Fixed Point Theorems for kg -Contractive Mappings in a Complete Strong Fuzzy
Metric Space
Kandala Kanakamahalakshmi Sarma
?and Yohannes Gebru Aemro
Abstract
In this paper, we introduce a new class of contractive mappings in fuzzy metric spaces and we present some fixed point results for this class of maps.
Keywords:Fixed points, strong fuzzy metric space, generalizedkg-contractive mappings.
2010 Mathematics Subject Classification: Primary 47H10; Secondary 54H25.
How to cite this article
K. K. M. Sarma, Y. G. Aemro, Fixed point theorems forkg-contractive mappings in a complete strong fuzzy metric space,Math. Interdisc. Res.
3(2018) 15-29.
1. Introduction
The concept of a fuzzy metric space was introduced by Kramosil and Micálek [7]. Afterwards, George and Veeramani [1] modified the concept of fuzzy metric space due to [7]. Later on, Gregori and Sapene [4] introduced fuzzy contraction mappings and proved a fixed point theorem in fuzzy metric space in the sense of George and Veeramani. In particular, Miheţ enlarged the class of fuzzy contrac- tive mappings of Gregori and Sapene [4] in a complete non-Archimedean(strong) fuzzy metric space and proved a fuzzy Banach contraction theorem using a strong condition for completeness, now called the completeness in the sense of Grabiec, or G-completeness. Motivated by the “Generalized weak contractions” introduced by Singh et al. [13] in metric spaces. In this paper, we introducekg−contractive
?Corresponding author (E-mail: [email protected]) Academic Editor: Ali Reza Ashrafi
Received 24 July 2017, Accepted 10 October 2017 DOI: 10.22052/mir.2017.93427.1070
2018 University of Kashanc This work is licensed under the Creative Commons Attribution 4.0 International License.
maps in fuzzy metric spaces and we prove the existence of fixed points in complete strong fuzzy metric spaces.
2. Preliminaries
We begin with some basic definitions and results which will be used in the main part of our paper.
Definition 2.1. [12] A binary operation ∗: [0,1]×[0,1]→[0,1]is said to be a continuous t-norm if it satisfies the following conditions :
(T1) ∗ is associative and commutative;
(T2) ∗ is continuous;
(T3) a∗1 =afor alla∈[0,1];
(T4) a∗b≤c∗dwhenevera≤c andb≤dfor alla, b, c, d∈[0,1].
Remark 1. A t-norm∗ is called positive ifa∗b >0 for alla, b∈(0,1).
Some examples of continuous t-norms are Lukasievicz t-norm, i.e. a∗Lb = max{a+b−1,0}, product t-norm, i.e. a∗b = ab, and minimum t-norm, i.e.
a∗Mb= min{a, b}, fora, b∈[0,1].
The concept of fuzzy metric space is defined by George and Veeramani [1] as follows.
Definition 2.2. [1] Let X be a nonempty set and ∗ be a continuous t-norm.
Assume that, for eachx, y, z∈X andt, s >0, a fuzzy setM :X×X×(0,∞)→ [0,1]satisfies the following conditions:
(M1) M(x, y, t)>0,
(M2) M(x, y, t) = 1if and only if x=y, (M3) M(x, y, t) =M(y, x, t),
(M4) M(x, y, t)∗M(y, z, s)≤M(x, z, t+s), (M5) M(x, y,·) : (0,∞)→[0,1]is continuous.
Then, we callM a fuzzy metric onX, and we call the 3-tuple(X, M,∗)a fuzzy metric space.
Definition 2.3. [5] Let (X, M,∗) be a fuzzy metric space. The fuzzy metricM is said to be strong (non- Archimedean) if, for eachx, y, z∈X and eacht >0, it satisfies
(M40) :M(x, z, t)≥M(x, y, t)∗M(y, z, t).
Remark 2. The axiom(M40)cannot replace the axiom(M4) in the definition of fuzzy metric since in that caseM could not be a fuzzy metric onX (see Example 8 in [11]).
Note that it is possible to define a strong fuzzy metric by replacing (M4) by (M40) and demanding in (M5) that the function M(x, y,·) be an increasing continuous function on t, for each x, y ∈ X (in fact, in such a case we have M(x, z, t+s)≥M(x, y, t+s)∗M(y, z, t+s)≥M(x, y, t)∗M(y, z, s)).
