We generalize the concept of fuzzy metric space in the sense of George and Veeramani by presenting the definition of generated fuzzy metric space and generated fuzzy bonded metric space. We prove also some known results of metric spaces including Baire’s theorem for generated fuzzy metric spaces through α-defective diameter.
In 1965, the theory of fuzzy sets was discovered by Zadeh [16]. Later, many writings on the fuzzy logic and fuzzy sets have been published as in [4, 14]. Making some modifications George and Veeramani redefined the fuzzy metric space, which was introduced by Kramosil and Michalek in [10], and gived some new results on the fuzzy metric spaces [5, 6]. On the other hand in 1972 Grossman and Katz established a new calculus, called non-Newtonian calculus, alternatively to the classical calculus [8]. Creating this calculus, they used increasing generators. As a special case of generators the exponential function exp gene-rated multiplicative (or geometric) calculus. Bashirov et al. studied on the multiplicative calculus and restated some classical properties of derivatives and integrals in the realm of multiplicative calculus [2]. Çakmak and Başar constructed the field of non-Newtonian real numbers and the concept of non-Newtonian metric [5]. Binbaşıoğlu et.al [3] and Kirişci [9] defined some topological properties related to non-Newtonian metric spaces.
Thinking any generator may not be increasing, we reinterpret the definition of non-Newtonian metric and call generated metric. After that, we generalize the concept of fuzzy metric space by presenting the definition of generated fuzzy metric and generated fuzzy bonded metric. In this case, it is obviously that every fuzzy metric is a generated fuzzy metric when we take the generator as the identity function I. We also give the concept of α-defective diameter for any subset of generated fuzzy metric space based on [11, 12]. Finally, we examine some theorems about convergence of sequences and Baire’s theorem which are located in functional analysis and topological spaces [1, 13] for generated fuzzy metric spaces, and prove with help of the concept of α-defective diameter as different from [6] and [7].
Preliminaries
A generator α is a one-to-one map whose domain is and whose range is (or ) a subset of . Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. There are four basic binary operations on generated by α. The operations α-addition and α-multiplication on are defined by
for all , respectively. For every , let . Then the operation α-subtraction is defined by
for all . Let kα = α (k) for all . For every , we define . Then the operation α-division is defined by
for all and . Let the generator α be monotone. We define a total order (called α-order) with
for all . Then the range is an ordered field. Hence, the generator α generates an arithmetic on called α-arithmetic.
Remark 2.1. If the generator α is monotone, then (1) it is continuous on and (2) for any two elements x and y in , iff x ⩽ y. Despite this, we will remain faithful to the definition of α-order.
Example 2.2. (a) The identity map I generates classical arithmetic on .(b) The exponential map exp generates geometric arithmetic on , and is an ordered field with the operations
and with the total order for all .(c) The inverse tangent map arctan generates a arithmetic on such that is an ordered field with the operations
and with the total order for all .(d) Some of the infinite number of arithmetics on are called q -arithmetics by Grossman and Katz [8]. For any nonzero real number p, the generator q from to itself defined as
for all . If p = 1 then the q-arithmetic is the classical arithmetic. For p = 2, the map q generates quadratic arithmetic and then is an ordered field with the operations
and with the total order for all . (e) Let α be a map on defined as for all . The image is an ordered field with the operations
and with the total order for all x, y ∈ (-1, 1).
In situation that the α is not monotone, an α-order can also be defined on .
Example 2.3. (a) In Example 2.2(d), for p = -1 the generator q defined as
generates harmonic arithmetic and then is an ordered field with the operations
and with the total order
for all . (b) A map α defined as
generate an arithmetic on with the operations
with the total order
Depending on the monotonicity of α, the intervals of may be different from those of . For any , we define α-open interval with
and similarly, we define α-closed interval with
We say that α- if for every there is an integer n0 = n0 (ɛ, α)∈ such that whenever n > n0. If the generator α is monotone, then α-. The set of α-positive real numbers is . The Table 1 describes these concepts for some generators.
