# Cyclicb-Multiplicative A,B-Hardy–Rogers-Type Local Contraction and Related Results in b-Multiplicative and b-Metric Spaces

Authors: Abdullah Eqal Al-Mazrooei; Abdullah Shoaib; Jamshaid Ahmad
Journal: Journal of Mathematics
 (2020)

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## Abstract

The aim of this paper is to define cyclicb-multiplicative Hardy–Rogers-type local contraction in the context of generalized spaces named as b-multiplicative spaces to extend various results of the literature including the main results of Yamaod et al. In this way, we apply a new generalized contractive condition only on a closed set instead of a whole set and by using b-multiplicative space instead of multiplicative metric space. We apply our results to obtain new results in b-metric spaces. Examples are given to show the usability of our results, when others cannot.


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## Body

## 1. Introduction and Preliminaries

Bakhtin [1] was the first who gave the idea of b-metric. After that, Czerwik [2] gave an axiom and formally defined a b-metric space. For further results on the b-metric space, see [3, 4]. Ozaksar and Cevical [5] investigated the multiplicative metric space and proved its topological properties. Mongkolkeha and Sintunavarat [6] described the concept of multiplicative proximal contraction mapping and proved some results for such mappings. Recently, Abbas et al. [7] proved some common fixed point results of quasi-weak commutative mappings on a closed ball in the setting of multiplicative metric spaces. For further results on the multiplicative metric space, see [8–12]. In 2017, Ali et al. [13] introduced the notion of the b-multiplicative space and proved some fixed point results. As an application, they established an existence theorem for the solution of a system of Fredholm multiplicative integral equations. For further results on the b-multiplicative space, see [14]. Shoaib et al. [4] discussed some results for the mappings satisfying contraction condition on a closed ball in a b-metric space. For further results on a closed ball, see [15–25]. In this paper, we generalized the results in [12] by using cyclic b-multiplicative A,B-Hardy–Rogers-type local contraction on a closed ball in a b-multiplicative space. Moreover, we show that our results can be applied on those mappings where the other results cannot be applied. The following definitions and results will be used to understand this paper.Definition 1 (see [13]).
LetW be a nonempty set, and let s≥1 be a given real number. A mapping mb:W×W⟶1,∞ is called b-multiplicative with coefficient s, if the following conditions hold:(i)
mbw,μ>1 for all w,μ∈W with w≠μ and mbw,μ=1 if and only if w=μ(ii)
mbw,μ=mbμ,w for all w,μ∈W(iii)
mbw,z≤mbw,μ.mbμ,zs for all w,μ,z∈W
The pairW,mb is called a b-multiplicative space. If r>1, u∈W, then Bmbu,r¯=v:mbu,v≤r and Bmbu,r¯=v:mbu,v<r are called the closed ball and the open ball in W,mb, respectively. Note that if r≤1, then we obtain empty open balls because mbu,v≥1.Example 1.
LetW=0,∞. Define a mapping ma:W×W⟶1,∞:(1)maw,μ=aw,μ2,where a>1 is any fixed real number. Then, for each a, ma is b-multiplicative on W with s=2. Note that ma is not a multiplicative metric on W. Considering a=2, r=216, and u=1, then Bmbu,r¯=0,5 is a closed ball in W.Definition 2 (see [13]).
LetW,mb be a b-multiplicative space.(i)
A sequencewn is called b-multiplicative Cauchy iff(2)mbwm,wn⟶1,asm,n⟶+∞.(ii)
A sequencewn is b-multiplicative convergent iff there exists w∈W such that(3)mbwn,w⟶1,asn⟶+∞.(iii)
Ab-multiplicative space W,mb is said to be b-multiplicative complete if every b-multiplicative Cauchy sequence in W is b-multiplicative convergent to some μ∈W.Example 2.
The spaceW,mb defined in Example 1 is a b-multiplicative complete space. Taking wn=1/n, then mb1/m,1/n⟶1, as m,n⟶+∞. Hence, wn is a b-multiplicative Cauchy sequence. Now, mb1/n,0⟶1, as n⟶+∞ implies that wn is b-multiplicative convergent to 0∈W.Definition 3 (see [3]).
