# Fixed Point and Subfixed Point for Fuzzy Mappings in Generalized Metric Fuzzy Spaces

Authors: Mila Stojaković; Ljiljana Gajić; Biljana Carić
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

A generalized metric in space of set of fuzzy sets is introduced. We prove some common fixed point for contractive iterate at the point and orbitally contractive at the point fuzzy mappings and subfixed point results for family of mappings satisfying generalized contractive conditions in generalized metric fuzzy spaces.


---

## Body

## 1.  Introduction

Uncertainty regarding some experiments may essentially have two origins. It may arise from randomness due to the natural variability of observation or it may be caused by imprecisions due to partial information, for example, expert opinions or sparse data sets. An incomplete data set delivers an imprecise assessment of the information which should be expressed by a[0,1]-fuzzy set instead of a number. In other words, the system is complemented with extra dimension of uncertainty provided by fuzzy set theory. Fuzzy logic is the principal component of an array of methodologies for dealing with problems in which uncertainty and imprecision play important roles.Fixed point theory in uncertain systems can be treated in different ways and one of them is by using the fuzzy logic. Depending on which segment of the problem is plagued with some sort of uncertainty the appropriate structure of the basic space in which the problem is considered could be used. If the distance between elements is imprecise, then the fuzziness is included in metric, as it was done in the definition of fuzzy metric spaces by Kaleva and Seikkala [1]. This model has a lot of similarities with probabilistic metric spaces (Menger spaces). Fixed point techniques, methods, and results are closely related in these two structures. Some of fixed point results in fuzzy metric space could be found in [2–7].If affiliations of elements are imprecise, then the system could be treated as a fuzzy system. Appropriate structure is introduced depending on the related problem. The treatment of the problem involving fixed point theory has to be placed in spaces with adequate topological structure.We denote by𝒦(X) the set of compact subsets of X and by ℱ(X) the set of fuzzy sets with compact α-levels defined over X, where X has some metric structure. Mustafa and Sims [8] introduced the definition of a generalized metric space, briefly, G-metric spaces. In [9] the G-metric is introduced in 𝒦(X) and, in [10], the similar construction is made to establish the G-metric in the set ℱ(X). In both cases the structure of the basic G-metric space [3–5] is used to define the Hausdorff G-metric by the metric dG derived from G-metric G. In our paper the basic space is the metric space (instead of G-metric space) and, using the same idea as in [9, 10], the Hausdorff G-metric 𝒢 is introduced, but metric d of the original metric space (X,d) is used instead of the derived dG metric. In spite of the fact that the relation of the basic and the derived spaces is simpler than in [10] the structure of generalized metric space (ℱ(X),𝒢) is not reduced. Further, we analyse the existence and uniqueness of a common fixed point for the family {fi} of self-mappings in the set of fuzzy sets ℱ(X) endowed with generalized metric 𝒢. The different type of generalized contractive condition is considered using a nondecreasing, right continuous function Φ:[0,∞)→[0,∞). Putting different additional conditions on Φ, we can follow how Φ influences the other conditions, related technique of proving, and the final fixed point result.For more fixed point results for mappings defined inG-metric spaces of fuzzy sets, we refer the reader to [10–12].


