Letter to the Editor Comment on “Common Fixed Point Theorems for Commutating Mappings in Fuzzy Metric Spaces”
نویسندگان
چکیده
and Applied Analysis 3 Definition 2.2 Kramosil and Michálek 3 . The triple X,M, ∗ is called a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm, andM is a fuzzy set onX×X× 0,∞ satisfying the following conditions: for all x, y, z ∈ X and s, t > 0, FM-1 M x, y, 0 0, FM-2 M x, y, t 1 if and only if x y, FM-3 M x, y, t M y, x, t , FM-4 M x, y, t ∗M y, z, s ≤ M x, z, t s , FM-5 andM x, y, · : 0, ∞ → 0, 1 is left-continuous. Remark 2.3. According to FM-2 and FM-4 , it can easily be seen that M x, y, · is nondecreasing for all x, y ∈ X see Lemma 4 in 9 . Similar to the case in 1 , in this note, we suppose that X,M, ∗ is a fuzzy metric space with the following additional condition: FM-6 limt→ ∞M x, y, t 1, for all x, y ∈ X. Definition 2.4 Grabiec 9 , George and Veeramani 4 . Let X,M, ∗ be a fuzzy metric space. Then i a sequence {xn} in X is said to be convergent to a point x ∈ X, denoted by limn→∞xn x, if limn→∞M xn, x, t 1, for any t > 0; ii a sequence {xn} in X is called a G-Cauchy sequence if and only if limn→∞M xn p, xn, t 1 for any t > 0 and p > 0; iii a sequence {xn} in X is called anM-Cauchy sequence if and only if for each ∈ 0, 1 and t > 0, there exists n0 ∈ N such that M xm, xn, t > 1 − , for any m,n ≥ n0; iv a fuzzy metric space X,M, ∗ is said to be G-complete (M-complete) if every GCauchy sequence M-Cauchy sequence is convergent; v a map f : X → X is said to be continuous at x0 ∈ X if {f xn } converges to f x0 for each {xn} converging to x0. The authors have proved the following conclusion see the proof of Theorem 2.2 in 1 . Lemma 2.5 Zheng and Lian 1 . Let ψ : 0, ∞ → 0, ∞ be an increasing and left-continuous function with ψ t > t for all t > 0. Then
منابع مشابه
Common fixed point theorems for occasionally weakly compatible mappings in Menger spaces and applications
In 2008, Al-Thaga and Shahzad [Generalized I-nonexpansive self-maps and invariant approximations, Acta Math. Sinica 24(5) (2008), 867{876]introduced the notion of occasionally weakly compatible mappings (shortly owcmaps) which is more general than all the commutativity concepts. In the presentpaper, we prove common xed point theorems for families of owc maps in Mengerspaces. As applications to ...
متن کاملOn Fixed Point Theorems for Contractive-type Mappings in Fuzzy Metric Spaces
In this paper, we provide two different kinds of fixed pointtheorems in fuzzy metric spaces. The first kind is for the fuzzy$varepsilon$-contractive type mappings and the second kind is forthe fuzzy order $psi$-contractive type mappings. They improve thecorresponding conclusions in the literature.
متن کاملSOME FIXED POINT THEOREMS FOR SINGLE AND MULTI VALUED MAPPINGS ON ORDERED NON-ARCHIMEDEAN FUZZY METRIC SPACES
In the present paper, a partial order on a non- Archimedean fuzzymetric space under the Lukasiewicz t-norm is introduced and fixed point theoremsfor single and multivalued mappings are proved.
متن کاملOn the fixed point theorems in generalized weakly contractive mappings on partial metric spaces
In this paper, we prove a fixed point theorem for a pair of generalized weakly contractive mappings in complete partial metric spaces. The theorems presented are generalizations of very recent fixed point theorems due to Abdeljawad, Karapinar and Tas. To emphasize the very general nature of these results, we illustrate an example.
متن کاملNon-Archimedean fuzzy metric spaces and Best proximity point theorems
In this paper, we introduce some new classes of proximal contraction mappings and establish best proximity point theorems for such kinds of mappings in a non-Archimedean fuzzy metric space. As consequences of these results, we deduce certain new best proximity and fixed point theorems in partially ordered non-Archimedean fuzzy metric spaces. Moreover, we present an example to illustrate the us...
متن کامل