# On Fixed Point Findings for Diverse Contractions inb Dislocated-Multiplicative Metric Spaces

Authors: A. Kamal; Asmaa M. Abd-Elal
Journal: Journal of Mathematics
 (2021)

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## Abstract

In our present research study, we present the idea ofb dislocated-multiplicative metric space (abbrev. bd-multiplicative metric space) that is generalization of b-multiplicative metric space and dislocated-multiplicative metric space. Furthermore, we prove some of the fixed point theorems in bd-multiplicative metric spaces. Also, we get common fixed point findings for fuzzy mappings in these spaces. Our findings are improved and more generalized form of several findings (see, e.g., [5, 6]).


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## Body

## 1. Introduction

In 2008, the idea of multiplicative calculus was defined by Bashirov et al. [1] and then the conception of multiplicative metric spaces (multiplicative distance) was introduced by Çevikel and Özava̧sar [2]. Czerwik [3] presented concepts of b-metric space that is the popularization of metric space. Dosenovic et al. in section Future Work in [4] presented the idea of b-multiplicative metric spaces. After that, Ali et al. in [5] studied fixed point theorems for single-valued and multivalued mappings on b-multiplicative metric spaces.Furthermore, several authors obtained some fixed point findings for mappings satisfying different contractive conditions (see, e.g., [6–8]). The idea of fuzzy mappings was initially studied by Weiss [9] and Butnariu [10]. Then, the concept of fuzzy mappings was studied by Heilpern [11]. Many of the fixed point theorems for fuzzy contraction mappings in the metric linear space were proved (e.g., [12–15]), which are the fuzzy extension for the Banach contraction principle. The concept of b-multiplicative metric spaces, as one of the useful generalizations of multiplicative metric spaces, was first used by Dosenovic et al. in [4],and Ali et al. in [5] study fixed point theorems for single-valued and multivalued mappings on b-multiplicative metric spaces.Our findings inb-multiplicative metric space, d-multiplicative metric space, and multiplicative metric space can be obtained as corollaries of our findings.In this part, we list some of the concepts which we will use in our major findings.The definition ofb-multiplicative metric space is given as follows.Definition 1 (see [4, 5]).
Suppose thatX is a nonempty set and s≥1 is a given real number. A function d:X×X⟶1,∞ is considered as a b-multiplicative metric if it satisfies the following conditions: ∀η,ξ,z∈X,(i)
dη,ξ≥1(ii)
dη,ξ=1 iff η=ξ(iii)
dη,ξ=dξ,η(iv)
dη,ξ≤dη,zs⋅dz,ξsExample 1 (see [5]).
LetX=0,∞. Define a function d:X×X⟶1,∞, dη,ξ=aη−ξ2, where a>1 is any fixed real number. Then, X,d is a b-multiplicative metric with s=2.In [11, 16], an element in any fuzzy set has a degree of belonging, a membership function may be used in order to introduce the value of degree of belonging for any element to a set, and the value of degree of belonging takes real values on the whole closed interval 0,1. The membership function is(1)μA:X⟶0,1.SupposeX,d is a metric linear space. In X, a fuzzy set is a function A:X⟶0,1. Thus, it is an element of IX, where I=0,1. If A is a fuzzy set and η∈X, then the function value Aη is considered as the grade of membership of η in A.IX denotes to the collection of all fuzzy sets in X. The α-level set of A is defined by(2)Aα=η:Aη≥αwithα∈0,1andA0=η:Aη>0¯,whenever ¯ is the closure of set (nonfuzzy) .Definition 2 (see [17]).
A fuzzy setA in X is an approximate quantity if its α-level set is a nonempty compact subset (nonfuzzy) of X for each α∈0,1.
The set of all an approximate quantities denoted byW∗X is a subcollection of FX.Ozavsar and Cevikel [2] prove that every multiplicative contraction in a complete multiplicative metric space has a unique fixed point.Definition 3 (see [2]).
Assume thatX,d is a multiplicative metric space. A mapping g:X⟶X is called multiplicative contraction if(3)∃λ∈0,1:dgη1,gη2≤dη1,η2λ∀η1,η2∈X.Theorem 1 (see [2]).
Assume thatX,d is a multiplicative metric space. A mapping g:X⟶X is called multiplicative contraction. Then, g has a unique fixed point.Theorem 2 (see [4]).
Suppose thatX,d is a complete multiplicative metric space and a continuous function g:X⟶X, λ∈0,1 such that(4)dgη,gξ≤maxdη,gη,dξ,gξλ.Then,g has a unique fixed point.In 2015, Kang et al. [18] introduced the concept of compatible mappings as follows.Definition 4.
LetX,d be a multiplicative metric space. The mappings f,F:X⟶X; then, f,F is called compatible if and only if ft=Ft for some t in X implying fFt=Fft.Many authors studied many fixed point theorems for compatible mappings in multiplicative metric space and employed it to prove a common fixed point theorem (see [4, 18]).In this paper, we introduce the new notion ofbd-multiplicative metric space. We prove fixed point theorems for single mappings and a common fixed point for fuzzy mappings in bd-multiplicative metric space. As illustrative application, we state some of our theorems on Cartesian product in these spaces.


