact mapping with
. Let
be a sequence defined as
follows:
with
. Then
converges to
attractive point of
.
Proof:
is a bounded sequence and
by Theorem 2.3.
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Also,
a subsequence
of
and
such that
by definition 2.4.
Moreover, since
is uniformly continuous and
, we get:
Now, return to the definition 2.1 (2) and by the
uniform continuity of
we have:
And so:
Therefore,
since
Hence,
.
By Theorem (2.3) if
, then
.
3 Numerical Results
Example 3.1
Let
(the set of real numbers) be the space
modulared as
Let
, define
as
.
is a nonempty convex subset of
that satisfies
conditions.
is a uniformly continuous function
and
holds.
is nonempty.
where,
.
Choose
using Matlab program we get
the results in Table 2 below.
We see that the sequence
converges to
and
we can increase the speed of convergence by
changing the values of
.
Note that when
are both closed to zero
then
converge to
more rapidly.
Figure 1 below shows the differences between
choosing
in finding the sequence
.
Example 3.2
Let
(the set of real numbers) be the space
modulared as
Let
, define
as
.
is a nonempty convex subset of
that satisfies
conditions.
is a uniformly continuous function
and
holds.
is nonempty.
where,
.
Choose
using Matlab program we get the
results in Table 3 below.
We see that the sequence
converges to
and
we can increase the speed of convergence by
changing the values of
.
Note that when
are both closed to zero
then
converge to
more rapidly.
Figure 2 below shows the differences between
choosing
in finding the sequence
.
4 Conclusion and Future Work
In this paper, firstly, we introduced two new classes
of mapping called ραand ραnon-
spreading mappings. Specifically, these classes are
of high importance as they are based on Modular
Function Spaces (MFS). Moreover, in the following
sections, we have proved the existence and
uniqueness of attractive elements for these
classes. Furthermore, we have introduced various
numerical examples to find the attractive elements
based on our proven theorems. As for future works
of our study, we are planning to consider recent
studies of other mappings researches on Modular
Function Spaces.
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Table 2. The values of
when
with different values of
as follows:
n
,
,
,
,
,
,
,
1
0.3
0.3
0.3
0.3
0.3
0.3
0.3
2
0.13125
0.1078125
0.1359375
0.08925
0.14325
0.0764925
0.1492575
3
0.057421875
0.038745117
0.06159668
0.026551875
0.068401875
0.019503675
0.074259338
4
0.02512207
0.013924026
0.027910995
0.007899183
0.032661895
0.00497295
0.036945877
5
0.010990906
0.005003947
0.01264717
0.002350007
0.015596055
0.001267978
0.018381497
6
0.004808521
0.001798293
0.005730749
0.000699127
0.007447116
0.000323303
0.009145255
7
0.002103728
0.000646262
0.002596746
0.00020799
0.003555998
8.24341E-05
0.004549993
8
0.000920381
0.00023225
0.00117665
6.18771E-05
0.001697989
2.10186E-05
0.002263735
9
0.000402667
8.3465E-05
0.00053317
1.84084E-05
0.00081079
5.35923E-06
0.001126265
10
0.000176167
2.99952E-05
0.000241593
5.47651E-06
0.000387152
1.36647E-06
0.000560345
11
7.70729E-05
1.07795E-05
0.000109472
1.62926E-06
0.000184865
3.48415E-07
0.000278786
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Table 3. The values of
when
with different values of
as follows:
n
,
,
,
,
1
6
6
6
6
2
4.8
4.352
4.85
4.928
3
3.888
3.247181
3.9645
4.085837
4
3.19488
2.50651
3.282665
3.424233
5
2.668109
2.009964
2.757652
2.904478
10
1.422953
1.136765
1.475758
1.56988
20
1.027191
1.002508
1.034857
1.051027
30
1.001748
1.000046
1.002554
1.004569
40
1.000112
1.000001
1.000187
1.000409
50
1.000007
1
1.000014
1.000037
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.10
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Volume 22, 2023
Fig. 1: Differences between choosing
in finding the sequence
of example 3.1
Fig. 2: Differences between choosing
in finding the sequence
of example 3.2
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
1 2 3 4 5 6 7 8 9 10 11
a=0.5,b=0.5
a=0.25,b=0.25
a=0.25,b=0.75
a=0.1,b=0.1
a=0.1,b=0.9
a=0.01,b=0.01
a=0.01,b=0.99
a=0.5,b=0.5
0
1
2
3
4
5
6
1357911 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
a=0.5,b=0.5
a=0.1,b=0.1
a=0.25,b=0.75
a=0.1,b=0.9
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DOI: 10.37394/23206.2023.22.10
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Volume 22, 2023
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DOI: 10.37394/23206.2023.22.10
Mohammad Amro, Abdalla Tallafha, Wasfi Shatanawi
E-ISSN: 2224-2880
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Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
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Conflict of Interest
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that are relevant to the content of this article.
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