Remark 3. Every fuzzy metric space is not strong fuzzy metric space.
The following example shows that there are non -strong fuzzy metric spaces.
Example 2.4. [6] Let X = {x, y, z}, ∗ = · (usual product) and M : X×X × (0,∞)→[0,1] defined for eacht >0as M(x, x, t) =M(y, y, t) = M(z, z, t) = 1, M(x, z, t) =M(z, x, t) =M(y, z, t) =M(z, y, t) = t+1t , M(x, y, t) =M(y, x, t) =
t2
(t+2)2.Then, (X, M,∗)is a non-strong fuzzy metric space.
Lemma 2.5. [2] Let(X, M,∗)be a fuzzy metric space. For allx, y∈X,M(x, y,·) is non-decreasing function on(0,∞).
Remark 4. We observe that0< M(x, y, t)<1 provided x6=y, for all t >0(see [8]). Let (X, M,∗) be a fuzzy metric space. For t > 0, the open ball B(x, r, t) with a center x ∈ X and radius 0 < r < 1 is defined by B(x, r, t) = {y ∈ X : M(x, y, t)>1−r}.
A subsetA⊂Xis called open if for eachx∈A, there existt >0and0< r <1 such thatB(x, r, t)⊂A. Letτ denote the family of all open subsets ofX. Then τ is a topology onX, called the topology induced by the fuzzy metric M. This topology is metrizable (see [3]).
Definition 2.6. [1] Let(X, M,∗)be a fuzzy metric space.
1. A sequence{xn} in X is said to be convergent to a pointx∈ X if, for all t >0,limn→∞M(xn, x, t) = 1.
2. A sequence{xn} inX is called a Cauchy sequence if, for each0< <1and t >0, there exitsn0∈Nsuch thatM(xn, xm, t)>1−, for eachn, m≥n0. 3. A fuzzy metric space in which every Cauchy sequence is convergent is said
to be complete.
4. A fuzzy metric space in which every sequence has a convergent subsequence is said to be compact.
Remark 5. In a fuzzy metric space, the limit of a convergent sequence is unique.
Definition 2.7. [9] Let(X, M,∗)be a fuzzy metric space. Then, the mappingM is said to be continuous onX×X×(0,∞)if
n→∞lim M(xn, yn, tn) =M(x, y, t),
when{(xn, yn, tn)}is a sequence inX×X×(0,∞)which converges to a point (x, y, t)∈X×X×(0,∞),i.e.,
n→∞lim M(xn, x, t) = lim
n→∞M(yn, y, t) = 1 and lim
n→∞M(x, y, tn) =M(x, y, t).
Lemma 2.8. [10] If(X, M,∗)is a fuzzy metric space, then M is a continuous function onX×X×(0,∞).
Definition 2.9. [4] A fuzzy contractive mapping on a fuzzy metric space in the sense of George and Veeramani(X, M,∗)is a self-mappingf ofX with the prop- erty
1
M(f x, f y, t)−1≤k( 1
M(x, y, t)−1) for allx, y∈X, for allt >0. (1) First, we define the following :
Definition 2.10. Letψ: (0,1]→[1,∞)be a function which satisfies the following conditions:
1. ψis continuous and non-increasing;
2. ψ(x) = 1if and only if x= 1.
We denote byΨthe class of all functions which satisfies the above conditions.
Note thatΨ6=∅, in fact the mapψ: (0,1]→[1,∞)defined byψ(t) = 1t is a member ofΨ.
Definition 2.11. Letφ: (0,1]→(0,1]be a function which satisfies the following conditions:
1. φis upper semi continuous;
2. φ(s) = 1 if and only ifs= 1.
We denote byΦthe class of all functions which satisfies the above conditions.
Note that Φ6=∅, in fact the map φ: (0,1]→(0,1]defined byφ(t) =√ tis a member ofΦ.
Now, we introduce akg-contractive mapping in a fuzzy metric space.
Definition 2.12. Let(X, M,∗)be a fuzzy metric space. We say that a mapping T :X →X is a kg-contractive mapping if there exists(ψ, φ)∈Ψ×Φsuch that
M(x, y, t)∗M(y, T y, t)≤M(x, T x, t) implies
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)),
(2) for allx, yin X andt >0, where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}.