## 2. Fixed Point Theorems inbd-Multiplicative Metric Spaces

In this part, we present the conception ofbd-multiplicative metric space. Also, we introduce some of the fixed point theories and show our main findings with the help of some examples in this space.Definition 5.
Suppose thatX≠ϕ and s≥1 is a given real number. A function d:X×X⟶ℝ+ is called bd-multiplicative metric space if it satisfies the following conditions: ∀η,ξ,z∈X,(i)
dη,ξ≥1(ii)
dη,ξ=1 implies η=ξ(iii)
dη,ξ=dξ,η(iv)
dη,ξ≤dη,z⋅dz,ξsExample 2.
LetX=ℝ+∪0. Define d:X×X⟶1,∞ as(5)dη,ξ=aη+ξ2,∀η,ξ∈X,a≥1.
Then,X,d is bd-multiplicative metric space with s=2.Example 3.
LetX=1,∞. Define d:X×X⟶1,∞ as(6)dη,ξ=aη−1+ξ−12,∀η,ξ∈X,a≥1.
Then,X,d is bd-multiplicative metric space with s=2.Definition 6.
LetX,d be an bd-multiplicative metric space. We say that ηn converges to η if and only if(7)dηn,η⟶bd1,asn⟶∞.Definition 7.
LetX,d be an bd-multiplicative metric space. We say that ηn is bd-multiplicative Cauchy if and only if(8)dηn,ηm⟶bd1,asn,m⟶∞.Definition 8.
Anbd-multiplicative metric space X,d is complete if every multiplicative Cauchy sequence in X is convergent.Now, we state the following lemma without proof.Lemma 1.
Suppose thatX,d is bd-multiplicative metric space. Then, any subsequence of convergent sequence in X is convergent.The following theorem is the generalization of Theorem 3.2 in [2].Theorem 3.
Suppose thatX,d is a complete bd-multiplicative metric space and a continuous function g:D⟶X; D⊆X satisfies(9)dgη,gξ≤dη,ξk,where η,ξ∈D and k∈0,1/s. Then, g has a unique fixed point.Proof.
Letη0 be an arbitrary point in X; then by hypothesis, there exists η1 such that η1=gη0. In a similar way, one can obtain a sequence ηn⊆X such that(10)ηn=gηn−1=gnη0,dηn,ηn+m≤dηn,ηn+1sn⋅dηn+1,ηn+2sn+1…dηn+m−1,ηn+msn+m−1=dgnη0,gnη1sn⋅dgn+1η0,gn+1η1sn+1…dgn+m−1η0,gn+m−1η1sn+m−1≤dη0,η1ksn⋅dη0,η1ksn+1…dη0,η1ksn+m−1≤dη0,η1ksn/1−ks.
Then,(11)dηn,ηn+m≤dη0,η1ksn/1−ks.
Asn⟶∞ in (11) and k<1/s⇒ks<1, then ηn is a multiplicative Cauchy sequence.
SinceX,d is complete, then ηn is convergent such that limn⟶∞ηn=η∗. However,(12)η∗=limn⟶∞ηn+1=limn⟶∞gηn=glimn⟶∞ηn=gη∗.
Therefore,η∗ is a fixed point of g. Suppose that gη∗=η∗, gη¯=η¯, and η¯≠η¯.(13)dη∗,η¯=dgη∗,gη¯≤dη∗,η¯k≤dη∗,η¯.
This is a contradiction with assumption; then,η∗=η¯. Then, g has a unique fixed point.Example 4.
Suppose thatX=1,∞, X,d is bd-multiplicative metric space and dη,ξ=aη−1+ξ−12, where a=2 with s=2. Define g:X⟶X such that gη=η+1/2:(14)dgη,gξ=2η−1/2+ξ−1/22=21/4η−1+ξ−12=dη,ξ1/4.