Table1
α
0α
1α
[0α, 1α] α
I
(- ∞ , ∞)
-∞
0
1
∞
(0, ∞)
[0, 1]
-I
(- ∞ , ∞)
∞
0
-1
-∞
(- ∞ , 0)
[-1, 0]
exp
(0, ∞)
0
1
e
∞
(1, ∞)
[1, e]
arctan
0
arccot
(0, π)
π
0
The α-square of a number x in is denoted by . Since for any
for any integer m, by induction the m-th α-power of x is xmα = α ([α-1 (x)] m). The square α-root is denoted by for all . Generally, for any positive integer n ⩾ 2, the n-th α-root (if there exists) of a number x in is . Besides, the α-absolute value of x in is
Consider the generator α may not be increasing, let us update the definition of non-Newtonian metric in [5] with new notations.
Definition 2.4. Let X be a non-empty set. The map is called generated metric (or α-metric) if it satisfies the following properties: (gm1) (gm2) dα (x, y) =0α if and only if x = y, (gm3) dα (x, y) = dα (y, x) and (gm4) for all x, y, z ∈ X . Also the pair (X, dα) is said to be a generated metric space.
Each metric is a generated metric while the generator α is the unit function I.
Remark 2.5. Non-negativity of any α-metric is directly related to the order given. For example, let -I be the inverse of unit function I and . Then the map d-I defined as d-I (x, y) = - |x - y| for all x, y is an -I-metric. Since -I is an decreasing generator, and
for all .
Generated Fuzzy Metric Spaces
We start with the definitions required for the issue.
Definition 3.1. Let α and β be two generator. The function f from to is called (α, β)-continuous if for every and for every there exists such that and} implies . In particular, when β = I, then the function f is called α-continuous.
Definition 3.2. [9] A binary operation ∗ : [0, 1] 2 → [0, 1] is a t-norm if it is commutative, associative, increasing and 1 is its neutral element, i.e.,
The above definition is a special case following definition when it be taken α = I.
Definition 3.3. A binary operation is a generated t-norm if it is commutative, associative, increasing and 1α is its neutral element, i.e.,
The double of α from to defined by Λ (a, b) = (α (a) , α (b)) for all . Let us give the following result without proof.
Corollary 3.4.For any generatorαon(1) Ifis a generated t-norm, thenis a t-norm,(2) If ∗ is at-norm, thenis a generated t-norm.
Let be a generated t-norm. For the t-norm we have for all a, b ∈ [0, 1]. Similarly, let ∗ be a t-norm. For the generated t-norm we have for all a, b ∈ [0α, 1α] α.
Example 3.5. Let α be the exponential function exp. (a) For the product t-norm a ∗ b = ab, the map defined by is a generated t-norm on [1, e] 2. (b) The binary operation is a generated t-norm on [1, e] 2. The map ∗ defined by is a t-norm (Łukasiewicz t-norm).
Now let us give the fuzzy metric space in the sense of George and Veeramani[5].
Definition 3.6. Let X be a non-empty set and ∗ be a continuous t-norm. The triple (X, M, ∗) is called fuzzy metric space if the fuzzy set M on satisfies the following properties: (fm1) M (x, y, t) >0, (fm2) M (x, y, t) =1 if and only if x = y, (fm3) M (x, y, t) = M (y, x, t) , (fm4) M (x, y, t) ∗ M (y, z, s) ⩽ M (x, z, t + s) and (fm5) is continuous for all x, y, z ∈ X and all t, s > 0. If (X, M, ∗) is a fuzzy metric space, we will say that (M, ∗) or simply M is a fuzzy metric on X.
Let be an ordered field generated by a generator α on . We define the generated fuzzy metric space and obtain some topological properties of this space.
