Aggregation of Fuzzy Metric Spaces: A Fixed Point Theorem
ELİF GÜNER,
HALİS AYGÜN
Kocaeli University,
Department of Mathematics,
Umuttepe Campus, 41380,
TURKEY
Abstract: In the last years, fuzzy (quasi-) metric spaces have been used as an important mathematical tool to
measure the similarities between the two points with respect to a real parameter. For the reason of the
importance of these structures, different kinds of methods have been investigated for use in the applied
sciences. So generating new fuzzy (quasi-) metrics from the old ones with aggregation functions has been a
research topic. In this paper, we provide a general fixed point theorem using residuum operators for
contractions obtained through aggregation functions. We show that there are some necessary conditions and
also we provide some examples to show that these conditions cannot be omitted.
Key–Words: - Aggregation function, contraction, fuzzy metric, fixed point theorem, residuum operator,
t-norm.
Received: April 14, 2023. Revised: November 11, 2023. Accepted: December 7, 2023. Published: March 8, 2024.
1 Introduction
In applied sciences, aggregation functions have
become important tools in merging a collection of
data (information) into a single one. The necessity
of merging the information usually presented
through numerical values has led to growing
attention to studying numerical functions.
Especially, the need for this merge is imposed in
fields such as machine learning, data mining, image
processing, and decision-making pattern
recognition. Many methods to merge the numerical
information are based on aggregation functions.
Another different use of the aggregation function is
for aggregating the metric spaces. In this way,
aggregation functions allow us to generate new
metric spaces from the existing ones. We observe
that this problem has been mainly studied in [1], [2],
[3]. Then, the authors obtained a characterization of
those functions that aggregate a collection of quasi-
metrics into a single one by means as in the papers,
[4], [5], [6]. It should be pointed out that a function
that aggregates quasi-metrics (on products) also
aggregates metrics (on products) but the converse is
not true in general.
One of the other application areas where the
aggregation functions are used successfully is used
for fuzzy binary relations which are provided by
fuzzy databases. In light of this fact, the
aggregation of fuzzy relations has been studied in
the papers, [7], [8], [9], [10]. Also, the authors have
studied the aggregation of fuzzy equivalence
relations (or indistinguishability operators) in [11].
Some important works for aggregation of
indistinguishability operators have been studied in
[12], [13], [14].
In cases where the similarity degree between
the elements must be measured concerning a
positive real parameter, indistinguishability
operators are not a suitable tool. To handle these
situations, the notion of fuzzy (pseudo-) metric was
initiated in the literature, [15], [16]. This new type
of similarity measures can be interpreted in a similar
way as fuzzy equivalence relations but in this case,
the obtained measures are always relative to the
parameter value. The detailed studies related to
structures of fuzzy (pseudo-) metrics can be found
in [17], [18], [19], [20], [21], [22], [23], [24], [25],
[26]. Nowadays, researchers have given their
attention to the question of whether the aggregation
function must satisfy which conditions to guarantee
that merging the fuzzy quasi-metrics generates a
new one. The authors have studied in detail which
functions allow us to merge a collection of fuzzy
(quasi-) metrics into a single one in [27]. They also
presented a characterization of such functions in
terms of triangular triplets, isotonicity, and
supmultiplicativity.
In this paper, we provide a general fixed point
theorem using residuum operators for contractions
obtained through aggregation functions. We show
that there are some necessary conditions and also
we provide some examples to show that these
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conditions cannot be omitted.
2 Preliminaries
This section is devoted to compiling definitions and
results useful throughout the paper.
Definition 1
,
[28],
A triangular norm (briefly, t-
norm) is a binary operation
on [0,1] such that,
for all
, the following axioms are
satisfied:
(T1)
(T2)
(T3)
(T4)
If, in addition,
is continuous (with respect to
the usual topology) as a function defined on
, we will say that
is a continuous
t-norm.