Example 2.13. Let X = [0,∞) and M(x, y, t) = (t+1t )d(x,y), where d(x, y) =
|x−y|, and ∗ be the product continuous t-norm. Here, (X, M,∗) is a complete fuzzy metric space. LetT :X →X be a map defined by
T(x) = x
4, ifx≤1 0, ifx >1 we can easily see thatT is a kg−contractive map.
In Section 3, we prove the existence of fixed points ofkg-contractive mappings in a complete strong fuzzy metric space.
3. Main Results
The following proposition is important to prove our main result.
Proposition 3.1. Let (X, M,∗) be a strong fuzzy metric space. Let T :X →X be a kg-contractive mapping. Fix x0 ∈ X. Define a sequence {xn} in X by xn+1=T xn forn= 0,1,2, . . .. If limn→∞M(xn, xn+1, t) = 1 for allt >0, and the sequence{M(xn, xn+1, t)} is increasing in [0,1]for all t >0, then {xn} is a Cauchy sequence.
Proof. Since the mappingT is akg-contractive map, there exists(ψ, φ)∈Ψ×Φ such that
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t))), (3) for allx, yin X andt >0.
Suppose that sequence {xn} is not a Cauchy sequence. Then there exist∈ (0,1)andt0>0such that for allk≥1, there are positive integersm(k), n(k)∈N withm(k)> n(k)≥ksuch that
M(xn(k), xm(k), t0)≤1−. (4) We assume that m(k) is the least integer exceedingn(k) and satisfies the above inequality, that is equivalent to
M(xn(k), xm(k)−1, t0)>1−andM(xn(k), xm(k), t0)≤1−. Now we have
1−≥M(xn(k), xm(k), t0) ≥ M(xn(k), xm(k)−1, t0)∗M(xm(k)−1, xm(k), t0)
> (1−)∗M(xm(k)−1, xm(k), t0).
Thenlimk→∞(1−)∗M(xm(k)−1, xm(k), t0) = 1−. Hence, lim
k→∞M(xn(k), xm(k), t0) exists andlimk→∞M(xn(k), xm(k), t0) = 1−.
First we prove that
(i) limk→∞M(xm(k)−1, xn(k)−1, t0) = 1−; (i) limk→∞M(xm(k)−1, xn(k), t0) = 1−; (iii) limk→∞M(xn(k)−1, xm(k), t0) = 1−. Since
M(xm(k), xn(k), t0) ≥M(xm(k), xm(k)−1, t0)
∗M(xm(k)−1, xn(k)−1, t0)
∗M(xn(k)−1, xn(k), t0)
(5) and
M(xm(k)−1, xn(k)−1, t0) ≥M(xm(k)−1, xm(k), t0)
∗M(xm(k), xn(k), t0)
∗M(xn(k), xn(k)−1, t0),
(6) by taking the limit superior in (5) and the limit inferior in (6), we get
1−≥lim sup
k→∞
M(xm(k)−1, xn(k)−1, t0) (7) and
lim inf
k→∞ M(xm(k)−1, xn(k)−1, t0)≥1−. (8) Since the limit superior is always greater than or equal to the limit inferior, from (7) and (8), we obtain
lim sup
k→∞
M(xm(k)−1, xn(k)−1, t0) = 1−, lim inf
k→∞ M(xm(k)−1, xn(k)−1, t0) = 1−.
Thus, limk→∞M(xm(k)−1, xn(k)−1, t0)exists and equal to 1−. Thus (i) holds.
We now prove (ii). By condition(M40)of the strong fuzzy metric space, we have M(xm(k)−1, xn(k), t0)≥M(xm(k)−1, xm(k), t0)∗M(xm(k), xn(k), t0), (9) and
M(xm(k), xn(k), t0)≥M(xm(k), xm(k)−1, t0)∗M(xm(k)−1, xn(k), t0). (10) Taking the limit inferior in (9) and the limit superior in (10) ask→ ∞, we have
lim inf
k→∞ M(xm(k)−1, xn(k), t0)≥1−, and
1−≥lim sup
k→∞
M(xm(k)−1, xn(k), t0).
This implies 1−≥lim sup
k→∞
M(xm(k)−1, xn(k), t0)≥lim inf
k→∞ M(xm(k)−1, xn(k), t0)≥1−.