LetW≠ϕ and s≥1 be a real number. A mapping d:W×W⟶ℝ+∪0 is said to be b-metric with coefficient “s,” if for all w,μ,z∈W, the following assertions hold:(i)
dw,μ=0⟺w=μ(ii)
dw,μ=dμ,w(iii)
dw,z≤dw,μ+dμ,z
The pairW,d is said to be a b-metric space. If r>0, u∈W, then Bdu,r¯=v:mbu,v<r is called a closed ball in W,d.Remark 1 (see [13]).
Everyb-metric space W,d generates a b-multiplicative space W,mb defined as(4)mbþ,μ=edþ,μ.Remark 2.
LetW,mb be a b-multiplicative space generated by a b-metric space W,d, r>0 and þ°∈W. If Bdþ°,r¯ and Bmbþ°,er¯ are closed balls in W,d and W,mb, respectively, then Bdþ°,r¯=Bmbþ°,er¯.


## 2. Results onb-Multiplicative Spaces

Definition 4.
LetK≠ϕ, A,B:K⟶0,∞, and g:K⟶K. We say that g is a locally cyclic A,B-admissible mapping on set H, if(5)þ∈H,Aþ≥1⟹Bgþ≥1,þ∈H,Bþ≥1⟹Agþ≥1.Definition 5.
LetK,mb be a b-multiplicative space with s≥1 and Bmbþ°,r¯ be a closed set. A self-mapping g is said to be cyclic b-multiplicative A,B-Hardy–Rogers-type local contraction on Bmbþ°,r¯, if the following conditions hold:(1)
Aþ°≥1 and Bþ°≥1(2)
g is a locally cyclic A,B-admissible mapping on Bmbþ°,r¯(3)
AþBμ≥1 implies(6)mbgþ,gμ≤mbþ,μλmbþ,gþ.mbμ,gμθmbþ,gμ.mbμ,gþν,for þ,μ∈Bmbþ°,r¯, where λ,θ,ν∈0,1 and sλ+s+1θ+s2+sν<1.Example 3.
LetK=0,∞ and g:X⟶X be a mapping defined as(7)gx=3x10,if0≤x≤3,x7+x+6,otherwise.
Define a completeb-multiplicative metric with s=2 as(8)mbx,y=1ifx=y2x+y2ifx≠y.
Consideringþ°=1 and r=216, then Bmbþ°,r¯=0,3. A,B:K⟶0,+∞ is defined by(9)Aþ=1ifþ∈0,300otherwise,Bþ=1ifþ∈0,30otherwise,now, Aþ°≥1 and Bþ°≥1. Clearly, g is a locally cyclic A,B-admissible mapping on Bmbþ°,r¯. Note that, if we take þ=20, then Aþ≥1 but Bgþ≱1(gþ) so g is a not-cyclic A,B-admissible mapping on K, and the results in [12] cannot be applied. Taking λ=9/100, θ=1/40, and v=3/70, now, λ,θ,ν∈0,1 and sλ+s+1θ+s2+sν<1. For each þ,μ∈Bmbþ°,r¯ with AþBμ≥1, we have(10)23þ/10+3μ/102≤2þ+μ29/100.2þ+3þ/102.2μ+3μ/1021/40.2þ+3μ/102.2μ+3þ/1023/70ormbgþ,gμ≤mbþ,μλmbþ,gþ.mbμ,gμθmbþ,gμ.mbμ,gþν.
Thus, all conditions of Definition5 hold. Therefore, g is a cyclic b-multiplicative A,B-Hardy–Rogers-type local contraction on Bmbþ°,r¯. Note that(11)mbgþ,gμ≰mbþ,μλmbþ,gþ.mbμ,gμθmbþ,gμ.mbμ,gþν,for all þ,μ∈K, so again the results in [12] cannot be applied because g cannot satisfy any definition in [12].Definition 6.
If we takeθ=ν=0 in Definition 5, then we say it is cyclic b-multiplicative A,B-Banach-type local contraction on Bmbþ°,r¯. If we take λ=ν=0 in Definition 5, then we say it is cyclic b-multiplicative A,B-Kannan-type local contraction on Bmbþ°,r¯. If we take θ=λ=0 in Definition 5, then we say it is b-multiplicative cyclic A,B-Chatterjea-type local contraction on Bmbþ°,r¯. If we take s=1 in Definition 5, then we say it is cyclic multiplicative A,B-Hardy–Rogers-type local contraction on Bmbþ°,r¯. If we exclude the role of functions A and B in Definition 5, that is, if we exclude conditions (1) and (2) and the restriction AþBμ≥1 from Definition 5, then we say it is b-multiplicative Hardy–Rogers-type local contraction on Bmbþ°,r¯.Example 4.