Then, (1) holds such thatk=1/4. Therefore, g has a unique fixed point 1∈X.Corollary 1.
Suppose thatX,d is a complete multiplicative metric space and a continuous function g:X⟶X satisfies(15)dgη,gξ≤dη,ξk,where η,ξ∈D and k∈0,1. Then, g has a unique fixed point.Corollary 2.
Suppose thatX,d is a complete b-multiplicative metric space and a continuous function g:X⟶X satisfies(16)dgη,gξ≤dη,ξk,where η,ξ∈X and k∈0,1/s. Then, g has a unique fixed point.The following theorem is the generalization of Theorem 2.32 in [4].Theorem 4.
Suppose thatX,d is a complete bd-multiplicative metric space and a continuous function g:D⟶X, k∈0,1/s, such that(17)dgη,gξ≤maxdη,ξ,dη,gη,dξ,gξk.Then,g has a unique fixed point.Proof.
Letη0 be an arbitrary point in X; then by hypothesis, there exists η1 such that η1=gη0.
In a similar way, one can obtainη2∈X such that η2=gη1.(18)dη1,η2=dgη0,gη1≤maxdη0,η1,dη0,gη0,dη1,gη1k=maxdη0,η1,dη0,η1,dη1,η2k=maxdη0,η1,dη1,η2k=dη0,η1k.
Otherwise, we have a contradiction, that is,dη1,η2≤dη1,η2k.(19)dη2,η3=dgη1,gη2≤maxdη1,η2,dη1,gη1,dη2,gη2k=maxdη1,η2,dη1,η2,dη2,η3k=maxdη1,η2,dη2,η3k=dη0,η1k2.
Continuing in this way, we produce a sequenceηn in X such that ηn=gηn−1 and(20)dηn,ηn+1=dgηn−1,gηn≤maxdηn−1,ηn,dηn−1,gηn−1,dηn,gηnk=maxdηn−1,ηn,dηn−1,ηn,dηn,ηn+1k=maxdηn−1,ηn,dηn,ηn+1k=dη0,η1kn−1k=dη0,η1kn,for each n∈ℕ. It follows by induction that dηn,ηn+1≤dη0,η1kn. However,(21)dηn,ηn+m≤dηn,ηn+1sn⋅dηn+1,ηn+2sn+1…dηn+m−1,ηn+msn+m−1=dgnη0,gnη1sn⋅dgn+1η0,gn+1η1sn+1…dgn+m−1η0,gn+m−1η1sn+m−1≤dη0,η1ksn⋅dη0,η1ksn+1…dη0,η1ksn+m−1≤dη0,η1ksn1+ks+ks2+…≤dη0,η1ksn/1−ks.
Ask∈0,1/s, m,n⟶∞, ks<1, and X,d is complete, then ηn is a multiplicative Cauchy sequence in X and there exists η∗∈X such that dηn,η∗⟶bd1.
From Lemma1, η∗=limn⟶∞ηn+1=limn⟶∞gηn=glimn⟶∞ηn=gη∗.
Then,η∗ is fixed point of g, and gη∗=η∗. Suppose that g has another fixed point η¯ such that gη¯=η¯ and η¯≠η¯.(22)dη∗,η¯=dgη∗,gη¯≤maxdη∗,η¯,dη∗,gη∗,dη¯,gη¯k=maxdη∗,η¯,dη∗,η∗,dη¯,η¯k≤maxdη∗,η¯,dη∗,ηns⋅dηn,η∗s,dη¯,ηns⋅dηn,η¯sk=maxdη∗,η¯,1k=dη∗,η¯k≤dη∗,η¯.
This is a contradiction with assumption; then,η∗=η¯.
Then,g has a unique fixed point.Corollary 3.
Suppose thatX,d is a complete multiplicative metric space and a continuous function g:X⟶X satisfies(23)dgη,gξ≤maxdη,ξ,dη,gη,dξ,gξk,where η,ξ∈X and k∈0,1. Then, g has a unique fixed point.Corollary 4.
Suppose thatX,d is a complete b-multiplicative metric space and a continuous function g:X⟶X satisfies(24)dgη,gξ≤maxdη,ξ,dη,gη,dξ,gξk,where η,ξ∈X and k∈0,1/s. Then, g has a unique fixed point.