Thus,
lim sup
k→∞
M(xm(k)−1, xn(k), t0) = lim inf
k→∞ M(xm(k)−1, xn(k), t0) = 1−. Hence, limk→∞M(xm(k)−1, xn(k), t0) exists and limk→∞M(xm(k)−1, xn(k), t0) = 1−, which proves (ii). We now prove (iii). By the condition(M40)in a strong fuzzy metric space, we have
M(xn(k)−1, xm(k), t0)≥M(xn(k)−1, xn(k), t0)∗M(xn(k), xm(k), t0), (11) and
M(xn(k), xm(k), t0)≥M(xn(k), xn(k)−1, t0)∗M(xn(k)−1, xm(k), t0). (12) Taking the limit inferior in (11) and the limit superior in (12) ask→ ∞, we obtain
lim inf
k→∞ M(xn(k)−1, xm(k), t0)≥1−, and
1−≥lim sup
k→∞
M(xn(k)−1, xm(k), t0).
This implies 1−≥lim sup
k→∞
M(xn(k)−1, xm(k), t0)≥lim inf
k→∞ M(xn(k)−1, xm(k), t0)≥1−. Thus,
lim sup
k→∞
M(xn(k)−1, xm(k), t0) = lim inf
k→∞ M(xn(k)−1, xm(k), t0) = 1−. Hence limk→∞M(xn(k)−1, xm(k), t0) exists and limk→∞M(xn(k)−1, xm(k), t0) = 1−,so (iii) holds.
Now, since m(k) + 1 > m(k) ≥ n(k) + 1 > n(k) and {M(xn, xn+1, t)} is increasing for anyt, it follows thatM(xm(k), xm(k)+1, t0)≥M(xn(k), xn(k)+1, t0).
On the other hand, from the property of the fuzzy metric space we have M(xm(k), xn(k), t0)∗M(xn(k), xn(k)+1, t0)≤M(xn(k), xn(k)+1, t0).
Thus,
M(xm(k), xn(k), t0)∗M(xn(k), xn(k)+1, t0)≤M(xm(k), xm(k)+1, t0).
From (3) we have
ψ(M(xm(k)+1, xn(k)+1, t0)) = ψ(M(T xm(k), T xn(k), t0)) (13)
≤ ψ(N(xn(k), xm(k), t0))φ(N(xm(k), xn(k), t0)),
where
N(xm(k), xn(k), t0) = min{M(xm(k), xn(k), t0), M(xm(k), T xm(k), t0), M(xn(k), T xn(k), t0),
max{M(xm(k), T xn(k), t0), M(xn(k), T xm(k), t0)}}.
Equivalently
N(xm(k), xn(k), t0) = min{M(xm(k), xn(k), t0), M(xm(k), xm(k)+1, t0), M(xn(k), xn(k)+1, t0),
max{M(xm(k), xn(k)+1, t0), M(xn(k), xm(k)+1, t0)}}.
Ask→ ∞,
k→∞lim N(xm(k), xn(k), t0) = 1−. (14) Sinceψis continuous andφis upper semi continuous, by taking the limit superior ask→ ∞in (13), it follows that
ψ(1−)≤ψ(1−)φ(1−).
So that, φ(1−) = 1. Hence from the property of φ, we have = 0, which contradicts that0< <1. Therefore,{xn}is a Cauchy sequence in X.
Theorem 3.2. Let (X, M,∗) be a strong fuzzy metric space and T : X →X be kg-contractive mapping. Then,T has a unique fixed point inX.
Proof. Letx0∈Xbe an arbitrary element ofX. We define a sequencexn+1=T xn
for n= 0,1,2, . . .. If there exists n0 ∈ N∪ {0} such that xn0 =xn0+1 =T xn0, thenxn0 is the fixed point ofT.
Assume thatxn6=xn+1, for alln= 0,1,2,3, . . ..
SinceT iskg-contractive mapping, there exists(φ, ψ)∈Ψ×Φsuch that M(x, y, t)∗M(y, T y, t)≤M(x, T x, t)
implies that
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t))
for allx, yin X andt >0. Thus, forxn−16=xn, we have
M(xn−1, xn, t)∗M(xn, xn+1, t)≤M(xn−1, xn, t).