## 2.  Preliminaries

Definition 1 (see [8]).
LetX be a nonempty set, and let G:X×X×X→ℝ+ be a function satisfying the following properties:(G1)
G(x,y,z)=0 if x=y=z;(G2)
0<G(x,x,y), for all x,y∈X, with x≠y;(G3)
G(x,x,y)≤G(x,y,z), for all x,y,z∈X, with z≠y;(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯, (symmetry in all three variables);(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z), for all x,y,z,a∈X. Then function G is called a generalized metric, abbreviated G-metric, on X and the pair (X,G) is called a G-metric space. If G(x,y,y)=G(y,x,x) for all x,y∈X, then (X,G) is symmetric.Clearly, these properties are satisfied whenG(x,y,z) is the perimeter of the triangle with vertices at x, y, and z∈ℝ2; moreover taking a in the interior of the triangle shows that (G5) is the best possible.Example 2 (see [8]).
Let(X,d) be an ordinary metric space; then (X,d) defines G-metrics on X by
(1)Gs(x,y,z)=d(x,y)+d(y,z)+d(x,z),Gm(x,y,z)=max{d(x,y),d(y,z),d(x,z)}.The following useful properties of aG-metric are readily derived from the axioms.Proposition 3 (see [8]).
Let(X,G) be a G-metric space; then for any x,y,z, and a from X it follows that(1)
if  G(x,y,z)=0, then  x=y=z,(2)
G(x,y,z)≤G(x,x,y)+G(x,x,z),(3)
G(x,y,y)≤2G(y,x,x),(4)
G(x,y,z)≤G(x,a,z)+G(a,y,z),(5)
G(x,y,z)≤(2/3)(G(x,y,a)+G(x,a,z)+G(a,y,z)),(6)
G(x,y,z)≤G(x,a,a)+G(y,a,a)+G(z,a,a).Definition 4 (see [8]).
Let(X,G) be a G-metric space; and let {xn} be a sequence of points of X. A point x∈X is said to be the  limit of the sequence {xn} if limn,m→∞G(x,xn,xm)=0, and one says that the sequence {xn} is G-convergent to x.Proposition 5 (see [8]).
Let(X,G) be a G-metric space, then for a sequence {xn}⊆X and a point x∈X the following are equivalent:  (1)
{xn}  is  G-convergent to  x,(2)
G(xn,xn,x)→0  as  n→∞,(3)
G(xn,x,x)→0  as  n→∞.Definition 6 (see [8]).
Let(X,G) be a G-metric space; a sequence {xn} is called G-Cauchy if, for every ϵ>0, there is N∈ℕ such that G(xn,xm,xl)<ϵ, for all n,m,l≥N, that is, if G(xn,xm,xl)→0 as n,m,l→∞.Proposition 7 (see [8]).
In aG-metric space (X,G), the following are equivalent:  (1)
the sequence  {xn}  is  G-Cauchy,(2)
for every  ɛ>0, there exists an  n0∈ℕ  such that  G(xn,xm,xm)<ɛ, for all  n,m≥n0.AG-metric space (X,G) is G-complete (or complete G-metric) if every G-Cauchy sequence in (X,G) is G-convergent in (X,G).Proposition 8 (see [8]).
EveryG-metric space (X,G) defines a metric space (X,dG) by
(2)dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.
Note that if (X,G) is a symmetric G-metric space, then
(3)dG(x,y)=2G(x,y,y),∀x,y∈X.However, if(X,G) is nonsymmetric, then by G-metric properties
(4)32G(x,y,y)≤dG(x,y)≤3G(x,y,y),∀x,y∈X,
and in general these inequalities cannot be improved.Proposition 9 (see [8]).
Let(X,G) be a G-metric space; then the function G(x,y,z) is jointly continuous in all three of its variables.In [13] it was shown that if (X,G) is a G-metric space, putting ϱ(x,y)=G(x,y,y), (X,ϱ) is a quasi-metric space (generally, ϱ is not symmetric). It is well known that any quasi-metric induces different metrics and mostly used are(μ)
μ(x,y)=ϱ(x,y)+ϱ(y,x),(ρ)
ρ(x,y)=max{ϱ(x,y),ϱ(y,x)}.The following result is an immediate consequence of the above definitions and relations.Theorem 10.
Let(X,G) be a G-metric space and let D∈{μ,ρ}. Then  (1)
{xn}⊂X is G-convergent to x∈X if and only if {xn} is convergent to x in (X,D);(2)
{xn}⊂X is G-Cauchy if and only if {xn} is Cauchy in (X,D);(3)
(X,G) is G-complete if and only if (X,D) is complete.Recently, Samet et al. [14] and Jleli and Samet [13] observed that some fixed point theorems in context of G-metric space can be proved (by simple transformation) using related existing results in the setting of (quasi) metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before; see [15]. Karapinar and Agarval in [16] continued to develop Jleli-Samet technique in G-metric space, but, on the other side, they proved fixed point theorems on the context of G-metric space for which Jleli-Samet technique is not applicable. So, in some cases, as it is noticed even in Jleli and Samet paper [13], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper where in fixed point results the use of Jleli-Samet technique does not give satisfactory results. If, for instance, the function f in (17) is not independent of the variable μ∈ℱ(X), then n(μ), μ∈ℱ(X), the exponent factor in contraction conditions in our theorems, is not the constant function (as it is the case in our paper), implying that conditions which the contractor Φ in related metric space must satisfy become significantly more restrictive if the Jleli-Samet technique is used. But, using directly G-metric G, the proofs of theorems in our paper are given. The conclusion is that results from our paper cannot be deduced from the usual one in metric or quasi-metric space and cannot be derived from the results of Samet et al. [14] and Jleli and Samet [13].For some fixed point results inG-metric spaces we refer to [17–22].


