Abstract
In the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order -contractive type mappings. Then a common fixed point theorem is given for noncontractive type mappings. Kirk's problem on an extension of Caristi's theorem is also discussed.
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1. Introduction and Preliminaries
After the definition of the concept of fuzzy metric space by some authors [1–3], the fixed point theory on these spaces has been developing (see, e.g., [4–9]). Generally, this theory on fuzzy metric space is done for contractive or contractive-type mappings (see [2, 10–13] and references therein). In this paper we introduce the concept of fuzzy order -contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order
-contractive type mappings. Then, using an idea in [14], we will provide a common fixed point theorem for weakly increasing single-valued mappings in a complete fuzzy metric space endowed with a partial order induced by an appropriate function. Some fixed point results on ordered probabilistic metric spaces can be found in [15].
For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper.
Definition 1.1 (see [16]).
A binary operation is called a continuous
-norm if
is an Abelian topological monoid with the unit
such that
whenever
and
for all
.
A continuous t-norm is of Hadžić-type if there exists a strictly increasing sequence
such that
for all
Definition 1.2 (see [3]).
A fuzzy metric space (in the sense of Kramosil and Michálek) is a triple , where
is a nonempty set,
is a continuous
-norm and
is a fuzzy set on
, satisfying the following properties:
(KM-1), for all
(KM-2), for all
if and only if
(KM-3) for all
and
(KM-4) is left continuous, for all
(KM-5) for all
for all
If, in the above definition, the triangular inequality (KM-5) is replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ1_HTML.gif)
then the triple is called a non-Archimedean fuzzy metric space. It is easy to check that the triangular inequality (NA) implies (KM-5), that is, every non-Archimedean fuzzy metric space is itself a fuzzy metric space.
Example 1.3.
Let be an ordinary metric space and let
be a nondecreasing and continuous function from
into
such that
. Some examples of these functions are
,
and
. Let
for all
. For each
, define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ2_HTML.gif)
for all . It is easy to see that
is a non-Archimedean fuzzy metric space.
Let be a fuzzy metric space. A sequence
in
is called an M-Cauchy sequence, if for each
and
there exists
such that
for all
. A sequence
in a fuzzy metric space
is said to be convergent to
if
for all
. A fuzzy metric space
is called M-complete if every
-Cauchy sequence is convergent.
Definition 1.5 (see [7]).
Let be a fuzzy metric space. A sequence
in
is called G-Cauchy if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ3_HTML.gif)
for all The space
is called G- complete if every G-Cauchy sequence is convergent.
Lemma 1.6 (see [11]).
Each M -complete non-Archimedean fuzzy metric space with
of Hadžić-type is G-complete.
Theorem 2.10in the next section is related to a partial order on a fuzzy metric space under the ukasiewicz t-norm. We will refer to [14].
Lemma 1.7 (see [14]).
Let be a non-Archimedean fuzzy metric space with
and
Define the relation "
" on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ4_HTML.gif)
Then is a (partial) order on
named the partial order induced by
.
2. Main Results
The first two theorems in this section are related to Theorem in [17]. We begin by giving the following definitions.
Definition 2.1.
Let be an order relation on
. A mapping
is called nondecreasing w.r.t
if
implies
.
Definition 2.2.
Let be a partially ordered set, let
be a fuzzy metric space, and let
be a function from
to
. A mapping
is called a fuzzy order
-contractive mapping if the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ5_HTML.gif)
Theorem 2.3.
Let be a partially ordered set and
be an
-complete non-Archimedean fuzzy metric space with
of Hadžić-type. Let
be a continuous, nondecreasing function and let
be a fuzzy order
-contractive and nondecreasing mapping w.r.t
. Suppose that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ6_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ7_HTML.gif)
hold. If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ8_HTML.gif)
for each , then
has a fixed point.
Proof.
Let for
. Since
and
is nondecreasing w.r.t
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ9_HTML.gif)
Then, it immediately follows by induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ10_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ11_HTML.gif)
By taking the limit as we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ12_HTML.gif)
for all that is,
is G-Cauchy. Since
is
-complete (Lemma 1.6), then there exists
such that
.
Now, if is continuous then it is clear that
, while if the condition (2.3) hold then, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ13_HTML.gif)
and letting it follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ14_HTML.gif)
hence
Theorem 2.4.
Let be a partially ordered set, let
be an
-complete non-Archimedean fuzzy metric space, and let
be a continuous mapping such that
for all
. Also, let
be a nondecreasing mapping w.r.t
, with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ15_HTML.gif)
Suppose that either (2.2) or (2.3) holds. If there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ16_HTML.gif)
for all , then
has a fixed point.
Proof.