If we takeθ=ν=0 in Example 3, then we obtain an example of cyclic b-multiplicative A,B-Banach-type local contraction on Bmbþ°,r¯. If we take λ=ν=0 in Example 3, then we obtain an example of cyclic b-multiplicative A,B-Kannan-type local contraction on Bmbþ°,r¯. If we take θ=λ=0 in Example 3, then we obtain an example of cyclic b-multiplicative A,B-Chatterjea-type local contraction on Bmbþ°,r¯. If we take s=1 in Example 3, then we obtain an example of cyclic multiplicative A,B-Hardy–Rogers-type local contraction on Bmbþ°,r¯. If we exclude the role of functions A and B in Example 3, then we obtain an example of b-multiplicative Hardy–Rogers-type local contraction on Bmbþ°,r¯.Definition 7.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K and ρ:K⟶0,∞ be two mappings. We say that g is cyclic regular on a closed ball Bmbþ°,r¯ with respect to ρ, if one of the following conditions holds:(a)
IfBmbþ°,r¯ contains a sequence Pn such that ρPn≥1 for all n∈ℕ and þn⟶þ∗∈Bmbþ°,r¯ as n⟶∞, then ρþ∗≥1(b)
g is continuous on Bmbþ°,r¯Example 5.
The mappingg in Example 3 is cyclic regular on a closed ball Bmbþ°,r¯=0,3 with respect to Bþ=1ifþ∈0,30otherwise because for any sequence þn in 0,3 such that Bþn≥1 for all n∈ℕ and þn⟶þ∗∈Bmbþ°,r¯ as n⟶∞, then Bþ∗≥1. Also, g is continuous on 0,3.Theorem 1.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a cyclic b-multiplicative A,B-Hardy–Roger-type local contraction mapping on Bmbþ°,r¯. Suppose that(12)mbþ°,gþ°≤r1−sh/S,where h=λ+θ+sν/1−θ−sν. Then, there exists a b-multiplicative convergent sequence in Bmbþ°,r¯. Also, if g is cyclic regular on Bmbþ°,r¯ with respect to B, then there exists a fixed point of g in Bmbþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.Proof.
Consider thatAþ°≥1 and Bþ°≥1 and þ°∈Bmbþ°,r¯. As g is a cyclic A,B admissible mapping on Bmbþ°,r¯, we have(13)Aþ°≥1⟹Bþ1=Bgþ°≥1,Bþ°≥1⟹Aþ1=Agþ°≥1.
Note that(14)mbþ°,þ1=mbþ°,gþ°≤r1−sh/s.
By assumption, we have(15)sλ+s+1θ+s2+sν<1orsλ+sθ+s2ν<1−θ−sνorsλ+θ+sν1−θ−sν<1.
Sinceh=λ+θ+sν/1−θ−sν, so sh<1. Now, 1−sh/s<1 and r>1 implies r1−sh/s<r. Hence, þ1∈Bmbþ°,r¯. As(16)Aþ1≥1⟹Bþ2=Bgþ1≥1,Bþ1≥1⟹Aþ2=Agþ1≥1.
Assume thatþ2,þ3,…,þj∈Bmbþ°,r¯ for some j∈ℕ. By a similar method, as above for þ1 and þ2, we get(17)Aþi≥1andBþi≥1,fori=1,2,…,j+1.
This implies that(18)Aþj−1Bþj≥1,fori=1,2,…,j+1.
By using (6), we have(19)mbþj,þj+1=mbgþj−1,gþj≤Aþj−1Bþj.mbgþj−1,gþj≤mbþj−1,þjλmbþj−1,gþj−1.mbþj,gþjθmbþj−1,gþj.mbþj,gþj−1ν≤mbþj−1,þjλmbþj−1,þj.mbþj,þj+1θmbþj−1,þj+1.mbþj,þjν≤mbþj−1,þjλmbþj−1,þj.mbþj,þj+1θmbþj−1,þjs.mbþj,þj+1sνmbþj,þj+11−θ−sν≤mbþj−1,þjλ+θ+sνmbþj,þj+1≤mbþj−1,þjλ+θ+sν/1−θ−sνmbþj,þj+1≤mbþj−1,þjh,whereh=λ+θ+sν1−θ−sν.