## 3.  Fuzzy Generalized Metric Space

Let(X,d) be a metric space and let 𝒦(X) be the set of all nonempty compact subsets of X. For all x∈X and all A,B∈𝒦(X), let d(x,A)=infa∈Ad(x,a) and d(A,B)=infa∈A,b∈Bd(a,b). The Hausdorff metric on 𝒦(X) is defined by H(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)}. By ℱ(X) we denote the set of all fuzzy sets μ:X→[0,1] with compact α-levels μα, α∈(0,1], where μα={x∈X:μ(x)≥α} and bounded support μ0=∪α∈(0,1]μα¯. In the set ℱ(X) the metric is introduced by δ(μ,ν)=supα∈(0,1]H(μα,να). If μ,ν∈ℱ(X), then (μ⊆ν)⇔(μ(x)≤ν(x)) for all x∈X and (μ=ν)⇔(μ⊆ν∧ν⊆μ).The function𝒢:ℱ(X)×ℱ(X)×ℱ(X)→ℝ+ is defined by
(5)𝒢(μ,ν,η)=max{ℒ(μ,ν,η),ℒ(η,μ,ν),ℒ(ν,η,μ)},
where
(6)ℒ(μ,ν,η)=supα∈(0,1]{supx∈μα{d(x,να)+d(να,ηα)+d(x,ηα)}}.Lemma 11.
For allμ,ν∈ℱ(X), δ(μ,ν)<𝒢(μ,ν,ν)≤2δ(μ,ν).Proof.
Since(7)ℒ(μ,ν,ν)=supα∈(0,1]{2supx∈μαd(x,να)},ℒ(ν,μ,ν)=ℒ(ν,ν,μ)=supα∈(0,1]{supy∈ναd(y,μα)+d(μα,να)},
from (5) and (6), we obtain
(8)𝒢(μ,ν,ν)=max{supα∈(0,1]{2supx∈μαd(x,να)},supα∈(0,1]{supy∈ναd(y,μα)+d(μα,να)}}=supα∈(0,1]{max{2supx∈μαd(x,να),supy∈ναd(y,μα)supy∈να+d(μα,να)}}.
The next two inequalities,
(9)𝒢(μ,ν,ν)=supα∈(0,1]{max{2supx∈μαd(x,να),supy∈ναd(y,μα)supy∈να+d(μα,να)}}≤2supα∈(0,1]{max{supx∈μαd(x,να),supy∈ναd(y,μα)}}=2supα∈(0,1]{H(μα,να)}=2δ(μ,ν),𝒢(μ,ν,ν)=supα∈(0,1]{max{2supx∈μαd(x,να),supy∈ναd(y,μα)supx∈μα+d(μα,να)}}>supα∈(0,1]{max{supx∈μαd(x,να),supy∈ναd(y,μα)}}=supα∈(0,1]{H(μα,να)}=δ(μ,ν),
imply the relation δ(μ,ν)<𝒢(μ,ν,ν)≤2δ(μ,ν), what we had to prove.Proposition 12.
If(X,d) is a complete metric space, then (ℱ(X),𝒢) is a complete G-metric fuzzy space.Proof.
Properties (G1), (G2), and (G4) from Definition 1 are obvious, so the proof is omitted.
(G3) Since d(x,A)≤d(x,B)+d(A,B) for all x∈X and all A,B∈𝒦(X), we have
(10)ℒ(μ,ν,ν)=supα∈(0,1]{supx∈μα{d(x,να)+d(x,να)+d(να,να)}}≤supα∈(0,1]{supx∈μα{d(x,να)+d(x,ηα)+d(ηα,να)}}=ℒ(μ,ν,η).
By the same way we show that
(11)ℒ(ν,μ,ν)≤ℒ(η,μ,ν),ℒ(ν,ν,μ)≤ℒ(ν,η,μ),
which implies that 𝒢(μ,ν,ν)≤𝒢(μ,ν,η).
(G5) To prove that 𝒢(μ,ν,η)≤𝒢(μ,θ,θ)+𝒢(θ,ν,η), we consider the related inequality for ℒ:
(12)ℒ(μ,ν,η)=supα∈(0,1]{supx∈μα{d(x,να)+d(x,ηα)+d(ηα,να)}}≤supα∈(0,1]{supx∈μα{d(x,θα)+d(θα,να)+d(x,θα)supx∈μα+d(θα,ηα)+d(ηα,να)}}≤supα∈(0,1]{supx∈μα{d(x,θα)+supt∈θαd(t,να)+d(x,θα)+supt∈θαd(t,ηα)+d(ηα,να)}}=supα∈(0,1]{supx∈μα{d(x,θα)+d(x,θα)+d(θα,θα)}supηαt∈θα+supt∈θα{d(t,να)+d(t,ηα)+d(ηα,να)}}=ℒ(μ,θ,θ)+ℒ(θ,ν,η).
Analogously, ℒ(η,μ,ν)≤ℒ(θ,μ,θ)+ℒ(η,θ,ν) and ℒ(ν,η,μ)≤ℒ(θ,θ,μ)+ℒ(ν,η,θ). All three inequalities together imply that 𝒢(μ,ν,η)≤𝒢(μ,θ,θ)+𝒢(θ,ν,η).
The completeness of(ℱ(X),𝒢) is a consequence of the completeness of (ℱ(X),δ) and inequality from the previous lemma.Proposition 13.
G-metric fuzzy space (ℱ(X),𝒢) is not symmetric.Proof.
We prove that𝒢(μ,ν,ν)≠𝒢(ν,μ,μ). If
(13)supα∈(0,1]supx∈μαd(x,να)=A,supα∈(0,1]supy∈ναd(y,μα)=b,supα∈(0,1]d(να,μα)=C,
then(14)𝒢(μ,ν,ν)=max{2A,B+C}≠𝒢(ν,μ,μ)=max{2B,A+C}.


