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 ).
Definition 1.2 (see ).
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.
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 ).
Lemma 1.6 (see ).
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 ).
2. Main Results
The first two theorems in this section are related to Theorem in . We begin by giving the following definitions.
In order to state our next theorem, we give the concept of weakly comparable mappings on an ordered space.
Note that two weakly comparable mappings need not to be 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
Corollary 2.11 ([Caristi fixed point theorem in non-Archimedean fuzzy metric spaces]).
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.
On the other hand, from the triangular inequality (NA), the inequality
From the above theorem we can immediately obtain the following generalization of Corollary 2.11.
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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