Now, using inequality (19), we get(20)mbþj,þj+1≤mbþj−1,þjh≤mbþj−2,þj−1h2≤⋯≤mbþ°,þ1hj.
By using triangle inequality and inequality (20), we get(21)mbþ°,þj+1≤mbþ°,þ1s.mbþ1,þ2s2,…,mbþj,þj+1sj+1≤mbþ°,þ1s.mbþ°,þ1s2h,…,mbþ°,þ1sj+1hj≤mbþ°,þ1s1+sh+⋯+sjhjmbþ°,þj+1≤mbþ°,þ1s1−shj/1−sh.
By using inequality (12), we have(22)mbþ°,þj+1≤r1−sh/s.s1−shj/1−shmbþ°,þj+1≤r1−shj≤r,for allj∈ℕ.
This implies thatþj+1∈Bmbþ°,r¯. By induction on n, we conclude that þn∈Bmbþ°,r¯ for all n∈ℕ. By a similar method, for all n∈ℕ, we get(23)Aþn≥1,Bþn≥1,for alln∈ℕ.
This implies that(24)Aþn−1Bþn≥1,for alln∈ℕ.
Now, inequality (20) implies that(25)mbþn,þn+1≤mbþ°,þ1hn.
Now, we prove thatþn is a b-multiplicative Cauchy sequence in K. Let m>n, so m=n+p; p∈ℕ. By using the triangle inequality, we have(26)mbþn,þm≤mbþn,þn+1s.mbþn+1,þn+2s2,…,mbþn+p−1,þn+psp.
By using inequality (25), we get(27)mbþn,þm≤mbþ°,þ1shn.mbþ°,þ1s2hn+1,…,mbþ°,þ1sphn+p−1≤mbþ°,þ1shn1+sh+⋯+sp−1hp−1≤mbþ°,þ1shn1+sh+⋯+shp−1<mbþ°,þ1shn1+sh+⋯mbþn,þm<mbþ°,þ1shn/1−sh.
Taking limit asm,n⟶∞, we get mbþn,þm⟶1. Hence, the sequence þn is a b-multiplicative Cauchy sequence. By the completeness of K,mb, it follows that þn⟶þ∗∈Bmbþ°,r¯. Suppose that g is continuous. Thus, we get þ∗=limn⟶∞þn+1=limn⟶∞gþn=glimn⟶∞þn=gþ∗. Now, we assume that condition (a) of Definition 7 holds. As Bþn≥1 and þn⟶þ∗∈Bmbþ°,r¯, so Bþ∗≥1. Then, we have(28)mbgþ∗,þ∗≤mbgþ∗,gþns.mbgþn,þ∗s≤AþnBþ∗.mbgþn,gþ∗s.mbgþn,þ∗s≤mbþn,þ∗sλmbþn,gþn.mbþ∗,gþ∗sθmbþn,gþ∗.mbþ∗,gþnsν.mbgþn,þ∗s≤mbþn,þ∗sλmbþn,þn+1.mbþ∗,gþ∗sθmbþn,gþ∗.mbþ∗,þn+1sν.mbþn+1,þ∗s.
Lettingn⟶∞, we get(29)mbgþ∗,þ∗≤mbþ∗,gþ∗sθmbþ∗,gþ∗sνmbgþ∗,þ∗1−sθ−sv≤1mbgþ∗,þ∗≤11/1−sθ−sv=1.
Hence,mbqþ∗,þ∗=1, that is, gþ∗=þ∗. This proves that þ∗ is a fixed point of g. Eventually we prove that þ∗ is the unique fixed point of g. Suppose that μ is another fixed point of g. By the hypothesis, we find that Aþ∗≥1 and Bμ≥1. Thus,(30)mbþ∗,μ=mbgþ∗,gμ≤Aþ∗Bμ.mbgþ∗,gμ≤mbþ∗,μλmbþ∗,gþ∗.mbμ,gμθmbþ∗,gμ.mbμ,gþ∗ν≤mbþ∗,μλmbþ∗,þ∗.mbμ,μθmbþ∗,μ.mbμ,þ∗νmbþ∗,μ1−λ−2ν≤1mbþ∗,μ≤11/1−λ−2ν=1.
This proves thatmbþ∗,μ=1 and then þ∗=μ. Thus, þ∗ is the unique fixed point of g.Example 6.