Therefore,
ψ(M(T xn−1, T xn, t))≤ψ(N(xn−1, xn, t))φ(N(xn−1, xn, t)), (15)
where
N(xn−1, xn, t) = min{M(xn−1, xn, t), M(xn−1, xn, t), M(xn, xn+1, t), max{M(xn−1, T xn, t), M(xn, T xn, t)}}
= min{M(xn−1, xn, t), M(xn, xn+1, t), max{M(xn−1, xn+1, t), M(xn, xn+1, t)}}
= min{M(xn−1, xn, t), M(xn+1, xn, t)}.
Ifmin{M(xn−1, xn, t), M(xn+1, xn, t)}=M(xn+1, xn, t)for somenandt >0, then from (15), we have
ψ(M(xn+1, xn, t))≤ψ(M(xn+1, xn, t))φ(M(xn+1, xn, t)). (16) This implies thatφ(M(xn+1, xn, t)) = 1. By the property ofφ, we haveM(xn+1, xn, t) = 1. Thus xn = xn+1. This contradicts xn 6= xn+1 for all n. Therefore min{M(xn−1, xn, t), M(xn+1, xn, t)} =M(xn−1, xn, t) for all n and t >0. Thus from (15) we have
ψ(M(xn+1, xn, t))≤ψ(M(xn, xn−1, t))φ(M(xn, xn−1, t))< ψ(M(xn, xn−1, t)).
Sinceψis non-increasingM(xn+1, xn, t)≥M(xn, xn−1, t)for eachnandt >0.
Therefore, for every t > 0, {M(xn, xn+1, t)} is an increasing sequence of real numbers in (0,1]. Since every bounded and monotone sequence is convergent, {M(xn, xn+1, t)} converges to some number in(0,1].
Letlimn→∞M(xn, xn+1, t) =lt.We Show thatlt= 1,for allt >0.
Let t > 0, by taking the limit superior as k → ∞ in the inequality (16), the continuity of ψand the upper semi continuity φ show thatφ(lt)≥1. Hence, by the property ofφ,φ(lt) = 1implieslt= 1. Now by Proposition 3.1, the sequence {xn}is Cauchy.
SinceX is a complete strong fuzzy metric space, there existsz∈X such that xn →zas n→ ∞. We show thatzis a fixed point ofT.
(i) Suppose there exist a positive integerkandtk >0 such that
M(xn, z, tk)∗M(z, T z, tk)> M(xn, xn+1, tk) (17) for alln≥k. Taking limit asn→ ∞in (17) we getM(z, T z, tk)≥1, which implies thatT z=z.
(ii) Suppose for each positive integerkandt >0, there existsnk≥ksuch that M(xnk, z, t)∗M(z, T z, t)≤M(xnk, xnk+1, t).
Thus, there exists a subsequence {xnk} of{xn}such that M(xnk, z, t)∗M(z, T z, t)≤M(xnk, xnk+1, t).
Hence, from (2) we have
ψ(M(T z, xnk+1, t))≤ψ(N(z, xnk, t))φ(N(z, xnk, t)), (18) where
N(z, xnk, t) = min{M(xnk, z, t), M(z, T z, t), M(xnk, xnk+1, t), max{M(z, xnk+1, t), M(xnk, T z, t)}}.
Sincexnk →zask→ ∞(being a subsequence of{xn}) andlimk→∞M(xnk, xnk+1, t) = 1, it follows thatlimk→∞N(z, xnk, t) =M(z, T z, t). Sinceψ is continuous andφupper semi continuous, by taking the limit superior in (18) we get
ψ(M(T z, z, t))≤ψ(M(T z, z, t))φ(M(T z, z, t)).
So thatφ(M(T z, z, t)) = 1, thusM(T z, z, t) = 1, and henceT z=z. There- forezis a fixed point ofT.
Now, we show the uniqueness of fixed points of T. Letuand v be two fixed points of T. Thus, T u = u and T v = v. Since T is a kg-contractive map for u, v∈X, andt >0 we have
M(u, v, t)∗M(u, T u, t)≤M(v, T v, t), (19) which implies that
ψ(M(u, v, t)) =ψ(M(T u, T v, t))≤ψ(N(u, v, t))φ(N(u, v, t)), where
N(u, v, t) = min{M(u, v, t), M(u, T u, t), M(v, T v, t), max{M(u, T v, t), M(v, T u, t)}}
= min{M(u, v, t),1,1, M(u, v, t)}=M(u, v, t). (20) From (19) and (20), we observe that
ψ(M(u, v, t))≤ψ(M(u, v, t))φ((M(u, v, t)).