Definition 3.7. Let X be a non-empty set and ∗ be a continuous t-norm. The triple (X, Mα, ∗) is called generated fuzzy metric space (or fuzzy α-metric space) if the fuzzy set Mα on satisfies the following properties:
(gfm1) Mα (x, y, t) >0, (gfm2) Mα (x, y, t) =1 if and only if x = y, (gfm3) Mα (x, y, t) = Mα (y, x, t) , (gfm4) , (gfm5) is α-continuous for all x, y, z ∈ X and all If (X, Mα, ∗) is a generated fuzzy metric space, we will say that (Mα, ∗) or simply Mα is a generated fuzzy metric on X.
As similar to the above definition, we can define a type of fuzzy metric for any α-membership function .
Definition 3.8. Let X be a non-empty set and be a continuous generated t-norm. The triple is called generated fuzzy bonded metric space if the fuzzy set αM on satisfies the following properties:
(gfbm1)
(gfbm2) αM (x, y, t) =1α if and only if x = y, (gfbm3) αM (x, y, t) = αM (y, x, t) ,
(gfbm4)
,
(gfbm5)
is (α, α)-continuous for all x, y, z ∈ X and all
If is a generated fuzzy bonded metric space, we will say that or simply αM is a generated fuzzy bonded metric on X .
Lemma 3.9.Let (Mα, ∗) be a generated fuzzy metric and letbe a generated fuzzy bonded metric onX. Then theMα (x, y,.) and theαM (x, y,.) are increasing for allx, y ∈ X. If the generatorαis increasing and, then(1) (α-1 ∘ αM, ∗) is a generated fuzzy metric,(2)is a generated fuzzy bonded metric.
Proof. First suppose that Mα (x, y,.) and αM (x, y,.) are not increasing; namely, for 0 < t < s, let Mα (x, y, t) > Mα (x, y, s) and let . Then
hold. By Def. 3.7(gfm2) and Def. 3.8(gfbm2),
respectively. Hence, the assumptions are wrong since
(1) Since the α-membership function
satisfies the conditions Def.?? (gfbm1)-(gfbm5) and the generator α is increasing (and hence the inverse generator α-1 is increasing), then for all x, y, z ∈ X and the followigs hold: (gfm1) 0 < α-1 (αM (x, y, t)) = (α-1 ∘ αM) (x, y, t) if and only if . (gfm2) Let x = y. Since αM (x, y, t) =1α, then (α-1 ∘ αM) (x, y, t) = α-1 (1α) =1 . To converse, let (α-1 ∘ αM) (x, y, t) =1. Since αM (x, y, t) = α [(α-1 ∘ αM) (x, y, t)] = α (1) =1α, then x = y. (gfm3) Since αM(x, y, t) = αM (y, x, t), then (α-1 ∘ αM) (x, y, t) = (α-1 ∘ αM) (y, x, t) . (gfm4) Since for all a, b ∈ [0α, 1α] α, we have
(gfm5) Since the generator α one to one and increasing, then α and so α-1 are continuous. Hence, (2) It is proved in a similar way to (1).
Example 3.10. For the generator exp, let (X, dexp) be a generated metric space and denote a generated t-norm with for all a, b ∈ [1, e]. The generated fuzzy bonded metric induced by dexp on X × X × (1, ∞) is defined as
Then the triple is a generated fuzzy bonded metric space. On the other hand, by Example 3.5(1) and Lemma 3.9(1), the map defined as
on X × X × (1, ∞) is the generated fuzzy metric induced by dexp and so is a generated fuzzy bonded metric space.
Theorem 3.11.LetXbe a non-empty set. For any generatorαonletube a map satisfyingu (x, y, k) = (x, y, α (k)) for allx, y ∈ Xandk > 0.(1) If (Mα, ∗) is a generated fuzzy metric onX, then (Mα ∘ u, ∗) is a fuzzy metric onX,(2) If (M, ∗) is a fuzzy metric onX, then (M ∘ u-1, ∗) is a generated fuzzy metric onX.
Proof. (1) Let Mα be a generated fuzzy metric. One can easily see that for the map Mα ∘ u the conditions (fm1), (fm2), (fm3) and (fm5). We only prove the triangular inequality (fm4). For all x, y, z ∈ X and t, s > 0
(2) Let M be a fuzzy metric. M ∘ u-1 The conditions (gfm1), (gfm2), (gfm3) and (gfm5) are easy hold for M ∘ u-1. Let us prove (gfm4). For all x, y, z ∈ X and
The following result is the outcome discussed above.