In Example3, we have proved that g is a cyclic b-multiplicative A,B-Hardy–Rogers-type local contraction on Bmbþ°,r¯. It has been proved in Example 5 that the mapping g in Example 3 is cyclic regular on a closed ball Bmbþ°,r¯=0,3 with respect to Bþ=1ifþ∈0,30otherwise. Now,(31)h=λ+θ+sν1−θ−sν=9/100+1/40+23/701−1/40−23/70=2811245.
Now,(32)mbþ°,gþ°=mb1,g1=mb1,3/10=21,3/102≈3.27<20.95≈r1−sh/s.
Hence, all the conditions of Theorem1 are satisfied, and zero is the unique fixed point of the mapping g. Note that the results in [12] cannot ensure the existence of a fixed point of mapping g because g cannot satisfy the contractive condition of any theorem in [12].
The following results for various other contractions onb-multiplicative spaces can be proved by following the proof of Theorem 1.Theorem 2.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a cyclic b-multiplicative A,B-Banach-type local contraction mapping on Bmbþ°,r¯. Suppose that(33)mbþ°,gþ°≤r1−sλ/s.Then, there exists ab-multiplicative convergent sequence in Bmbþ°,r¯. Also, if g is cyclic regular on Bmbþ°,r¯, then there exists a fixed point of g in Bmbþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.Theorem 3.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a cyclic b-multiplicative A,B-Kannan-type local contraction mapping on Bmbþ°,r¯. Suppose that(34)mbþ°,gþ°≤r1−sh/s,where h=θ/1−θ. Then, there exists a b-multiplicative convergent sequence in Bmbþ°,r¯. Also, if g is cyclic regular on BmbP°,r¯, then there exists a fixed point of g in Bmbþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.Theorem 4.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a cyclic b-multiplicative A,B-Chatterjea-type local contraction mapping on Bmbþ°,r¯. Suppose that(35)mbþ°,gþ°≤r1−sh/s,where h=sν/1−sν. Then, there exists a b-multiplicative convergent sequence in Bmbþ°,r¯. Also, if g is cyclic regular on Bmbþ°,r¯, then there exists a fixed point of g in Bmbþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.Theorem 5.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a b-multiplicative Hardy–Roger-type local contraction mapping on Bmbþ°,r¯. Suppose that(36)mbþ°,gþ°≤r1−sh/s,whereh=λ+θ+sν1−θ−sν.Then, there exists a unique fixed point inBmbþ°,r¯.Theorem 6.
LetK,mb be a b-multiplicative complete space with coefficient s≥1 and g:K⟶K be a cyclic b-multiplicative A,B-Hardy–Roger-type local contraction mapping on K . Then, there exists a b-multiplicative convergent sequence in K. Also, if g is cyclic regular on K, then there exists a fixed point of g in K. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.The following result is a multiplicative metric version of Theorem1.Theorem 7.
LetK,m be a complete multiplicative space and g:K⟶K be a multiplicative A,B-Hardy–Roger-type local contraction mapping on Bmbþ°,r¯. Suppose that(37)mbþ°,gþ°≤r1−h,where h=λ+θ+ν/1−θ−ν. Then, there exists a multiplicative convergent sequence in Bbþ°,r¯. Also, if one of the following conditions holds:(a)
IfBmþ°,r¯ contains a sequence þn such that Bþn≥1 for all n∈ℕ and þn⟶þ∗∈Bmþ°,r¯ as n⟶∞, then Bþ∗≥1(b)
g is continuous on Bmþ°,r¯
Then, there exists a fixed point ofg in Bmþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.As an application, we give an existence theorem for the Fredholm multiplicative integral equation of the following type:(38)gu=∫abQu,w,gwdw,u,w∈a,b,where Q:a,b×a,b×ℝ+⟶ℝ+ is an integrable function.LetK=Ca,b,ℝ+, a>0 and ℝ+=0,∞, be the space of all positive, continuous real-valued functions, endowed with the b-multiplicative:(39)mbg,h=supu∈a,bmaxguhu2,hugu2.Clearly, the setEg0,r=hu:supu∈a,bmaxg0u/hu2,hu/g0u2≤r is a closed ball Bmbg0,r¯ in K,mb.Theorem 8.