## 3. Common Fixed Point Theorems for Fuzzy Mappings inbd-Multiplicative Metric Spaces

Definition 9.
Suppose thatX is an arbitrary set and Y is bd-multiplicative-metric space. A mapping F is stated according to be a fuzzy mapping iff F is a function from the set X into W∗Y, i.e., Fη∈W∗Y, for each η∈X.Definition 10.
Suppose thatX,d is a bd-multiplicative metric space. The functions g:Y⊆X⟶X and G:Y⟶W∗Y. A hybrid pair g,G is called D-compatible iff gt⊂Gt for some t∈Y implies gGt⊂Ggt.Definition 11.
Suppose thatX,d is a bd-multiplicative metric space. Two maps G and g are said to be occasionally coincidentally idempotent if g2η=gη for some Cg,G, where Cg,G refers to the set of all coincidence points of two mappings g and G, i.e.,(25)Cg,G=η:gη=Gη.Now, we state the following lemma without proof.Lemma 2.
Suppose thatX,d is a bd-multiplicative metric space and M⊆W∗X. Then,(26)M¯=η∈X:dη,M=1.Corollary 5.
Suppose thatX,d is a bd-multiplicative metric space and M⊆W∗X and dη,M=1 if and only if η∈M¯=M.Lemma 3.
Suppose thatX,d is a bd-multiplicative metric space, G:X⟶W∗X is a fuzzy map, and η0∈X. Then, there exists η1∈X such that η1⊆Gη0.Theorem 5.
Suppose thatX,d is a complete bd-multiplicative metric space and two continuous mappings g,f:X⟶X satisfy(27)dfη,fξ≤dη,ξkanddgη,gξ≤dη,ξk,where η,ξ∈X and k∈0,1/s and two fuzzy mappings G,F:X⟶W∗X, such that(i)
GXα⊂fX, FXα⊂gX(ii)
The pairsG,g and F,f are D-compatible and occasionally idempotent mappings
Then, there existsη∗∈X such that η∗=fη∗=gη∗ and η∗∈Fη∗α∩Gη∗α.Proof.
Supposeη0 is an arbitrary point in X. Then, there is ξ1=gη1⊂Fη0α,from Lemma3.2;then, there exists ξ2=fη2⊂Gη1α, where(28)ξ2n+1=gη2n+1⊂Fη2nα,(29)ξ2n+2=fη2n+2⊂Gη2n+1α.From (11) in Theorem 3,(30)dξn,ξm≤dξ0,ξ1ksn/1−ks.
Asn⟶∞, k<1/s⇒ks<1 that implies ξn is a multiplicative Cauchy sequence.
SinceX,d is complete, then ηn is convergent such that(31)limn⟶∞dξn,ξm=1.
Next, we prove thatη∗∈Fzα.(32)dfz,fFzα=df2z,fFzα=dfη∗,fFzα≤dη∗,Fzαk≤dη∗,fzks⋅dfz,Fzαks=dη∗,fzks⋅dη∗,Fzαks=dη∗,η∗ks⋅dη∗,Fzαks.⇓dη∗,Fzαk−ks≤dη∗,η∗ks≤dη∗,ξ2n+2ks2⋅dξ2n+2,η∗ks2=dξ2n+2,η∗ks2⋅dξ2n+2,η∗ks2.
From Lemma1, ξ2n+2 is a convergent sequence, i.e., dξ2n+2,η∗⟶1 as n⟶∞,i.e.,(33)dη∗,Fzαk−ks≤1.dη∗,Fzα≤1.
Then, we havedη∗,Fzα=1 and Corollary 5 illustrates that η∗∈Fzα¯=Fzα.
Sinceη∗=fz∈Fzα⊂gX, there exists w∈Xsuchthatη∗=gw.
Similar to the previous steps, we can prove thatη∗=gw∈Gwα.