## 4. Fixed Point for Contractive Iterate at the Point and Orbitally Contractive at the Point Fuzzy Mappings

A generalization of the contraction principle can be obtained using different type of a nondecreasing right continuous functionΦ:[0,∞)→[0,∞). The most usual additional properties imposed on Φ are given using a combination of the next seven conditions:(ϕ1)
Φ(0)=0,(ϕ2)
Φ(x)<x, for all x>0,(ϕ3)
limi→∞Φi(x)=0, for all x>0,(ϕ4)
{xi}⊂[0,∞) is a sequence such that xi+1≤Φ(xi); then limi→∞xi=0.(ϕ5)
for anyx≥0 there exists a y(x)≥0,   y(x)=supy≥0{y≤x+Φ(y)},(ϕ6)
limx→∞(x-Φ(x))=∞,(ϕ7)
∑i=1∞‍Φi(x)<∞, for all x>0.Some of the noted properties ofΦ are equivalent, some of them imply other, and some of them are incompatible. The next lemma discusses some of the relations between properties (ϕ1)–(ϕ7), especially those which are used in this paper to define a generalized contraction.Lemma 14.
LetΦ:[0,∞)→[0,∞) be a nondecreasing right continuous function. Then (i)
(ϕ2)⇔(ϕ3)⇔(ϕ4),(ii)
(ϕ7)⇒(ϕk)⇒(ϕ1), where k∈{2,3,4},(iii)
(ϕ5)+(ϕ2)⇏(ϕ7) and (ϕ6)+(ϕ2)⇏(ϕ7),(iv)
(ϕ7)⇏(ϕ5) and (ϕ7)⇏(ϕ6).Proof.
(i) It is enough to prove that(ϕ4)⇒(ϕ2)⇔(ϕ3)⇒(ϕ4).
(
ϕ
2
)
⇒
(
ϕ
3
). We assume that for some x>0, limi→∞Φi(x)=a>0. Since {Φi(x)} is nonincreasing sequence, by the right continuity of Φ, Φ(a)=Φlimi→∞Φi(x)=limi→∞Φi+1(x)=a>0; that is, 0<a=Φ(a) which contradicts (ϕ2).
(
ϕ
3
)
⇒
(
ϕ
2
). If for some x>0, Φ(x)≥x, then, knowing that Φ is nondecreasing, Φi(x)≥Φi-1(x)≥⋯≥Φ(x)≥x>0. It means that limi→∞Φi(x)≠0, which contradicts (ϕ3).
(
ϕ
3
)
⇒
(
ϕ
4
). Let {xi}⊂[0,∞) be any sequence such that xi+1≤Φ(xi). Then xi≤Φ(xi-1)≤Φ2(xi-2)≤⋯≤Φi(x0) and limi→∞xi≤limi→∞Φi(x0)=0.
(
ϕ
4
)
⇒
(
ϕ
2
). If (ϕ2) does not hold, that is, there exists a y>0, y≤Φ(y), then putting xi=y, for all i=0,1,…, we have the sequence with y=xi+1≤Φ(y)=Φ(xi), but that sequence does not converge to 0.
(ii) is obvious, so the proof is omitted.
(iii) Function(15)Φ(x)={(n+2)-1,(n+1)-1≤x<n-1,n∈ℕ,3-1,1≤x
satisfies (ϕ5), (ϕ6) and (ϕ2) but not (ϕ7).
(iv) Function(16)Φ(x)={(x2)2,0≤x<2,x-1,2≤x
satisfies (ϕ7) but not (ϕ5) nor (ϕ6).Theorem 15.
Let(X,d) be a complete metric space, let (ℱ(X),𝒢) be related G-metric fuzzy space, and let ℬ⊆ℱ(X). Further, let {fi} be the sequence of self-mappings of ℱ(X) such that, for all i∈ℕ, fi(ℬ)⊆ℬ and for each μ∈ℱ(X) there exists an n(μ)∈ℕ such that
(17)𝒢(fin(μ)(ν),fjn(μ)(μ),fjn(μ)(μ))≤Φ(max{𝒢(ν,μ,μ),fin(μ)2-1[𝒢(ν,fjn(μ)(μ),fjn(μ)(μ))+𝒢(μ,fjn(μ)(μ),fjn(μ)(μ))],2-1[𝒢(ν,fjn(μ)(μ),fjn(μ)(μ))+𝒢(fin(μ)(ν),μ,μ)]}),
for all i,j∈ℕ and all ν∈ℬ, where Φ satisfies (ϕ2). If there exists μ*∈ℬ such that fin(μ*)(μ*)=μ* for all i∈ℕ, then μ* is a unique common fixed point for {fi} in ℬ and, for every μ1∈ℱ(X), the sequence μj+1=fin(μ*)(μj), i∈ℕ, converges to μ*.Proof.
First we prove thatμ* is a unique point in ℬ with the property that fin(μ*)(μ*)=μ*, i∈ℕ. If ν∈ℬ, ν≠μ*, fin(μ*)(ν)=ν, i∈ℕ, then
(18)𝒢(ν,μ*,μ*)≤𝒢(fin(μ*)(ν),fjn(μ*)(μ*),fjn(μ*)(μ*))≤Φ(𝒢(ν,μ*,μ*)).
By the property Φ(t)<t, t>0, since 𝒢(ν,μ*,μ*)>0, we have the contradiction; that is, the assumption μ*≠ν is not correct.
Further, since(19)fi(μ*)=fi(fin(μ*)(μ*))=fin(μ*)+1(μ*)=fin(μ*)(fi(μ*)),
it follows that fi(μ*)=μ* for all i∈ℕ.
Now, for someμ1∈ℬ, we form the sequence μi+1=fin(μ*)(μi).
Ifμ1=μ*, then μi=fi-1n(μ*)(fi-2n(μ*)⋯(f1n(μ*)(μ*))⋯)=μ* and the sequence {μi} converges to μ*.