Proposition 2
,
[28], Let be a continuous t-norm,
then -residuum operator of is uniquely
determined by the formula
(1)
According to the above equation, the residuum
operator of the product t-norm (), the minimum t-
norm ) and the Lukasievicz t-norm () are as
follows:
To find a deeper treatment on the residuum
operator we refer the reader to [28]. Moreover,
following the results were provided in [4], on the
residuum operator, we can prove the following
proposition, which will be useful later.
Proposition 3 [28], Let be a continuous t-norm
then, for all .
An immediate corollary of the previous
propositions is the following one.
Corollary 4
,
[28], Let be a continuous t-norm
then, for all
Now, we are able to present the notion of fuzzy
metric space due to [16]. It is worth mentioning
that nowadays, such a concept is commonly used in
the literature following its reformulation given in
[18], which is given as follows. We obtain the
following results from Remark 2 and Remark 4:
Definition 5
,
[18], A fuzzy metric space is an
ordered triple such that is a (non-
empty) set, is a continuous t-norm and is a
fuzzy set on , satisfying the
following conditions, for all and
(KM1) ;
(KM2) for all iff ;
(KM3) ;
(KM4) ;
(KM5) The function is left-
continuous, where for each
Below, we can find the modification of the
preceding definition given in [15].
Definition 6 [15], A GV-fuzzy metric space is an
ordered triple such that is a (non-
empty) set, is a continuous t-norm and is a
fuzzy set on , satisfying, for all
and , conditions (KM3),
(KM4) and the following ones:
(GV1) ;
(GV2) iff ;
(GV5) The function is
continuous.
As usual, if is a (GV-)fuzzy metric
space, we say that , or simply , is a (GV-
)fuzzy metric on .
It should be noted that a GV-fuzzy metric
can be regarded as a fuzzy metric defining
for each . So, GV-fuzzy
metric spaces can be considered a particular case of
fuzzy metric.
Definition 7 [27], A function is
called a fuzzy metric aggregation function on
products if whenever is a t-norm and
is a family of fuzzy metric spaces
then is a fuzzy metric on where
is given by
for all and
.
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We refer to the paper [27], for the properties,
characterizations and examples of fuzzy metric
aggregation function on products.
An extension of Banach contraction principle for
fuzzy metric spaces using the residuum operator
was given as follows:
Theorem 8, [29], Let be a complete fuzzy
metric space and be a mapping. If there
exists a such that
for all and , and is Archimedean,
then has a unique fixed point in .
3 Fixed Point Theorems under
Aggregation Functions
Throughout this section, denotes the fuzzy
metric
induced by aggregation of the family of fuzzy
metric spaces through .
Definition 9 Let
be a family of
fuzzy metric spaces,
and let
be a fuzzy metric aggregation
function on products. A
mapping is said to be a
projective -contraction if there exist contractive
constants such that
for all , and .
A function is said to belong
to the class FMA if it fulfills the following
conditions:
(FMA1) for all .
(FMA2) for all
where .
The following lemma guarantees the completeness
for the fixed point theorem.
Lemma 10 Let be a fuzzy metric
aggregation function on products such that
. Let be a family of arbitrary
fuzzy metric spaces and
. Assume that
the fuzzy metric space is complete for all
. The fuzzy metric space is
complete.
Proof. Let be a Cauchy sequence in
. Then for each there exists
such that for each
and . Fix }. We will
show that the sequence is a Cauchy
sequence in . Let . We obtain
the next inequality:
since is isotone. So, is a Cauchy sequence
in . Also, since is arbitrary, we deduce
that is a Cauchy sequence in for all
. On the other hand, since is
complete for all , there exists
such that for all .
Then we have that for all and ,
there exists such that
for all . Let and . Since is a
continuous t-norm, there exists such
that . Also, it is
satisfied that for all
when . Thus, we obtain:
Hence, converges to x in which
means that is complete.
Proposition 11 Let
be a family of
fuzzy metric spaces,
, a
contraction for all and let
be a fuzzy metric aggregation
function on products. Then the mapping
defined by
is a projective -
contraction.
Proof. Since is a contraction there
exists a such that
For all and . Then, we obtain
which means that
is a projective -contraction.
