- Research Article
- Open access
- Published:
Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results
Fixed Point Theory and Applications volume 2010, Article number: 782680 (2010)
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.
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
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
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
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:
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:
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
or
hold. If there exists such that
for each , then has a fixed point.
Proof.
Let for . Since and is nondecreasing w.r.t , we have
Then, it immediately follows by induction that
hence
By taking the limit as we obtain
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 ,
and letting it follows
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
Suppose that either (2.2) or (2.3) holds. If there exists such that
for all , then has a fixed point.
Proof.
Let for . Then, as in the proof of the preceding theorem we can prove that
Therefore, for every , is a nondecreasing sequence of numbers in . Let, for fixed , Then we have , since . Also, since
and is continuous, we have . This implies and therefore, for all
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
Let, for each , be the least integer exceeding satisfying the inequality (2.16), that is,
Then, for each ,
Letting and using (2.15), we have, for ,
Then, since , we have
Letting and using (2.15) and (2.19), we obtain
which is a contradiction. Thus is an M-Cauchy sequence. Since is -complete, then there exists such that
If is continuous, then from it follows that Also, if (2.3) holds, then (since ) we have
Letting , we obtain that
hence .
Example 2.5.
Let . Consider the following relation on :
It is easy to see that is a partial order on . Let and
Then is an M-complete non-Archimedean fuzzy metric space (see [18]) satisfying for all . Define a self map of as follows:
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
If with , then
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
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
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:
Note that, since and are weakly comparable, we have
By continuing this process we get
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
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
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
Then "" is a (partial) order on
Proof.
Since , then for all and ,
that is, "" is reflexive.
Let be such that and Then for all
implying that for all that is, . Thus "" is antisymmetric.
Now for , let and . Then, for given ,
By using (2.40) and (2.41) we get
On the other hand, from the triangular inequality (NA), the inequality
holds. This implies
As is nondecreasing, it follows that
and therefore
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
for all and If satisfies the property
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.
References
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 1994,64(3):395–399. 10.1016/0165-0114(94)90162-7
Kaleva O, Seikkala S: On fuzzy metric spaces. Fuzzy Sets and Systems 1984,12(3):215–229. 10.1016/0165-0114(84)90069-1
Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336–344.
Chang SS, Cho YJ, Lee BS, Jung JS, Kang SM: Coincidence point theorems and minimization theorems in fuzzy metric spaces. Fuzzy Sets and Systems 1997,88(1):119–127. 10.1016/S0165-0114(96)00060-7
Cho YJ: Fixed points in fuzzy metric spaces. Journal of Fuzzy Mathematics 1997,5(4):949–962.
Fang JX: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets and Systems 1992,46(1):107–113. 10.1016/0165-0114(92)90271-5
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 1988,27(3):385–389. 10.1016/0165-0114(88)90064-4
Gregori V, Sapena A: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems 2002,125(2):245–252. 10.1016/S0165-0114(00)00088-9
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.
Miheţ D: On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems 2007,158(8):915–921. 10.1016/j.fss.2006.11.012
Miheţ D: Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems 2008,159(6):739–744. 10.1016/j.fss.2007.07.006
Miheţ D: Fuzzy quasi-metric versions of a theorem of Gregori and Sapena. Iranian Journal of Fuzzy Systems 2010,7(1):59–64.
Mishra SN, Sharma N, Singh SL: Common fixed points of maps on fuzzy metric spaces. International Journal of Mathematics and Mathematical Sciences 1994,17(2):253–258. 10.1155/S0161171294000372
Altun I: Some fixed point theorems for single and multi valued mappings on ordered non-archimedean fuzzy metric spaces. Iranian Journal of Fuzzy Systems 2010,7(1):91–96.
Ćirić LB, Miheţ D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topology and Its Applications 2009,156(17):2838–2844. 10.1016/j.topol.2009.08.029
Schweizer B, Sklar A: Statistical metric spaces. Pacific Journal of Mathematics 1960, 10: 313–334.
Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151
Radu V: Some remarks on the probabilistic contractions on fuzzy Menger spaces. Automation Computers Applied Mathematics 2002,11(1):125–131.
Caristi J: Fixed point theory and inwardness conditions. In Applied Nonlinear Analysis. Academic Press, New York, NY, USA; 1979:479–483.
Khamsi MA: Remarks on Caristi's fixed point theorem. Nonlinear Analysis: Theory, Methods & Applications 2009,71(1–2):227–231. 10.1016/j.na.2008.10.042
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/782680