Ifμi≠μ*, in order to prove that the sequence {μi} converges to μ*, we consider the sequence 𝒢(μi+1,μ*,μ*), i∈ℕ:
(20)𝒢(μi+1,μ*,μ*)=𝒢(fin(μ*)(μi),fjn(μ*)(μ*),fjn(μ*)(μ*))≤Φ(max{𝒢(μi,μ*,μ*),2-1[𝒢(μi,μ*,μ*)+𝒢(μ*,μ*,μ*)],2-1[𝒢(μi,μ*,μ*)+𝒢(μi+1,μ*,μ*)]})=Φ(max{𝒢(μi,μ*,μ*),2-1[𝒢(μi,μ*,μ*)+𝒢(μi+1,μ*,μ*)]}).
If we choose the option that
(21)max{𝒢(μi,μ*,μ*),2-1[𝒢(μi,μ*,μ*)+𝒢(μi+1,μ*,μ*)]}=2-1[𝒢(μi,μ*,μ*)+𝒢(μi+1,μ*,μ*)],
it implies that
(22)𝒢(μi,μ*,μ*)≤𝒢(μi+1,μ*,μ*).
On the other hand, in that case
(23)𝒢(μi+1,μ*,μ*)≤Φ(2-1[𝒢(μi,μ*,μ*)+𝒢(μi+1,μ*,μ*)])<2-1𝒢(μi,μ*,μ*)+2-1𝒢(μi+1,μ*,μ*);
that is
(24)𝒢(μi+1,μ*,μ*)<𝒢(μi,μ*,μ*).
It is obvious that (22) contradicts (24). So,
(25)𝒢(μi+1,μ*,μ*)≤Φ(𝒢(μi,μ*,μ*)).
Now, applying that procedure i times and letting i→∞, we get
(26)𝒢(μi+1,μ*,μ*)≤Φ(𝒢(μi,μ*,μ*))≤⋯≤Φi(𝒢(μ1,μ*,μ*)).
Since μ1≠μ*, 𝒢(μi,μ*,μ*)>0 and limi→∞𝒢(μi+1,μ*,μ*)=0. The last relation proves that the sequence {μi} converges to μ*.Theorem 16.
Let(X,d) be a complete metric space, f:ℱ(X)→ℱ(X), where (ℱ(X),𝒢) is related G-metric fuzzy space, and let Φ:[0,∞)→[0,∞) be a subadditive mapping satisfying (ϕ7). If for some μ0∈ℱ(X) the orbit 𝒪(f;μ0) is complete and for each μ∈𝒪(f;μ0) there exists an n(μ)∈ℕ such that
(27)𝒢(fn(μ)(ν),fn(μ)(μ),fn(μ)(μ))≤Φ(𝒢(ν,μ,μ)),
for all ν∈𝒪(f;μ0), then the sequence μi+1=fn(μi)(μi), i∈ℕ0, converges to some μ*∈ℱ(X).
If inequality (27) holds for all μ∈𝒪(f;μ0)¯, then fn(μ*)(μ*)=μ* and fi(μ)→μ* for all μ∈𝒪(f;μ0)¯. If f(𝒪(f;μ0))¯⊆𝒪(f;μ0)¯, then μ* is the fixed point of f.Proof.
First, we show that{μi}i∈ℕ0⊂ℱ(X) is a Cauchy sequence. For sufficiently large m∈ℕ, there exist k,r∈ℕ, 1≤r<n(μ0) such that m=k·n(μ0)+r. Using (27), we get
(28)𝒢(fm(μ0),μ0,μ0)≤𝒢(fkn(μ0)+r(μ0),fn(μ0)(μ0),fn(μ0)(μ0))+𝒢(fn(μ0)μ0,μ0,μ0)≤Φ(𝒢(f(k-1)n(μ0)+r(μ0),μ0,μ0))+𝒢(fn(μ0)(μ0),μ0,μ0)≤Φ(𝒢(f(k-1)n(μ0)+r(μ0),fn(μ0)(μ0),fn(μ0))+𝒢(fn(μ0)(μ0),μ0,μ0))+𝒢(fn(μ0)(μ0),μ0,μ0)≤Φ2(𝒢(f(k-2)n(μ0)+r(μ0),μ0,μ0))+Φ(𝒢(fn(μ0)(μ0),μ0,μ0))+𝒢(fn(μ0)(μ0),μ0,μ0)≤⋯≤Φk(𝒢(fr(μ0),μ0,μ0))+∑i=1k-1‍Φi(𝒢(fn(μ0),μ0,μ0)).
Putting A=max{𝒢(fp(μ0),μ0,μ0):1≤p≤n(μ0)}, for all m∈ℕ, the next inequality holds:
(29)𝒢(fm(μ0),μ0,μ0)≤∑s=1k‍Φs(A)≤∑s=1∞‍Φs(A)=B<∞,
and, consequently,
(30)𝒢(μm,μm,μm+1)=𝒢(fn(μm-1)(μm-1),fn(μm-1)(μm-1),fn(μm)fn(μm-1)(μm-1))≤Φ(𝒢(μm-1,μm-1,fn(μm)(μm-1)))≤⋯≤Φm(𝒢(μ0,μ0,fn(μm)(μ0)))≤Φm(B),
for all m∈ℕ. Using the last inequality, for every i,j∈ℕ, i<j, we have
(31)𝒢(μi,μi,μj)≤𝒢(μi,μi,μi+1)+⋯+𝒢(μk-1,μk-1,μk)≤∑s=ij‍Φs(B),
implying that {μi}i∈ℕ is a Cauchy sequence. Since (ℱ(X),𝒢) is a complete G-metric fuzzy space, there exists an μ*∈ℱ(X) such that limi→∞μi=μ*.
In the second part of the theorem, inequality (27) holds for all μ∈𝒪(f;μ0)¯. Then, the elements μi of the sequence {μi}i∈ℕ from the previous part of the proof satisfy the next two relations:
(32)𝒢(fn(μ*)(μ*),fn(μ*)(μ*),fn(μ*)(μi))≤Φ(𝒢(μ*,μ*,μi)),(33)𝒢(fn(μ*)(μi),μi,μi)=𝒢(fn(μ*)fn(μi-1)(μi-1),fn(μi-1)(μi-1),fn(μ*)fn(μi-1)(μi-1))≤Φ(𝒢(fn(μ*)(μi-1),μi-1,μi-1))≤Φi(𝒢(fn(μ*)(μ0),μ0,μ0)).
By (32)
(34)limi→∞fn(μ*)(μi)=fn(μ*)(μ*)
and by (33)
(35)limi→∞𝒢(fn(μ*)(μi),μi,μi)=𝒢(fn(μ*)(μ*),μ*,μ*)=0.
Hence, fn(μ*)(μ*)=μ*.
To show thatμ* is a unique fixed point of fn(μ*) in 𝒪(f;μ0)¯, we assume that there exists another point x**∈𝒪(f;μ0)¯ with the same property. Then
(36)𝒢(x**,μ*,μ*)=𝒢(fn(μ*)x**,fn(μ*)μ*,fn(μ*)μ*)≤Φ(𝒢(x**,μ*,μ*)).
That is x**=μ*. Further, if f(𝒪(f;μ0))¯⊆𝒪(f;μ0)¯, then fμ*=f(fn(μ*)μ*)=fn(μ*)(fμ*), implying that fμ*=μ*.