This implies φ((M(u, v, t)) = 1. Thus, M(u, v, t) = 1, which implies u = v.
Therefore the fixed point ofT is unique.
Corollary 3.3. Let(X, M,∗)be a strong fuzzy metric space andT be a self map ofX. If there exists(ψ, φ)∈Ψ×Φsuch that
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)) for allx, y∈X andt >0, where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}.
Then, T has a unique fixed point.
If we take ψ(t) = 1t and φ(s) = √
s in Theorem 3.2, we get the following corollary.
Corollary 3.4. Let(X, M,∗)be a strong fuzzy metric space andT be a self map ofX. Forx, y∈X andt >0ifM(x, y, t)∗M(y, T y, t)≤M(x, T x, t)implies that M(T x, T y, t)≥p
N(x, y, t), where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}.
Then, T has a unique fixed point.
Corollary 3.5. Let(X, M,∗)be a strong fuzzy metric space andT be a self map ofX. Forx, y∈X andt >0ifM(x, y, t)∗M(y, T y, t)≤M(x, T x, t)implies that M(T x, T y, t)≥p
M(x, y, t), thenT has a unique fixed point.
Proof. Letx, y∈X t >0. Suppose M(x, y, t)∗M(y, T y, t)≤M(x, T x, t). Then M(T x, T y, t)≥p
M(x, y, t). SinceN(x, y, t)≤M(x, y, t), where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}, we haveM(T x, T y, t)≥p
N(x, y, t). Therefore, by Corollary 3.4,T has a unique fixed point.
By takingψ(t) =1t andφ(t) =k(1−t) +t, for0< k <1we draw the following corollary.
Corollary 3.6. Let(X, M,∗)be a strong fuzzy metric space andT be a self map of X. For x, y ∈ X and t > 0 if there exists k ∈ (0,1) such that M(x, y, t)∗ M(y, T y, t)≤M(x, T x, t)implies (M(T x,T y,t)1 −1)≤k(N(x,y,t)1 −1), where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}},
thenT has a unique fixed point.
Corollary 3.7. Let(X, M,∗)be a strong fuzzy metric space andT be a self map of X. For x, y ∈ X and t > 0 if there exists k ∈ (0,1) such that M(x, y, t)∗ M(y, T y, t)≤M(x, T x, t) implies(M(T x,T y,t)1 −1)≤k(M(x,y,t)1 −1). ThenT has a unique fixed point.
Proof. Letx, y∈X ,t >0, andM(x, y, t)∗M(y, T y, t)≤M(x, T x, t). Then
( 1
M(T x, T y, t)−1)≤k( 1
M(x, y, t)−1).
SinceN(x, y, t)≤M(x, y, t)where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}.
Butk(M(x,y,t)1 −1)≤k(N(x,y,t)1 −1), thus(M(T x,T y,t)1 −1)≤k(N(x,y,t)1 −1).By Corollary 3.6,T has a unique fixed point.
4. Examples
In this section, we provide some examples in support of the main results of Section 3. The following example is in support of Theorem 3.2.
Example 4.1. LetX ={(1,1),(1,4),(4,1)} andM(x, y, t) = (t+1t )d(x,y), where d(x, y) = |x1−y1|+|x2−y2|, and x= (x1, y1), y = (x2, y2), ∗ be the product continuous t-norm. Here(X, M,∗)is a complete strong fuzzy metric space. Let T :X →X be a map defined by
T(x1, y1) =
(1,1), ifx1≤y1
(1,4), ifx1> y1
.
Then,T is akg-contractive map forψ(t) = 1t andφ(t) =√
t. Clearly,(ψ, φ)∈ Ψ×Φ. Now we wish to show
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)). (21) (i) Let(x, y) = ((1,1),(1,1)), T(1,1) = (1,1)and leta=t+1t .
