Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results
© I. Altun and D. Miheţ. 2010
Received: 2 July 2009
Accepted: 9 February 2010
Published: 14 February 2010
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 , 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 .
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 ).
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 ).
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
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.
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 ).
for all The space is called G- complete if every G-Cauchy sequence is convergent.
Lemma 1.6 (see ).
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 .
Lemma 1.7 (see ).
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 . We begin by giving the following definitions.
Let be an order relation on . A mapping is called nondecreasing w.r.t if implies .
for each , then has a fixed point.
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 ,
for all , then has a fixed point.
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
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.
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.
Then it is obvious that and for all . Thus and are weakly comparable mappings. Note that both and are not nondecreasing.
Let and be coordinate-wise ordering, that is, and . Let be defined by and , then and . Thus and are weakly comparable mappings.
then and for all Thus and are weakly comparable mappings. Note that, but then is not nondecreasing. Similarly is not nondecreasing.
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
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]).
for all and Then has a fixed point in
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  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
Then " " is a (partial) order on
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 ,
On the other hand, from the triangular inequality (NA), the inequality
This shows that , that is, " " is transitive.
From the above theorem we can immediately obtain the following generalization of Corollary 2.11.
then has a fixed point in
The reader is referred to the nice paper  for some discussion of Kirk's problem on an extension of Caristi's fixed point theorem.
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