Corollary 3.12.LetXbe a non-empty set. For a generated fuzzy metric (Mα, ∗) and a generated fuzzy bonded metric on X, let . If the generator α is increasing, then (1) (α-1 ∘ αM ∘ u, ∗) is a fuzzy metric, (2) is a generated fuzzy bonded metric.
Now, let us illustrate some topological concepts in a generated fuzzy metric space.
Definition 3.13. Let (X, Mα, ∗) be a generated fuzzy metric space. We define open ball Bα (x, r, t) for with centre x ∈ X and radius r, 0 < r < 1 as
Now, similarly we describe the closed ball as
In the following definition we present a novel concept, which is called α-defective diameter with the aid of [15].
Definition 3.14. Let (X, Mα, ∗) be a generated fuzzy metric space and A is non-empty subset of X. We define α-defective diameter for as
As a particular case, if we take α as the identity function I, is called defective diameter of A. The set A is called F (α)-bounded if .
Remark 3.15. (1) If A is a singleton set, then . Unlike in crisp set, we see that when , the set A does not have to be a singleton set. Let α be the identity function I. Suppose (X, Md, ∗) be a fuzzy metric space where Md is defined by Md (x, y, t) = t/(t + d (x, y)) for all x, y ∈ X and t > 0 with a metric d. For a two-element subset A = {x0, y0} of X
(2) If the set A is F (α)-bounded, then for every x, y ∈ A and there exists an r ∈ (0, 1) such that Mα (x, y, t) ⩾1 - r, that is, for every and for every there exists an r ∈ (0, 1) such that A ⊂ Bα [x, r, t].
Definition 3.16. (a) A sequence (xn) in a generated fuzzy metric space (X, Mα, ∗) is converges to x ∈ X, if for every ɛ ∈ (0, 1) and there exists such that Mα (xn, x, t) >1 - ɛ for all n ⩾ n0 and it is denoted by
(b) A sequence (xn) in a generated fuzzy metric space (X, Mα, ∗) is called Cauchy if for each ɛ ∈ (0, 1) and there exists such that Mα (xn, xm, t) >1 - ɛ for all n, m ⩾ n0. (c) A generated fuzzy metric space (X, Mα, ∗) is called complete if every Cauchy sequence in X converges. (d) A subset Y of a generated fuzzy metric space (X, Mα, ∗) is called closed if (xn) ⊂ Y and imply x ∈ Y .
Theorem 3.17.Let (X, Mα, ∗) be a generated fuzzy metric space and (xn) ⊂ X be a convergent sequence. Then, (1) (xn) is F (α)-bounded and its limit x is unique, (2) (xn) is a Cauchy sequence in X, (3) Every subsequence of (xn) converges to x.
Proof. (1) First, we show that the convergent sequence (xn) is F (α) -bounded. Then, for every ɛ ∈ (0, 1) and there exists such that Mα (xn, x, t/2) >1 - ɛ for all n ⩾ n0. Now for x0 ∈ X and 0 < s < 1, let
and for 0 < k < 1, let
Then we can find a r ∈ (0, 1) such that
Hence, for all we have
Thus for all (xn) ⊂ Bα [x0, r, t], that is, (xn) is bounded. Now we show that the limit of (xn) is unique. We suppose that the convergent sequence (xn) has two different limits x and y. Let and ɛ ∈ (0, 1). Then we can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ⩾1 - ɛ. Let ɛ = 1 - Mα (x, y, t). According to our assumption there exists such that Mα (xn, x, t/2) >1 - r for all n ⩾ n1 and there exists such that Mα (xn, y, t/2) >1 - r for all n ⩾ n2. When we take n0 = max {n1, n2}, then for n ⩾ n0
that is, Mα (x, y, t) > Mα (x, y, t), which is not possible. (2) Let and ɛ ∈ (0, 1). We can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) > (1 - ɛ). Since (xn) is a convergent sequence, there exists such that Mα (xn, x, t/2) >1 - r for all n ⩾ n0. For all n, m ⩾ n0 we get
(3) Let (xnk) ⊂ (xn). Since (xn) is a convergent sequence, there exists for each ɛ ∈ (0, 1) and such that Mα (xn, x, t/2) >1 - ɛ for all n ⩾ n0. If k ⩾ n0, since n0 ⩽ k ⩽ nk, we get Mα (xnk, x, t) >1 - ɛ.