## 5. Subfixed Point for Generalized Contraction Family of Fuzzy Mappings

We say thatμ*∈ℱ(X) is a subfixed point of the mapping f:ℱ(X)→ℱ(X) if and only if μ*⊆f(μ*).The proof of the next two propositions is the same as in [10]; only, instead of the derived metric dG, the metric d from original metric space (X,d) is used.Proposition 17.
Ifμ1,μ2,ν2∈ℱ(X) and μ1≤μ2, then there exists a ν1∈ℱ(X) such that 𝒢(μ1,ν1,ν1)≤𝒢(μ2,ν2,ν2).Proposition 18.
Ifμ,ν,η∈ℱ(X) and μ⊆ν, then  (i)
ℒ(μ,η,η)≤ℒ(ν,η,η),(ii)
ℒ(η,ν,ν)≤ℒ(η,μ,μ),(iii)
ℒ(μ,ν,ν)=0⇔μ⊆ν.In the next two theorems we consider the existence of a fuzzy setμ* which represents a common subfixed point for the family of self-mappings {fi}, that is, the point such that μ*⊆fi(μ*) for all i∈ℕ.Let(X,d) be a complete metric space, and let (ℱ(X),𝒢) be related G-metric fuzzy space. Further, let for all μ,ν∈ℱ(X) and all i,j∈ℕ, i≠j,
(37)𝒢(fi(ν),fj(μ),fj(μ))≤Φ(max{(𝒢(ν,μ,μ),ℒ(ν,fi(ν),fi(ν)),fjℒ(μ,fj(μ),fj(μ))),ℒ(ν,fj(μ),fj(μ)),ℒ(μ,fi(ν),fi(ν))}),
where Φ:[0,∞)→[0,∞).Ifμ0 is any element from ℱ(X) and a μ1∈ℱ(X) is chosen such that μ1⊆f1(μ0), then, by Proposition 17, there exists a μ2⊆f2(μ1), such that
(38)𝒢(μ1,μ2,μ2)≤𝒢(f1(μ0),f2(μ1),f2(μ1)).
By the same principle as for the first three members, the sequence {μi} is formed such that
(39)μi+1⊆fi+1(μi),∀i∈ℕ,(40)𝒢(μi,μi+1,μi+1)≤𝒢(fi(μi-1),fi+1(μi),fi+1(μi)).Lemma 19.
IfΦ in (37) satisfies (ϕ5) together with (ϕ2) or (ϕ3) or (ϕ4), the sequence {μi} defined in (39) is a Cauchy sequence.Proof.
By (37), for any i,j∈ℕ, we have
(41)𝒢(μi,μj,μj)≤𝒢(fi(μi-1),fj(μj-1),fj(μj-1))≤Φ(max{𝒢(μi-1,μj-1,μj-1),ℒ(μi-1,fi(μi-1),fi(μi-1)),ℒ(μj-1,fi(μi-1),fi(μi-1)),ℒ(μj-1,fj(μj-1),fj(μj-1)),ℒ(μi-1,fj(μj-1),fj(μj-1))})≤Φ(max{𝒢(μi-1,μj-1,μj-1),ℒ(μi-1,fi(μi-1),fi(μi-1)),ℒ(μj-1,fi(μi-1),fi(μi-1)),ℒ(μj-1,fj(μj-1),fj(μj-1)),ℒ(μi-1,fj(μj-1),fj(μj-1))})≤Φ(max{𝒢(μi-1,μj-1,μj-1),ℒ(μi-1,μi,μi),ℒ(μj-1,μi,μi),ℒ(μj-1,μj,μj),ℒ(μi-1,μj,μj)})≤Φ(max{𝒢(μi-1,μj-1,μj-1),𝒢(μi-1,μi,μi),𝒢(μj-1,μi,μi),𝒢(μj-1,μj,μj),𝒢(μi-1,μj,μj)}).
Considering relation (41) for different values k≤i, j≤n, we get
(42)supk≤i,j≤n{𝒢(μi,μj,μj)}≤Φ(supk-1≤i,j≤n{𝒢(μi,μj,μj)}),
for all k,n∈ℕ, 2≤k<n. Knowing that
(43)sup1≤i,j≤n{𝒢(μi,μj,μj)}≤max{𝒢(μ1,μ2,μ2),𝒢(μ2,μ1,μ1)}+sup2≤i,j≤n{𝒢(μi,μj,μj)}
and using (42) with k=2, we obtain
(44)sup1≤i,j≤n{𝒢(μi,μj,μj)}≤max{𝒢(μ1,μ2,μ2),𝒢(μ2,μ1,μ1)}+Φ(sup1≤i,j≤n{𝒢(μi,μj,μj)}).
Putting x=max{𝒢(μ1,μ2,μ2),𝒢(μ2,μ1,μ1)} and applying the property (ϕ5) in (44), there exists a y(x)>0 such that
(45)sup1≤i,j≤n{𝒢(μi,μj,μj)}≤y(x).
Taking k=2 in (42) again, using the last relation, we get
(46)sup2≤i,j≤n{𝒢(μi,μj,μj)}≤Φ(y(x)).
Continuing this process, we obtain
(47)supk≤i,j≤n{𝒢(μi,μj,μj)}≤Φk-1(y(x)),
where k,n∈ℕ, 2≤k<n. Now, letting k→∞ and using the properties of Φ, finally we show that {μi} is a Cauchy sequence; that is
(48)limk→∞supk≤i,j≤n{𝒢(μi,μj,μj)}≤limk→∞Φk-1(y(x))=0.