LetK=Ca,b,ℝ+, g0u∈K, r>1, a>0, A,B:K⟶0,∞, and S:K⟶K:(40)Sgu=∫abQu,w,gwdw,where Q:a,b×a,b×ℝ+⟶ℝ+ is an integrable function. Assume that the following conditions hold:(1)
Ag°u≥1 and Bg°u≥1;(2)
S is a cyclic A,B-admissible mapping on Eg0,r;(3)
For eachu,w∈a,b and for gu, hu belongs to closed set Eg0,r, such that AguBhu≥1; then, this implies(41)Qu,w,gwQu,w,h,w≤gwhwλ;(4)
The constantλ is such that 2λ<1/b−a and(42)supu∈a,bmaxg0ug1u2,g1ug0u2≤r1−2λb−a/2;Also, if one of the following conditions holds:(5)
S is continuous on Eg0,r or(6)
Ifgnu is a sequence in Eg0,r such that gnu⟶g∗u∈Eg0,r as n⟶∞ and Bgnu≥1 for all n∈ℕ, then Bg∗u≥1.Then, the integral equation (38) has a solution. Moreover, if Ag≥1 and Bg≥1 for all g in the set of fixed points of S, then equation (38) has a unique solution.Proof.
Letgu,hu∈Eg0,r. Now, we have(43)SguShu2≤∫abQu,w,gwQu,w,hwdw2≤∫abgwhwλdw2≤∫abmbg,hλ/2dw2=mbg,hb−aλ/22=mbg,hλb−a,for eachu∈a,b.
Thus, we getmbSg,Sh≤mbg,hα, α=λb−a. As 2λ<1/b−a, so wα<1. Also, hypothesis (4) implies(44)mbg0,Sg0≤r1−wα/w.
Therefore, by Theorem2, there exists a unique fixed point of the operator S. Hence, the integral equation (38) has a unique solution.


## 3. Results onb-Metric Spaces

Definition 8.
LetK,b be a b-metric space and Bbþ°,r¯ be a closed set. A self-mapping g is said to be cyclic b-A,B-Hardy–Rogers-type local contraction on Bbþ°,r¯, if the following conditions hold:(1)
Aþ°≥1 and Bþ°≥1(2)
g is a locally cyclic A,B-admissible mapping on Bbþ°,r¯(3)
AþBμ≥1 implies(45)bgþ,gμ≤λbþ,μ+θbþ,gþ+bμ,gμ+νbþ,gμ+bμ,gþ,for þ,μ∈Bmbþ°,r¯, where λ,θ,ν∈0,1 and sλ+s+1θ+s2+sν<1.Theorem 9.
LetK,b be a complete b-metric space with coefficient s≥1 and g:K⟶K be a cyclic b-A,B-Hardy–Roger-type local contraction mapping on Bbþ°,r¯. Suppose that(46)bþ°,gþ°≤r1−shs,where h=λ+θ+sν/1−θ−sν. Then, there exists a convergent sequence in Bbþ°,r¯. Also, if one of the following conditions holds:(a)
IfBbþ°,r¯ contains a sequence þn such that Bþn≥1 for all n∈ℕ and þn⟶þ∗∈Bbþ°,r¯ as n⟶∞, then Bþ∗≥1(b)
g is continuous on Bbþ°,r¯.Then, there exists a fixed point ofg in Bbþ°,r¯. Moreover, if Bz≥1 and Az≥1, for all z in the set of fixed points of g, then the fixed point of g will be unique.Proof.
Definingmbþ,μ=ebþ,μ. Then, by Remark 1W,mb is a b-multiplicative space. By taking exponential on both sides of inequality (46), we have(47)ebþ°,gþ°≤er1−sh/sormbþ°,gþ°≤ε1−sh/s,where ε=er>1. Now, by taking exponential on both sides of inequality (45) and by using Remark 2, we have(48)ebgþ,gμ≤eλbþ,μeθbþ,gþ+bμ,gμeνbþ,gμ+bμ,gþ,for all þ,μ belong to the closed set Bbþ°,r¯. Now, by using Remarks 1 and 2, we have(49)mbgþ,gμ≤mbþ,μλmbþ,gþ.mbμ,gμθmbþ,gμ.mbμ,gþν,for all þ,μ belong to the closed set Bmbþ°,ε¯. Now, by Theorem 1, g has a unique fixed point in Bmbþ°,ε¯ or Bbþ°,r¯.Example 7.
LetK=ℝ endowed with the b-metric bþ,μ=þ−μ for all þ,μ∈K and g:K⟶K be defined by(50)gþ=−12þifþ∈−13,132þifþ∈ℝ\−13,13,and A,B:K⟶0,+∞ be given by(51)Aþ=1ifþ∈−3,00otherwise,Bþ=1ifþ∈0,10otherwise.