Given two generated fuzzy metric spaces and , for (x1, x2) , (y1, y2) ∈ X1 × X2, t > 0, if we define
then it easy to prove that Mα is generated fuzzy metric on X1 × X2.
Theorem 3.18.Let and be generated fuzzy metric spaces. If (xn) ⊂ X1, (yn) ⊂ X2 are convergent sequences such that and , then .
Proof. Let and ɛ ∈ (0, 1). We can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) > (1 - ɛ). Since (xn) ⊂ X1 and (yn) ⊂ X2 are convergent sequences, there exists such that for all n ⩾ n1 and there exists such that for all n ⩾ n2. Using the definition above mentioned for n0 = max {n1, n2} we obtain
This completes the proof of the theorem.
Theorem 3.19.For the generated metric space (X, Mα, ∗) the followings hold. (1) Every convergent sequence is a Cauchy sequence. (2) Every Cauchy sequence is F (α)-bounded. (3) If a Cauchy sequence (xn) has a subsequence (xnk) such that
Proof. (1) Let and ɛ ∈ (0, 1). We can find a number r ∈ (0, 1) such that (1 - r) ∗ (1 - r) >1 - ɛ. Since (xn) a convergent sequence in (X, Mα, ∗) there exists such that Mα (xn, x, t/2) >1 - r for all n ⩾ n0. Hence, for n, m ⩾ n0 we have
(2) Let (xn) be a Cauchy sequence. Then, for every ɛ ∈ (0, 1) and there exists such that Mα (xn, xm, t) >1 - ɛ for all n, m ⩾ n0. So, for n ⩾ n0 we have Mα (xn, xn0, t) >1 - ɛ. Let
If s = max {ɛ, r}, then {xn : n = 1, 2,. . . ⊂ Bα [xn0, s, t], that is, (xn) is bounded. (3) Let and ɛ ∈ (0, 1). We can find r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ⩾1 - ɛ. Since (xn) is a Cauchy sequence, there exists such that Mα (xn, xm, t/2) >1 - r for all m, n ⩾ n0. Since , there exists positive integer ip such that ip > n0, Mα (xip, x, t/2) >1 - r. For n ⩾ n0 we have
Then and hence (X, Mα, ∗) is complete.
Definition 3.20. Let (X, Mα, ∗) be a generated fuzzy metric space and (An) be a sequence of non-empty subsets of X .We say that (An) has appearing α-defective diameter if a sequence (An) has appearing α-defective diameter, then for every r ∈ (0, 1) and there exists such that Mα (x, y, t) >1 - r for all x, y ∈ An.
Theorem 3.21.Let (X, Mα, ∗) be a generated fuzzy metric space and (An) be any decreasing sequence of non-empty closed subsets of X with appearing α-defective diameter. X is complete if and only if there is exactly one point x ∈ X such that .
Proof. Let the generated fuzzy metric space X be complete. We can form a sequence (xn) by taking a point xn ∈ An for each . If we choose m ⩾ n, we get Am ⊂ An, so that all points {xm : m ⩾ n} of the sequence belong to the set An. So for each we obtain for all m ⩾ n and consequently . In this case (xn) is a Cauchy sequence. Since X is complete, there exists a point x ∈ X, such that . We take a set An0. The limit of the sequence {xn : n ⩾ n0} ⊂ An0 is of course the same point x. Futhermore, since An0 is closed, x ∈ An0. This means that x belongs to every member of the sequence (An).Hence we get . Now we consider another point . Then, for every the relation holds and this implies that Mα (x, x′, t) =1. This means that because of x = x′.