Using Lemma 14(ii), property (ϕ3) can be replaced by (ϕ2) or (ϕ4).Lemma 20.
IfΦ in (37) satisfies (ϕ6) together with (ϕ2) or (ϕ3) or (ϕ4), the sequence {μi} defined in (39) is a Cauchy sequence.Proof.
Using properties ofΦ satisfying (ϕ6) together with (ϕ4), we prove by contradiction that
(49)limn→∞sup1≤i,j≤n{𝒢(μi,μj,μj)}=sup1≤i,j{𝒢(μi,μj,μj)}<∞.
The assumption that (49) is not the case leads to
(50)limn→∞sup1≤i,j≤n{𝒢(μi,μj,μj)}=∞.
On the other hand, by (ϕ6) and (44) we obtain
(51)∞=limn→∞(sup1≤i,j≤n{𝒢(μi,μj,μj)}-ϕ(sup1≤i,j≤n{𝒢(μi,μj,μj)}))≤max{𝒢(μ1,μ2,μ2),𝒢(μ2,μ1,μ1)},
which is a contradiction implying that (49) is true. Now we define a decreasing sequence {xk}⊂(0,∞) by
(52)xk=supk≤i,j{𝒢(μi,μj,μj)}
and, using (41),
(53)xk≤ϕ(xk-1)≤⋯≤ϕk-1(x1).
Letting k→∞, finally we prove that
(54)limk→∞supk≤i,j{𝒢(μi,μj,μj)}=0.
So, {μi} is a Cauchy sequence. Using Lemma 14(ii), property (ϕ4) can be replaced by (ϕ2) or (ϕ3).Theorem 21.
If all assumptions from Lemma19 or from Lemma 20 are satisfied, then there exists a ν*∈ℱ(X) such that ν*⊆fi(ν*) for all i∈ℕ.Proof.
In Lemma19 or Lemma 20 it was proved that {μi} is a Cauchy sequence and, by the completeness of (ℱ(X),𝒢), limi→∞μi=μ*∈ℱ(X). To prove that μ*⊆fj(μ*), we proceed as follows:
(55)ℒ(μ*,fj(μ*),fj(μ*))≤limi→∞ℒ(μ*,μi,μi)+limi→∞ℒ(μi,fj(μ*),fj(μ*))=limi→∞𝒢(fi(μi-1),fj(μ*),fj(μ*))≤limi→∞Φ(max{𝒢(μi-1,μ*,μ*),fi(μi-1)ℒ(μi-1,fi(μi-1),fi(μi-1)),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μi-1,fj(μ*),fj(μ*)),ℒ(μ*,fi(μi-1),fi(μi-1))})≤limi→∞Φ(max{𝒢(μi-1,μ*,μ*),ℒ(μi-1,μi,μi),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μi-1,fj(μ*),fj(μ*)),ℒ(μ*,μi,μi)})≤Φ(max{𝒢(μ*,μ*,μ*),ℒ(μ*,μ*,μ*),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μ*,μ*,μ*)})=Φ(max{0,0,ℒ(μ*,fj(μ*),fj(μ*)),0+ℒ(μ*,fj(μ*),fj(μ*)),0})≤ℒ(μ*,fj(μ*),fj(μ*)).
Hence,ℒ(μ*,fj(μ*),fj(μ*))=0⇒μ*⊆fj(μ*), for all j∈ℕ, what we had to prove.Letφ:[0,∞)5→[0,∞) be nondecreasing continuous from the right function with respect to each of the five variables such that
(56)∑k∈ℕ‍φk(t,t,t,2t,0)<∞,
for all t>0. Obviously, for all t>0,
(57)limkφk(t,t,t,2t,0)=0,φ(t,t,t,2t,0)<t,φ(0,0,0,0,0)=0.
Further, let (X,d) be a complete metric space, let (ℱ(X),𝒢) be related G-metric fuzzy space, and the family of self-mapings {fi} satisfy the next inequality:
(58)𝒢(fi(ν),fj(μ),fj(μ))≤φ(𝒢(ν,μ,μ),ℒ(ν,fi(ν),fi(ν)),(ν,fj(μ),fj(μ))ℒ(μ,fj(μ),fj(μ)),ℒ(ν,fj(μ),fj(μ)),ℒ(μ,fi(ν),fi(ν))),
for all μ,ν∈ℱ(X) and all i,j∈ℕ.Using the same arguments as in (39), the sequence {μi} is formed:
(59)μi+1⊆fi+1(μi),∀i∈ℕ,
with property
(60)𝒢(μi,μi+1,μi+1)≤𝒢(fi(μi-1),fi+1(μi),fi+1(μi)).Lemma 22.
The sequence{μi} defined in (59) is a Cauchy sequence.Proof.
In order to prove that{μi} is a Cauchy sequence, we consider the sequence {𝒢(μi,μi+1,μi+1)}. Using relations (58) and (60) and the implication μi⊆fi(μi-1)⇒ℒ(μi,fi(μi-1),fi(μi-1))=0, we get