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However, the converse of the above proposition
may not be true in general as shown in the
following example:
Example 12 Let be the fuzzy metric
space where
for all
and . Consider the family of fuzzy metric
spaces
such that .
Define the function by
. Then is a -fuzzy metric aggregation
function on products.
Consider the mapping defined by
for all .
Now, we show that is a projective -contraction.
For , we obtain
and
which means that
where
for all and . We
have a similar result for . So, this implies that
F is a projective -contraction. But,
and are not contractions. If we take
and assume that there is a
such that
then we obtain
and this follows that
or which are
contradictions. So, we deduce that is not a
contraction. One can easily observe that is not a
contraction with a similar process.
Proposition 13 Let
be a family of
fuzzy metric spaces,
and
a fuzzy metric aggregation function
on products satisfying for
all . If ,
, is a projective -contraction
and is a mapping satisfying
for all and , then is a contraction with
some constant .
Poof. Let ,
, be a projective -
contraction. Then there exists such that
for all and .
If we take and
, then we have
which means that is contraction for all .
We note here that Example 12 shows that the
condition cannot be
deleted in the above theorem.
Theorem 14 Let
be a family of
fuzzy metric spaces and
. If
is a fuzzy metric aggregation
function on products satisfying and
is a projective -
contraction with constants ,
then is a contraction from
into itself.
Proof. Let and . Since is isotone
and F is projective -contraction, there exists
such that
Take . Then we have
since is antitone with respect to the first
argument. Then we obtain
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and hence is a contraction since .
The following example shows that the condition
of Theorem 14 cannot be
omitted.
Example 15, Let be the stationary
fuzzy metric space where
for
all and . Consider the family of
fuzzy metric spaces
such that
. Define the function
by
. Then is a -
stationary fuzzy metric aggregation function on
products. However does not satisfy since
for all .
Consider the mapping defined by
for all . It is
clear that is a projective -contraction since
there exists such that:
,
for all and
.
However, is not a contraction from
from itself. Take and . Then
we have
Assume that that there exists a such that
.
Then we obtain which
means that . So we have a contradiction and
hence is not a contraction from into
itself.
Now we give the fixed point theorem by using
Archimedean t-norm and aggregation functions:
Theorem 16 Let
be a family of
complete fuzzy metric spaces, be an Archimedean
t-norm and
. If is a
fuzzy metric aggregation function on products
satisfying and is a
projective -contraction with constants
, then has a unique fixed
point in .
Proof. Since and satisfies the conditions in
Theorem 14, we can guarantee that is a
contraction. Also, we obtain from Lemma 10 that
is a complete metric space since
is a family of complete fuzzy metric
space and belongs to FMA. In addition to these,
is an Archimedean t-norm as stated in the
hypothesis, so we conclude that from Theorem 8,
has a unique fixed point in .
In the following, we give an example that shows
there are contractions on which are not
projective -contraction. So this fact implies that
Theorem 16 is not a direct consequence of
Theorem 8.
Example 17 Let be the stationary fuzzy
metric space where
for all
and . Consider the family of fuzzy
metric spaces
such that
. Define the function
by
. Then, is a
-stationary fuzzy metric aggregation function.
Consider the mapping defined by
for all .
Now, we show that is a contraction mapping.
Let . Then, we obtain
for all and . Hence, is a
contraction on . Now, we will show that
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is not a projective -contraction.
Let . Then, we have
for all and which concludes
that F is not a projective -contraction.
4 Conclusion
Fixed point theory is an important tool when we
solve certain functional equations such as
differential equations, integral equations, fractional
differential equations, matrix equations, etc. We
can reformulate the considered problem in terms of
investigating the existence and uniqueness of a
fixed point of a function. Also, this theory has
several applications in many different fields such as
biology, physics, chemistry, economics, game
theory, optimization theory and etc. For future
work, we plan to investigate some applications of
the obtained result to the mentioned areas.
Acknowledgement:
The authors are thankful to the anonymous referees
for their valuable suggestions.
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Contribution of Individual Authors to the
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The authors equally contributed to 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
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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