As two pairsG,g and F,f are D-compatible,(34)η∗=fz⊂Fzαandη∗=gw⊂Gwα.
Moreover,(35)fη∗=ffz⊂fFzα⊂Ffzα=Fη∗α.(36)gη∗=ggw⊂gGwα⊂Ggwα=Gη∗α.
Now, we show thatη∗=fη∗ and η∗=gη∗. Since ξn and ηn are convergent sequences, from Lemma 1,(37)η∗=limn⟶∞ξ2n+2=limn⟶∞fη2n+2=flimn⟶∞η2n+2=fη∗.(38)η∗=limn⟶∞ξ2n+1=limn⟶∞gη2n+1=glimn⟶∞η2n+1=gη∗.
Then,η∗=fη∗=gη∗ and η∗∈Fη∗α∩Gη∗α.Example 5.
Suppose thatX=0,1, X,d is a bd-multiplicative metric space defined by dη,ξ=aη+ξ2, and a>1. Define maps g,f:X⟶X as gη=η2, fη=η3∀η,andξ∈X. Also, define two fuzzy mappings G,F:X⟶W∗X as(39)Fηξ=0if0≤ξ<1/51/3if1/5≤ξ≤η32/3ifη3<ξ<4/51if4/5≤ξ≤1,and(40)Gηξ=0if0≤ξ<1/41/6if1/4≤ξ≤η21/9ifη2<ξ<6/51if6/5≤ξ≤1.
Now, forα=1/3, fFη1/3=1/125,η9⊂1/5,η3=Ffη1/3 and for α=1/6, gGη1/6=1/16,η4⊂1/4,η4=Ggη1/6; i.e., g,G and f,F are D-compatible. Finally, f1=ff1∈1/5,1=Ff1 and gg1=g1∈1/4,1=Gg1; i.e., f,F and g,G are occasionally coincidentally idempotent. Then, 1=f1=g1∈1/5,1∩1/4,1=F11/3∩G11/6 is a common fixed point.Example 6.
Suppose thatX=2/5,3/2, X,d is a bd-multiplicative metric space defined by dη,ξ=2η−2/5+ξ−2/52. Define maps g,f:X⟶X as gη=1/2η+2/5, fη=η+2/6∀η,andξ∈X. Also, define two fuzzy mappings G,F:X⟶W∗X as(41)Fηξ=0if0≤ξ<1/51/3if1/5≤ξ<η+2/6,2/3ifη+2/6≤ξ≤1and(42)Gηξ=0if0≤ξ<1416if14≤ξ<12η+2514if12η+25≤ξ≤1.
Now, forα=2/3,(43)fFη2/3=η+14/36,1/2⊂η+14/36,1=Ffη2/3,and for α=1/4,(44)gGη1/4=1/4η+1/10,7/20⊂1/4η+1/10,1=Ggη1/4.
Hence,g,G and f,F are D-compatible. Finally,(45)f2/5=ff2/5∈4/10,1=Ff2/5and(46)gg2/5=g2/5∈2/10,1=Gg2/5.
Therefore,f,F and g,G are occasionally coincidentally idempotent. Furthermore,(47)dfη,fξ=2η+2/6−2/5+ξ+2/6−2/52=2136η−2/5+ξ−2/52=dη,ξ1/36.
Then,2/5=f2/5=g2/5∈4/10,1∩2/10,1=F2/52/3∩G2/51/4 is a common fixed point.We concluded the following corollary when we setf=g in Theorem 5.Corollary 6.
LetY⊂X, and suppose continuous mapping g:Y⟶X satisfies dgη,gξ≤dη,ξk, where η,ξ∈Y, k∈0,1/s and fuzzy mapping G:Y⟶W∗X such that(i)
GYα⊂gY(ii)
The pairG,g is D-compatible and occasionally idempotent mappings
Then, there existsη∗∈Y such that η∗=gη∗∈Gη∗α.Theorem 6.
LetY⊂X and suppose two continuous mappings g,f:Y⟶X, satisfy(48)dfη,fξ≤dη,ξkanddgη,gξ≤dη,ξk,where η,ξ∈Y, k∈0,1/s, and Fnα:Y⟶W∗X such that ∀η∈Y,(i)
FlYα⊂fY and FkYα⊂gY(ii)
The pairsFl,f and Fk,g are D-compatible and occasionally idempotent mappings
Then, there existsη∗∈Y such that η∗=fη∗=gη∗ and η∗∈∩n=0∞Fnη∗α.Proof.
