Abstract
Recently, Al-Fhaid and Mohiuddine (Adv. Differ. Equ. 2013:203, 2013) and Mohiuddine and Alghamdi (Adv. Differ. 2012:141, 2012) got some results in intuitionistic fuzzy normed spaces using ideas of intuitionistic fuzzy sets due to Atanassov and fuzzy normed spaces due to Saadati and Vaezpour. In this note, we show that the mentioned results follow directly from well-known theorems in fuzzy normed spaces.
MSC:54E40, 54E35, 54H25.
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1 Introduction
Intuitionistic fuzzy normed spaces were investigated by Saadati and Park [1]. They introduced and studied intuitionistic fuzzy normed spaces based both on the idea of intuitionistic fuzzy sets due to Atanassov [2] and the concept of fuzzy normed spaces given by Saadati and Vaezpour in [3]. Next Deschrijver et al. [4] modified the concept of intuitionistic fuzzy normed spaces and introduced the notation of ℒ-fuzzy normed space. Recently, Al-Fhaid and Mohiuddine [5] and Mohiuddine and Alghamdi [6] got some results in intuitionistic fuzzy normed spaces. In this note we prove that the topology generated by an intuitionistic fuzzy normed space coincides with the topology generated by the generalized fuzzy normed space , and thus, the results obtained in [5] and [6] are immediate consequences of the corresponding results for fuzzy normed spaces.
2 Preliminaries
A binary operation is a continuous t-norm if it satisfies the following conditions:
-
(a)
∗ is associative and commutative,
-
(b)
∗ is continuous,
-
(c)
for all ,
-
(d)
whenever and , for each .
Two typical examples of continuous t-norm are and .
A binary operation is a continuous t-conorm if it satisfies the following conditions:
-
(a)
⋄ is associative and commutative,
-
(b)
⋄ is continuous,
-
(c)
for all ,
-
(d)
whenever and , for each .
Two typical examples of a continuous t-conorm are and .
In 2005, Saadati and Vaezpour [3] introduced the concept of fuzzy normed spaces.
Definition 2.1 Let X be a real vector space. A function is called a fuzzy norm on X if for all and all ,
() for ;
() if and only if for all ;
() if ;
() ;
() is a non-decreasing function of ℝ and ;
() for , is continuous on ℝ.
For example, if for , is normed space and
for all and . Then μ is a (standard) fuzzy normed and is a fuzzy normed space.
Saadati and Vaezpour showed in [3] that every fuzzy norm on X generates a first countable topology on X which has as a base the family of open sets of the form where for all , and .
3 Intuitionistic fuzzy normed spaces
Saadati and Park [1] defined the notion of intuitionistic fuzzy normed spaces with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy normed space due to Saadati and Vaezpour [3].
Definition 3.1 The 5-tuple is said to be an intuitionistic fuzzy normed space if X is a vector space, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm, and μ, ν are fuzzy sets on satisfying the following conditions for every and :
-
(a)
,
-
(b)
,
-
(c)
if and only if ,
-
(d)
for each ,
-
(e)
,
-
(f)
is continuous,
-
(g)
and ,
-
(h)
,
-
(i)
if and only if ,
-
(j)
for each ,
-
(k)
,
-
(l)
is continuous,
-
(m)
and .
In this case is called an intuitionistic fuzzy norm.
Example 3.2 Let be a normed space. Denote and for all and let μ and ν be fuzzy sets on defined as follows:
for all . Then is an intuitionistic fuzzy normed space.
Saadati and Park proved in [1] that every intuitionistic fuzzy norm on X generates a first countable topology on X which has as a base the family of open sets of the form where for all , and .
Lemma 3.3 Let be an intuitionistic fuzzy normed space. Then, for each , and , we have .
Proof It is clear that .
Now, suppose that . Then , so, by condition (i) of Definition 3.1, we have
Hence , and consequently . The proof is finished. □
From Lemma 3.3, we deduce the following.
Theorem 3.4 Let be an intuitionistic fuzzy normed space. Then the topologies and coincide on X.
References
Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 2006, 27(2):331-344. 10.1016/j.chaos.2005.03.019
Atanassov KT: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20(1):87-96. 10.1016/S0165-0114(86)80034-3
Saadati R, Vaezpour SM: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 2005, 17(1-2):475-484. 10.1007/BF02936069
Deschrijver G, O’Regan D, Saadati R, Vaezpour SM: ℒ-Fuzzy Euclidean normed spaces and compactness. Chaos Solitons Fractals 2009, 42(1):40-45. 10.1016/j.chaos.2008.10.026
Al-Fhaid AS, Mohiuddine SA: On the Ulam stability of mixed type QA mappings in IFN-spaces. Adv. Differ. Equ. 2013., 2013: Article ID 203
Mohiuddine SA, Alghamdi MA: Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 141
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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Saadati, R., Park, C. A note on IFN-spaces. Adv Differ Equ 2014, 63 (2014). https://doi.org/10.1186/1687-1847-2014-63
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DOI: https://doi.org/10.1186/1687-1847-2014-63