Let for
. Then, as in the proof of the preceding theorem we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ17_HTML.gif)
Therefore, for every ,
is a nondecreasing sequence of numbers in
. Let, for fixed
,
Then we have
, since
. Also, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ18_HTML.gif)
and is continuous, we have
. This implies
and therefore, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ19_HTML.gif)
Now we show that is an M-Cauchy sequence. Supposing this is not true, then there are
and
such that for each
there exist
with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ20_HTML.gif)
Let, for each ,
be the least integer exceeding
satisfying the inequality (2.16), that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ21_HTML.gif)
Then, for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ22_HTML.gif)
Letting and using (2.15), we have, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ23_HTML.gif)
Then, since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ24_HTML.gif)
Letting and using (2.15) and (2.19), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ25_HTML.gif)
which is a contradiction. Thus is an M-Cauchy sequence. Since
is
-complete, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ26_HTML.gif)
If is continuous, then from
it follows that
Also, if (2.3) holds, then (since
) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ27_HTML.gif)
Letting , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ28_HTML.gif)
hence .
Example 2.5.
Let . Consider the following relation on
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ29_HTML.gif)
It is easy to see that is a partial order on
. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ30_HTML.gif)
Then is an M-complete non-Archimedean fuzzy metric space (see [18]) satisfying
for all
. Define a self map
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ31_HTML.gif)
Now, it is easy to see that is continuous and nondecreasing w.r.t
. Also, for
we have
. Now we can see that
is fuzzy order
-contractive with
.
Indeed, let with
. Now if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ32_HTML.gif)
If with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ33_HTML.gif)
Therefore is fuzzy order
-contractive with
. Hence all conditions of Theorem 2.4 are satisfied and so
has a fixed point on
.
In order to state our next theorem, we give the concept of weakly comparable mappings on an ordered space.
Definition 2.6.
Let be an ordered space. Two mappings
are said to be weakly comparable if
and
for all
.
Note that two weakly comparable mappings need not to be nondecreasing.
Example 2.7.
Let and
be usual ordering. Let
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ34_HTML.gif)
Then it is obvious that and
for all
. Thus
and
are weakly comparable mappings. Note that both
and
are not nondecreasing.
Example 2.8.
Let and
be coordinate-wise ordering, that is,
and
. Let
be defined by
and
, then
and
. Thus
and
are weakly comparable mappings.
Example 2.9.
Let and
be lexicographical ordering, that is,
or
if
then
. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ35_HTML.gif)
then and
for all
Thus
and
are weakly comparable mappings. Note that,
but
then
is not nondecreasing. Similarly
is not nondecreasing.
Theorem 2.10.
Let be an M -complete non-Archimedean fuzzy metric space with
be a bounded-from-above function, and let
be the partial order induced by
If
are two continuous and weakly comparable mappings, then
and
have a common fixed point in
Proof.
Let be an arbitrary point of
and let us define a sequence
in
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ36_HTML.gif)
Note that, since and
are weakly comparable, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ37_HTML.gif)
By continuing this process we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ38_HTML.gif)
that is, the sequence is nondecreasing. By the definition of
we have
for all
, that is, for even
, the sequence
is a nondecreasing sequence in
. Since
is bounded from above,
is convergent and hence it is Cauchy. Then, for all
there exists
such that for all
and
we have
. Therefore, since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ39_HTML.gif)
This shows that the sequence is M-Cauchy. Since
is M-complete, it converges to a point
. As
and
, by the continuity of
and
we get
.
Corollary 2.11 ([Caristi fixed point theorem in non-Archimedean fuzzy metric spaces]).
Let be an M -complete non-Archimedean fuzzy metric space with
let
be a bounded-from-above function and
be a continuous mapping, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ40_HTML.gif)
for all and
Then
has a fixed point in
Proof.
We take in the above theorem and note that the weak comparability of
and
reduces to (2.36).
The generalization suggested by Kirk of Caristi's fixed point theorem [19] is well known. A similar theorem in the setting of non-Archimedean fuzzy metric spaces is stated in the final part of our paper.
In what follows is nondecreasing, subadditive mapping (i.e.,
for all
), with
Theorem 2.12.
Let be a non-Archimedean fuzzy metric space with
and
Define the relation "
" on
through
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ41_HTML.gif)
Then "" is a (partial) order on
Proof.
Since , then for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ42_HTML.gif)
that is, "" is reflexive.
Let be such that
and
Then for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ43_HTML.gif)
implying that for all
that is,
. Thus "
" is antisymmetric.
Now for , let
and
. Then, for given
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ45_HTML.gif)
By using (2.40) and (2.41) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ46_HTML.gif)
On the other hand, from the triangular inequality (NA), the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ47_HTML.gif)
holds. This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ48_HTML.gif)
As is nondecreasing, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ49_HTML.gif)
and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ50_HTML.gif)
This shows that , that is, "
" is transitive.
From the above theorem we can immediately obtain the following generalization of Corollary 2.11.
Corollary 2.13.
Let be an M -complete non-Archimedean fuzzy metric space with
let
be a bounded-from-above function and
be a continuous mapping, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ51_HTML.gif)
for all and
If
satisfies the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F782680/MediaObjects/13663_2009_Article_1341_Equ52_HTML.gif)
then has a fixed point in
The reader is referred to the nice paper [20] for some discussion of Kirk's problem on an extension of Caristi's fixed point theorem.
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Altun, I., Miheţ, D. Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results. Fixed Point Theory Appl 2010, 782680 (2010). https://doi.org/10.1155/2010/782680
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DOI: https://doi.org/10.1155/2010/782680