The proof of this theorem is completed, when puttingFl=G and Fk=F in Theorem 5.Remark 1.
IfFl=G and Fk=F, then Theorem 6 implies Theorem 5.Theorem 7.
Suppose thatX,d is a complete bd-multiplicative metric space and two continuous mappings g,f:X⟶X satisfy(49)dfη,fξ≤maxdη,ξ,dη,fη,dξ,fξk,dgη,gξ≤maxdη,ξ,dη,gη,dξ,gξk,where η,ξ∈X, k∈0,1/s, and two fuzzy mappings F,G:X⟶W∗X, such that(i)
GXα⊂fX, FXα⊂gX(ii)
The pairsF,f and G,g are D-compatible and occasionally idempotent mappings
Then, there existsη∗∈X: η∗=fη∗=gη∗ and η∗∈Fη∗α∩Gη∗α.Proof.
Letη0 be an arbitrary point in X; then, there exists ξ1=gη1∈Fη0α and from Lemma 3, there exists ξ2=fη2⊂Gη1α such that(50)ξ2n+1=gη2n+1⊂Fη2nα,ξ2n+2=fη2n+2⊂Gη2n+1α.
Sinceξ1=gη1∈Fη0,there existsξ2=fη2⊂Gη1α, and from Theorem 4,(51)dξn,ξm≤dξ0,ξ1ksn/1−ks.
Asn⟶∞, k<1/s⇒ks<1 that implies ξn is a multiplicative Cauchy sequence. Since X,d is complete, then ηn is convergent such that(52)limn⟶∞dξn,ξm=1.
Asξ2n+2 is a Cauchy sequence in fX and fX is joint orbitally complete, then there exists η∗∈X such that(53)limn⟶∞ξ2n+2=η∗,η∗=fz∀z∈X.
We prove thatη∗∈Fzα and(54)dη∗,Fzα=dfz,Fzα≤dfz,η∗s⋅dη∗,Fzαs=dffz,fzs⋅dη∗,Fzαs=df2z,f2zs⋅dη∗,Fzαs≤maxdfz,fz,dfz,f2z,dfz,f2zks⋅dη∗,Fzαs=maxdfz,fz,dfz,fz⋅dfz,fzks⋅dη∗,Fzαs=maxdη∗,η∗,dη∗,η∗,dη∗,η∗ks⋅dη∗,Fzαs=dη∗,η∗ks⋅dη∗,Fzαs.dη∗,Fzα1−s≤dη∗,η∗ks≤dη∗,ξ2n+2s⋅dξ2n+2,η∗sks.
Asn⟶∞, then dη∗,ξ2n+2⟶bd1. However, ξ2n+2 is a convergent sequence; i.e.,(55)dη∗,Fzα1−s≤1,dη∗,Fzα≤1.
Then, we havedη∗,Fzα=1 and Corollary 5 illustrates that η∗∈Fzα¯=Fzα.
Sinceη∗=fz⊂Fzα⊂gX, there exists w∈Xsuch thatη∗=gw.
Similar to the previous steps, we can prove thatη∗=gw∈Gwα.
As two pairsG,g and F,f are D-compatible,(56)η∗=fz⊂Fzandη∗=gw⊂Gwα,and therefore(57)fη∗=ffz⊂fFzα⊂Ffzα=Fη∗α,(58)gη∗=ggw⊂gGwα⊂Ggwα=Gη∗α.
Now, we show thatη∗=fη∗ and η∗=gη∗. Since ξn and ηn are convergent sequences, then(59)η∗=limn⟶∞ξ2n+2=limn⟶∞fη2n+2=flimn⟶∞η2n+2=fη∗,(60)η∗=limn⟶∞ξ2n+1=limn⟶∞gη2n+1=glimn⟶∞η2n+1=gη∗.
Then,f,g,F,andG have a common fixed point.Theorem 8.
Suppose thatX,d is a complete bd-multiplicative metric space, Y⊂X, and two continuous mappings g,f:Y⟶X satisfy(61)dfη,fξ≤maxdη,ξ,dη,fη,dξ,fξk,dgη,gξ≤maxdη,ξ,dη,gη,dξ,gξk,where η,ξ∈Y, k∈0,1/s, and Fn:Y⟶W∗X, such that(i)
FlYα⊂fY, FkYα⊂gY, k=2n+1, l=2n+2, and n∈ℕ(ii)
The pairsFk,f and Fl,g are D-compatible and occasionally idempotent mappingsThen, there exist η∗∈Y such that η∗=fη∗=gη∗ and η∗∈∩n=0∞Fnη∗α.