(61)𝒢(μi,μi+1,μi+1)≤𝒢(fi(μi-1),fi+1(μi),fi+1(μi))≤φ(𝒢(μi-1,μi,μi),ℒ(μi-1,fi(μi-1),fi(μi-1)),ℒ(μi,fi+1(μi),fi+1(μi)),ℒ(μi-1,fi+1(μi),fi+1(μi)),ℒ(μi,fi(μi-1),fi(μi-1)))≤φ(𝒢(μi-1,μi,μi),ℒ(μi-1,μi,μi),ℒ(μi,μi+1,μi+1),ℒ(μi-1,μi+1,μi+1),0)≤φ(𝒢(μi-1,μi,μi),𝒢(μi-1,μi,μi),𝒢(μi,μi+1,μi+1),𝒢(μi-1,μi+1,μi+1),0).
Since 𝒢(μi-1,μi+1,μi+1)≤𝒢(μi-1,μi,μi)+𝒢(μi,μi+1,μi+1), the next relation holds:
(62)𝒢(μi,μi+1,μi+1)≤φ(𝒢(μi-1,μi,μi),𝒢(μi-1,μi,μi),𝒢(μi,μi+1,μi+1),𝒢(μi-1,μi,μi)+𝒢(μi,μi+1,μi+1),0).
The assumption 𝒢(μi,μi+1,μi+1)≰𝒢(μi-1,μi,μi), which is equivalent to 𝒢(μi,μi+1,μi+1)>𝒢(μi-1,μi,μi), leads to inequality
(63)𝒢(μi,μi+1,μi+1)≤φ(𝒢(μi,μi+1,μi+1),𝒢(μi,μi+1,μi+1),𝒢(μi,μi+1,μi+1),2𝒢(μi,μi+1,μi+1),0)<𝒢(μi,μi+1,μi+1),
which is a contradiction. In the last transformation we used nondecreasingness of ϕ and the property φ(t,t,t,2t,0)<t, t∈ℝ+. Hence, 𝒢(μi,μi+1,μi+1)≤𝒢(μi-1,μi,μi). With notation 𝒢(μi-1,μi,μi)=ti, we have
(64)ti+1≤φ(ti,ti,ti,2ti,0)<ti≤⋯≤φi(t1,t1,t1,2t1,0).
Ifμ0=μ1, then t1=𝒢(μ0,μ1,μ1)=0. Since φ(0,0,0,0,0)=0, from inequality (64) we get (μ0=μ1)⇒(μ1=μ2)⇒⋯⇒(μi-1=μi)⇒⋯. Further, μ0=μi⊆fi(μi-1)=fi(μ0), which means that μ*=μ0 and the proof is completed.
Ifμ0≠μ1, we prove that limi→∞𝒢(μi,μj,μj)=0 whenever i<j:
(65)𝒢(μi,μj,μj)≤𝒢(μi,μi+1,μi+1)+𝒢(μi+1,μi+2,μi+2)+⋯+𝒢(μj-1,μj,μj)=∑k=ij-1‍𝒢(μk,μk+1,μk+1)=∑k=ij-1‍tk+1≤∑k=ij-1‍φk(t1,t1,t1,2t1,0),
and, since t1=𝒢(μ0,μ1,μ1)>0, limi→∞𝒢(μi,μj,μj)=0. Also, 𝒢(μj,μi,μi)≤2𝒢(μi,μj,μj)→0. So, we have proved that {μi} is a Cauchy sequence and, consequently, there exists a μ*∈ℱ(X) such that limi→∞μi=μ*; that is, limi→∞𝒢(μi,μ*,μ*)=limi→∞𝒢(μ*,μi,μi)=0.Theorem 23.
If all assumptions from the previous lemma are satisfied, then there exists aν*∈ℱ(X) such that ν*⊆fi(ν*) for all i∈ℕ.Proof.
Ifℒ(μ*,fj(μ*),fj(μ*))=0 for all j∈ℕ, then μ*⊆fj(μ*). So, assume that, for some j∈ℕ, ℒ(μ*,fj(μ*),fj(μ*))>0. Then
(66)ℒ(μ*,fj(μ*),fj(μ*))≤limi→∞ℒ(μ*,μi,μi)+limi→∞ℒ(μi,fj(μ*),fj(μ*))≤limi→∞ℒ(fi(μi-1),fj(μ*),fj(μ*))≤limi→∞𝒢(fi(μi-1),fj(μ*),fj(μ*))≤limi→∞φ(𝒢(μi-1,μ*,μ*),ℒ(μi-1,fi(μi-1),fi(μi-1)),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μi-1,fj(μ*),fj(μ*)),ℒ(μ*,fi(μi-1),fi(μi-1)))≤limi→∞φ(𝒢(μi-1,μ*,μ*),ℒ(μi-1,μi,μi),ℒ(μ*,fj(μ*),fj(μ*)),ℒ(μi-1,fj(μ*),fj(μ*)),ℒ(μ*,μi,μi)).
If we put a=ℒ(μ*,fj(μ*),fj(μ*))>0, since limi→∞μi=μ*, for every ɛ∈(0,a) there exists an i0=max{i1,i2,i3,i4}, where  (i)
𝒢(μi-1,μ*,μ*)<ɛ<a for all i>i1,(ii)
ℒ(μi-1,μi,μi)<ɛ<a for all i>i2,(iii)
ℒ(μi-1,fj(μ*),fj(μ*))≤ℒ(μi-1,μi,μi)+ℒ(μi,fj(μ*),fj(μ*))<(ε/2)+a+(ε/2)<2a, for all i>i3,(iv)
ℒ(μ*,μi,μi)<ɛ for all i>i4. Now, relation (66) becomes
(67)ℒ(μ*,fj(μ*),fj(μ*))≤φ(ɛ,ɛ,a,ɛ+a,ɛ)≤φ(a,a,a,2a,ɛ)
and, letting ɛ→0, we get a≤φ(a,a,a,2a,0)<a, where a=ℒ(μ*,fj(μ*),fj(μ*)). Hence, ℒ(μ*,fj(μ*),fj(μ*))=0⇒μ*⊆fj(μ*), for all j∈ℕ, what we had to prove.



---

*Source: 254259-2013-12-19.xml*