Conversely, let (xn) be a Cauchy sequence in X and for every let An = {xm : m ⩾ n} be a non-empty closed subset of X. Since the sequence (An) is decreasing and since (xn) is a Cauchy sequence, we have . To our assumption the intersection of all these sets contains only a single point of the space X. Suppose that . For every ɛ ∈ (0, 1) there exists such that . Since x ∈ An0, for every ɛ ∈ (0, 1) and we have Mα (xn, x, t) >1 - ɛ for all n ⩾ n0. Therefore, the Cauchy sequence (xn) converges to the point x. Hence, the generated fuzzy metric space X is complete.
Now we give the Baire theorem for generated fuzzy metric spaces.
Theorem 3.22.(Baire’s theorem) Let X be a generated complete fuzzy metric space and let be a countable family of open subsets which are dense in X. Then the set is also dense in X.
Proof. In order to prove the theorem we have to show that for every x ∈ X, r ∈ (0, 1) and , the intersection of open balls Bα (x, r, t) and the set is non-empty. First we take the set A1. Since A1 is dense in X, Bα (x, r, t) ∩ A1 is open and non-empty. Let x1 ∈ Bα (x, r, t) ∩ A1, then there exist r1 ∈ (0, 1) and such that Bα [x1, r1, t1] ⊂ Bα (x, r, t) ∩ A1. Let B1 = Bα (x1, r1, t1). Since A2 is dense in X, B1 ∩ A2 is open and non-empty. Let x2 ∈ B1 ∩ A2, then there exist r2 ∈ (0, 1/2) and such that Bα [x2, r2, t2] ⊂ B1 ∩ A2. Let Bn = Bα (xn, rn, tn) for all n ⩾ 2. If we proceed inductively, we obtain a sequence (xn) in X and a sequence (rn) of positive real numbers such that
and rn ∈ (0, 1/n) for all n. Thus, the Theorem 3.21 guarantees that is a singleton set. From we obtain that .
BashirovA.E., KurpinarE.M. and ÖzyapiciA., Multiplicative calculus and its applications, Journal of Applied Mathematical Analysis and Applications337 (2008), 36–48.
3.
BinbaşioğluD., DemirizS. and TürkoğluD., Fixed points of non-Newtonian contraction mappings on non-Newtonian metric spaces, Journal of Fixed Point Theory and Applications18(1) (2016), 213–224.
4.
BuckleyJ.J., EslamiE., An Introduction to Fuzzy Logic and Fuzzy Sets, Advances in Soft Computing, Physica-Verlag, Heidelberg, 2002.
5.
ÇakmakA.F. and BaşarF., Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications2012(228) (2012), 17.
6.
GeorgeA. and VeeramaniP., On some results in fuzzy metric spaces, Fuzzy Sets and Systems64(3) (1994), 395–399.
7.
GeorgeA. and VeeramaniP., On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems90(3) (1997), 365–368.
8.
GrossmanM., KatzR., Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
9.
KirişciM., Topological structures of non-Newtonian metric spaces, Electronic Journal of Mathematical Analysis and Applications5(2) (2017), 156–169.
10.
KramosilI. and MichalekJ., Fuzzy metrics and statistical metric spaces, Kybernetica11(5) (1975), 336–344.
11.
SchweizerB. and SklarA., Statistical metric spaces, Pacific Journal of Mathematics10 (1960), 313–334.
12.
SchweizerB., SklarA., Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co, New York, 1983.
WangZ., YangR., LeungK.-S.Nonlinear Integrals and Their Applications in Data Mining, Advances in Fuzzy Systems-Applications and Theory, 24, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.
15.
XiaoJ.-Z., ZhuX.-H. and JinP.-P., Iterated function systems and attractors in the KM fuzzy metric spaces, Fuzzy Sets and Systems267 (2015), 100–116.
16.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.