## 4. Applications

In this section, we give some applications on our main results. We state some of our theorems on Cartesian product without proof.Theorem 9.
Suppose thatX,d is a complete bd-multiplicative metric space. The map g:D2=D×D⟶D2, D⊆X satisfies(62)dga,c,gb,d≤da,c,b,dk,where a,c,b,d∈D2 and k∈0,1/s. Then, g has a unique fixed point.The next example illustrates the previous theory.Example 7.
Suppose thatX=1,∞ and D=1,η:η∈X. Define d:D2⟶1,∞ as(63)da,c,b,d=ac/b−1+d/a−12,∀a,c,b,d∈D,a≥1. Then, X,d is a complete bd-multiplicative metric space with s=2.
Letg:D⟶D be a function defined by g1,η=1,η+1/2. Then, condition (3) holds. However,(64)dg1,η,g1,ξ=d1,1+η/2,1,1+ξ/2=aη−1/2+ξ−1/22=a1/4η−1+ξ−12=d1,η,1,ξ1/4,with k=1/4. It is obvious that 1,1∈D2 is a unique fixed point of a map g.Theorem 10.
Suppose thatX,d is a complete bd-multiplicative metric space and g:D2⟶D2, k∈0,1/s, such that(65)dga,c,gb,d≤maxda,c,b,d,da,c,ga,c,db,d,gb,dk.Then,g has a unique fixed point.Theorem 12.
Suppose thatX,d is a complete bd-multiplicative metric space and two continuous mappings g,f:D2⟶D2 satisfy(66)dfa,c,fb,d≤da,c,b,dk,dga,c,gb,d≤da,c,b,dk,where a,c,b,d∈D2, k∈0,1/s, and two fuzzy mappings G,F:D2⟶W∗X, such that(i)
GD2α⊂fD2 and FD2α⊂gD2(ii)
The pairsF,f and G,g are D-compatible and occasionally idempotent mappings
Thenf,g,F,andG have a common fixed point.Theorem 13.
Suppose thatX,d is a complete bd-multiplicative metric space, D⊂X, and two mappings g,f:D2⟶D2 satisfy(67)dfa,c,fb,d≤da,c,b,dk,dga,c,gb,d≤da,c,b,dk,where a,c,b,d∈D2, k∈0,1/s, and Fn:D2⟶W∗X such that(i)
FlD2α⊂fD2, FkD2α⊂gD2, k=2n+1, l=2n+2, and n∈ℕ(ii)
The pairsFk,f and Fl,g are D-compatible and occasionally idempotent mappingsThen, there exists η∗∈D×D such that η∗=fη∗=gη∗ and η∗∈∩n=0∞Fnη∗α.Theorem 14.
Suppose thatX,d is a complete bd-multiplicative metric space and two mappings g,f:D2⟶D2, D⊂X, satisfy(68)dfa,c,fb,d≤maxda,c,b,d,da,c,fa,c,db,d,fb,dk,dga,c,gb,d≤maxda,c,b,d,da,c,ga,c,db,d,gb,dk,where a,c,b,d∈D2, k∈0,1/s, Fnα:D2⟶W∗X, such that(i)
FlD2α⊂fD2, FkD2α⊂gD2, k=2n+1, l=2n+2, and n∈ℕ(ii)
The pairsFk,f and Fl,g are D-compatible and occasionally idempotent mappings
Then, there existsη∗∈D2 such that η∗=fη∗=gη∗ and η∗∈∩n=0∞Fnη∗α.


## 5. Conclusion

In this paper, we introduced the concept ofbd-multiplicative metric spaces. We studied some of the fixed point theorems in these spaces. Also, we obtain common fixed point theorems for fuzzy mappings in complete bd-multiplicative metric spaces. Finally, we get some of applications on our main findings. We hope that our presented idea herein will be a source of motivation for other researchers to extend and improve these findings for their applications.



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*Source: 4762281-2021-12-16.xml*