Letr=1/3 and þ°=0, then Bbþ°,r¯=−1/3,1/3 is closed. Now, Aþ°≥1 and Bþ°≥1. Also, g is a locally A,B-admissible mapping on Bbþ°,r¯. If þ,μ belong to Bbþ°,r¯ such that AþBμ≥1, then þ∈−1/3,0 and μ∈0,1/3. Taking λ=1/2, θ=1/9, and ν=1/18, we have(52)bgþ,gμ≤λbþ,μ+θbþ,gþ+bμ,gμ+νbþ,gμ+bμ,gþ.
So,g is cyclic b-A,B-Hardy–Rogers-type local contraction on Bbþ°,r¯. Also,(53)bþ°,gþ°≤r1−shs.
Now, ifþn is a sequence in Bbþ°,r¯ such that Bþn≥1 and þn⟶þ as n⟶∞. Then, þn∈0,1/3. Hence, þ∈0,1/3 and Bþ≥1. So, all hypotheses of Theorem 9 are satisfied, and therefore g has a unique fixed point.
Note that the results of Alizadeh et al. [26] and other results for A,B-admissible mapping cannot be applied. Since A−3≥1, but Bg−3=B−60≱1. Also, B1≥1, but Ag1=A2=0≱1. Therefore, g is not a cyclic A,B-admissible mapping.
Now, we give an example of a mappingg which is a cyclic A,B-admissible, but none of the previously defined contractions in other papers holds. Therefore, other results for A,B-admissible mapping fail to ensure the existence of a fixed point. However, g has a fixed point, and our result is valid for such mappings.Example 8.
LetK=ℝ endowed with the b-metric Bþ,μ=þ−μ for all þ,μ∈K and g:K⟶K be defined by(54)gþ=−12þifþ∈−13,13−þifþ∈−3,3−−13,132þifþ∈ℝ\−3,3,and A,B:K⟶0,+∞ be given by(55)Aþ=1ifþ∈−3,00otherwise,Bþ=1ifþ∈0,30otherwise.
By takingλ=1/2, θ=1/9, ν=1/18, r=1/3, and þ°=0, all hypotheses of Theorem 9 are satisfied, and therefore g has a unique fixed point. Note that, g is a cyclic A,B-admissible mapping on K, but all other results for A,B-admissible mapping cannot be applied. For example, defining ψ,φ:0,∞⟶0,∞ by ψt=t and φt=1/4t. Let AþBμ≥1. Then, þ∈−3,0 and μ∈0,3. If þ=1 and μ=2, then(56)ψdg1,g2=1>34=ψd1,2−φd1,2.
That is, Theorem 2.4 of [26] cannot be applied here.Definition 9.
In Definition8, if(1)
Ifθ=ν=0, then we say it is cyclic b-A,B-Banach-type local contraction on Bbþ°,r¯(2)
Ifλ=ν=0, then we say it is cyclic b-A,B-Kannan-type local contraction on Bbþ°,r¯(3)
Ifθ=λ=0, then we say it is cyclic b-A,B-Chatterjea-type local contraction on Bbþ°,r¯(4)
Ifs=1, then we say it is cyclic A,B-Hardy–Rogers-type local contraction on Bbþ°,r¯(5)
We exclude the role of functionsA and B; that is, if we exclude conditions (1) and (2) and the restriction AþBμ≥1 from Definition 8, then we say it is b-Hardy–Rogers-type local contraction on Bbþ°,r¯Example 9.
If we considerK=ℝ endowed with the b-metric Bþ,μ=þ−μ for all þ,μ∈K and define g:K⟶K as in Example 7, then by taking λ=1/2 and θ=ν=0, we can get cyclic b-A,B-Banach-type local contraction on Bbþ°,r¯. Similarly, if we consider λ=ν=0 and θ=1/9 in Example 7, we can get cyclic b-A,B-Kannan-type local contraction on Bbþ°,r¯. Also, if we take, θ=λ=0 and ν=1/18, then we can get cyclic b-A,B-Chatterjea-type local contraction on Bbþ°,r¯. If we exclude the role of functions A and B in Example 7, then we can get an example of b-Hardy–Rogers-type local contraction on Bbþ°,r¯.Remark 3.
By using Definition9, we can make five new theorems in b-metric spaces.



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