Here M(x, y, t) = 1, M(T x, T y, t) = 1, M(x, T x, t) = 1, M(y, T y, t) = 1, M(x, T y, t) = 1andM(y, T x, t) = 1. It follows thatM(x, y, t)∗M(y, T y, t)≤ M(x, T x, t)andN(x, y, t) = 1. Thenψ(M(T x, T y, t)) = 1,ψ(N(x, y, t)) = 1 andφ(N(x, y, t)) = 1, which implies that
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(ii) Let(x, y) = ((1,4),(1,4)), T(1,4) = (1,1)and leta=t+1t .
Here M(x, y, t) = 1, M(T x, T y, t) = 1, M(x, T x, t) = a3, M(y, T y, t) = a3, M(x, T y, t) = a3 and M(y, T x, t) = a3. It follows that M(x, y, t)∗ M(y, T y, t)≤M(x, T x, t)and N(x, y, t) =a3. Thenψ(M(T x, T y, t)) = 1, ψ(N(x, y, t)) = a−3 and φ(N(x, y, t)) = a32. Since 0 > −3 + 32 = −32 , we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(iii) Let(x, y) = ((4,1),(4,1)), T(4,1) = (1,4)and leta=t+1t .
Here M(x, y, t) = 1, M(T x, T y, t) = 1, M(x, T x, t) = a6, M(y, T y, t) = a6, M(x, T y, t) = a6 and M(y, T x, t) = a6. It follows that M(x, y, t)∗ M(y, T y, t) ≤ M(x, T x, t) and N(x, y, t) = a6. Then ψ(M(T x, T y, t)) = 1, ψ(N(x, y, t)) = a−6 and φ(N(x, y, t)) = a3. Since 0 > −3, we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(iv) Let(x, y) = ((1,1),(1,4)), T(1,1) = (1,1), T(1,4) = (1,1)and leta= t+1t . Here M(x, y, t) = a3, M(T x, T y, t) = 1, M(x, T x, t) = 1, M(y, T y, t) = a3, M(x, T y, t) = 1 and M(y, T x, t) = a3. It follows that M(x, y, t)∗ M(y, T y, t) ≤ M(x, T x, t) and N(x, y, t) = a3. Then ψ(M(T x, T y, t)) = 1, ψ(N(x, y, t)) = a−3 and φ(N(x, y, t)) = a32. Since 0 > −32 , we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(v) Let(x, y) = ((1,1),(4,1)), T(1,1) = (1,1), T(4,1) = (1,4)and leta= t+1t . Here M(x, y, t) = a3, M(T x, T y, t) = a3, M(x, T x, t) = 1, M(y, T y, t) = a6, M(x, T y, t) = a3 and M(y, T x, t) = a3. It follows that M(x, y, t)∗ M(y, T y, t) ≤ M(x, T x, t) and N(x, y, t) = a6. Then ψ(M(T x, T y, t)) = a−3, ψ(N(x, y, t)) = a−6 and φ(N(x, y, t)) = a3. Since −3 ≥ −6 + 3, we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(vi) Let(x, y) = ((1,4),(4,1)), T(1,4) = (1,1), T(4,1) = (1,4)and leta= t+1t . Here M(x, y, t) = a6, M(T x, T y, t) = a3, M(x, T x, t) = a3, M(y, T y, t) = a6, M(x, T y, t) = 1 and M(y, T x, t) = a3. It follows that M(x, y, t)∗ M(y, T y, t) ≤ M(x, T x, t) and N(x, y, t) = a6. Then ψ(M(T x, T y, t)) = a−3, ψ(N(x, y, t)) = a−6 and φ(N(x, y, t)) = a3. Since −3 ≥ −6 + 3, we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(vii) Let(x, y) = ((4,1),(1,4)), T(4,1) = (1,4), T(1,4) = (1,1)and leta= t+1t . Here M(x, y, t) = a6, M(T x, T y, t) = a3, M(x, T x, t) = a6, M(y, T y, t) = a3, M(x, T y, t) = a3 and M(y, T x, t) = 1. It follows that M(x, y, t)∗ M(y, T y, t) ≤ M(x, T x, t) and N(x, y, t) = a6. Then ψ(M(T x, T y, t)) = a−3, ψ(N(x, y, t)) = a−6 and φ(N(x, y, t)) = a3. Since −3 ≥ −6 + 3, we have ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)).
(viii) Let(x, y) = ((4,1),(1,1)), T(4,1) = (1,4), T(1,1) = (1,1)and leta= t+1t . Here M(x, y, t) = a3, M(T x, T y, t) = a3, M(x, T x, t) = a6, M(y, T y, t) = 1, M(x, T y, t) = a3 and M(y, T x, t) = a3. Now in this case M(x, y, t)∗ M(y, T y, t) > M(x, T x, t), but N(x, y, t) = a6, ψ(M(T x, T y, t)) = a−3, ψ(N(x, y, t)) = a−6 and φ(N(x, y, t)) = a3 imply that ψ(M(T x, T y, t)) ≤ ψ(N(x, y, t))φ(N(x, y, t)).Thus the relation (2) is true.
(ix) Let(x, y) = ((1,4),(1,1)), T(1,4) = (1,1), T(1,1) = (1,1)and leta= t+1t . Here M(x, y, t) =a3, M(T x, T y, t) = 1, M(x, T x, t) = a3, M(y, T y, t) = 1, M(x, T y, t) =a3andM(y, T x, t) = 1. It follows thatM(x, y, t)∗M(y, T y, t)
≤M(x, T x, t)andN(x, y, t) =a3. Thenψ(M(T x, T y, t)) = 1,ψ(N(x, y, t))
=a−3andφ(N(x, y, t)) =a32. Since−0≥ −3+32, we haveψ(M(T x, T y, t))≤ ψ(N(x, y, t))φ(N(x, y, t)).
From (Case i-Case ix) we observe that there exists(ψ, φ)∈Ψ×Φsuch that M(x, y, t)∗M(y, T y, t)≤M(x, T x, t)
implies
ψ(M(T x, T y, t))≤ψ(N(x, y, t))φ(N(x, y, t)),
(22)
for allx, yin X andt >0, where
N(x, y, t) = min{M(x, y, t), M(x, T x, t), M(y, T y, t),max{M(x, T y, t), M(y, T x, t)}}.
Therefore, T is a kg-contractive mapping and thus by Theorem 3.2, T has a unique fixed point. In fact,(1,1)is a fixed point ofT.
Acknowledgements. The authors would like to express their sincere thanks to the referees, whose comments and suggestions have been of crucial importance to improve this paper.
Conflicts of Interest. The authors declare that there is no conflicts of interest regarding the publication of this article.
References
[1] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64(3) (1994) 395–399.
[2] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27(3) (1988) 385–389.
[3] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces,Fuzzy Sets and Systems 115(3) (2000) 485–489.
[4] V. Gregori, A. Sapena,On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems Vol. 125, Issue 2, 2002, 245–252.
[5] V. Gregori, A. Sapena, Remarks to “On strong intuitionistic fuzzy metrics", submitted toJ. Nonlinear Sci. Appl..
[6] J. Gutiérrez García, S. Romaguera, Examples of non-strong fuzzy metrics, Fuzzy Sets and Systems 162(1) (2011) 91–93.
[7] I. Kramosil, J. Michálek, Fuzzy metric and statistical metric spaces, Kyber- netika 11(5) (1975) 336–344.
[8] D. Miheţ, A class of contractions in fuzzy metric spaces, Fuzzy Sets and Systems 161(8) (2010) 1131–1137.
[9] S. Phiangsungnoen, Y. J. Cho, P. Kumam, Fixed point results for modified various contractions in fuzzy metric spaces viaα-admissible,Filomat 30(7) (2016) 1869–1881.
[10] J. Rodríguez-López, S. Romaguera, The Hausdorff fuzzy metric on compact sets,Fuzzy Sets and Systems 147(2) (2004) 273–283.
[11] A. Sapena, S. Morillas, On strong fuzzy metrics, In: Proceedings of the Work- shop in Applied Topology WiAT’09 (2009) 135–141.
[12] B. Schweizer, A. Sklar, Statistical metric spaces,Pacific J. Math.10 (1960) 313–334.
[13] S. L. Singh, R. Kamal, M. De la Sen, R. Chugh, A fixed point theorem for generalized weak contractions,Filomat 29(7) (2015) 1481–1490.
Kandala Kanakamahalakshmi Sarma Collage of Science and Technology,
Andhra University, Department of Mathematics, Visakhapatnam-530 003, India
E-mail: [email protected]
Yohannes Gebru Aemro
Collage of Natural and Computational, Department of Mathematics,
P.O. Box 07,
Wolkite University, Wolkite, Ethiopia E-mail: [email protected]
