# Some Fixed Points Results inb-Metric and Quasi b-Metric Spaces

Authors: Reny George; Ivan D. Aranđelović; Vesna Mišić; Zoran D. Mitrović
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We present a fixed point result in quasib-metric spaces. Our result generalizes recent fixed point results obtained by Aleksić et al., Dung and Hang, Jovanović et al., Sarwar, and Rahman and classical results obtained by Hardy, Rogers, and Ćirić. Also, we obtain a common fixed point result in b-metric spaces. As a special case, we get a result of Ćirić and Wong.


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## Body

## 1. Introduction

The notion of a generalized contraction was presented by Ćirić in his dissertation [1]. In [1], Ćirić proved the first fixed point result for this class of mappings, which was published in [2]. Ćirić also published several papers on generalized contractions, such as for multivalued mappings in [3], on common fixed point of not necessarily commuting mappings in [4], for probabilistic metric spaces in [5, 6] and fixed point result of Meir-Keeler type in [7]. For further historical remarks of the papers of Ćirić, see [8].In 1973, Hardy and Rogers [9] proved a result of fixed point on metric space, which was extended to common fixed point result by Wong [10].The results of common fixed points of Wong [10] and Ćirić [4] are independent. More concepts of common fixed points can be seen in [11, 12].Also, Fréchet in the paper [13] introduced a class of metric spaces which are included in the class b-metric spaces. First, fixed point result in a b-metric space was presented by Bakhtin [14] and Czerwik [15] (for more on b-metric spaces see [16–23]). In the last few decades, many generalizations of a metric space appeared in literature. For some historical aspects of various generalizations of a metric space, the reader may refer to [24].In this paper, we present a fixed point theorem for a mapping defined on a quasib-metric space which generalizes recent fixed point results obtained by Aleksić et al. [16], Dung and Hang [18], Jovanović et al. [25], and Sarwar and Rahman [22]. Further, we obtain a result of common fixed point on a b-metric space. Our result generalizes the classical results presented by Ćirić [4] and Wong [10].


## 2. The Quasib-Metric Spaces

We start with definition of quasib-metric spaces, which was introduced by Shah and Hussain [23].Definition 1.
LetX be a nonempty set, d:X×X⟶0,+∞ and s∈0,+∞. Then, X,d,s is a quasi b-metric space if
(1)
dμ,ν=0 if and only if μ=ν(2)
dμ,ξ≤sdμ,ν+dν,ξ, for all μ,ν,ξ∈XClearly,X,d,1 is a quasi metric space.Remark 2.
LetX,d,s be a quasi b-metric space and dμ,ν=dν,μ for all μ,ν∈X. Then, X,d,s is a b-metric space.Lemma 3.
LetX,d,s be a quasi b-metric space. Then, s≥1.Proof.
Letμ,ν∈X. Then, dμ,ν≤sdμ,ν+dν,ν=sdμ,ν. So, s≥1.☐Remark 4.
Letrn be a sequence of nonnegative real numbers such that rn+1≤rn and limn⟶+∞rn=0. A quasi b-metric space is a topological space with Bnμn∈ℕ, as a base of neighborhood filter of the point μ where Bnμ=ν∈X:dμ,ν<rn.Definition 5.
LetX,d,s be a quasi b-metric space and a sequence μn⊆X.
(1)
Sequenceμn is a left Cauchy sequence, if dμn,μm⟶0 as m,n⟶+∞(2)
A quasib-metric space X,d,s is left complete if every left Cauchy sequence converges to some μ∈XDefinition 6.
LetX,d,s be a quasi b-metric space and the sequences μn, νn in X be such that limn⟶+∞μn=μandlimm⟶+∞νn=ν. A mapping d is sequentially continuous if limn,m⟶+∞dμn,νm=dμ,ν.We will use the following lemma in our main results.Lemma 7 (see [26]).
LetX,d,s be a quasi b-metric space and μn⊆X. If there exists λ∈0,1 such that
(1)dμn,μn+1≤λdμn−1,μn,
for alln∈ℕ, then μn is a left Cauchy sequence.


## 3. A Fixed Point Theorem in Quasib-Metric Spaces

LetX≠∅ and f:X⟶X be a given mapping. Then, μ∗∈X is a fixed point of mapping f if fμ∗=μ∗. Let μ0∈X, and consider the sequence μn defined by μn=fnμ0, i.e., μn is a sequence of Picard iterates of mapping f at point μ0.Now, we present our first result, which generalizes recent fixed point results obtained in [16, 18, 22, 25] for generalized contractive mappings defined on b-metric spaces.Theorem 8.
LetX,d,s be a left complete quasi b-metric space and a mapping f:X⟶X. If there exist α,β,γ∈0,1 such that α+β+γ<1, β≤γ and
(2)dfμ,fν≤αmaxdμ,ν,dμ,fμ,dν,fν,dμ,fν+dfμ,ν2s+βdμ,fνs+γdfμ,ν,
for anyμ,ν∈X, then for any μ0∈X sequence of Picard iterates μn defined by mapping f at μ0 is left Cauchy sequence. Moreover, if f is sequentially continuous or d is sequentially continuous, then, f has unique fixed point μ∗∈X and μn⟶μ∗ as n⟶+∞.Proof.
Letμ0∈X be arbitrary and μn sequence of Picard iterates defined by f at μ0. Then
(3)dμn+1,μn+2=dfμn,fμn+1≤αmaxdμn,μn+1,dμn,fμn,dμn+1,fμn+2,dμn,fμn+1+dfμn,μn+12s+βdμn,fμn+1s+γdfμn,μn+1≤αmaxdμn,μn+1,dμn+1,μn+2,dμn,μn+1+dμn+1,μn+22+βdμn,μn+2s≤αmaxdμn,μn+1,dμn+1,μn+2+βdμn,μn+1+βdμn+1,μn+2.
Ifdμn,μn+1<dμn+1,μn+2, then,
(4)1−α−βdμn+1,μn+2<βdμn,μn+1,which implies
(5)dμn+1,μn+2≤β1−α−βdμn,μn+1<dμn,μn+1.
So,dμn,μn+1≥dμn+1,μn+2 which implies
(6)1−βdμn+1,μn+2≤α+βdμn,μn+1.
Hence, we get that(7)dμn+1,μn+2≤α+β1−βdμn,μn+1<α+βα+γdμn,μn+1=λdμn,μn+1,where λ=α+β/α+γ<1. So, by Lemma 7, we obtain that μn is left Cauchy sequence. It is convergent because X,d,s is left complete. Thus, exists μ∗∈X such that μ∗=limμn.☐Case 9.
Let a mappingf be a sequentially continuous.
Then(8)μ∗=limμn=limfμn=fμ∗.Case 10.
Letd be a sequentially continuous.
Then(9)dfμn,fμ∗≤αmaxdμn,μ∗,dμn,μn+1,dμ∗,fμ∗,dμn,fμ∗+dfμn,μ∗2s+βdμn,μn+22s+γdfμn,μ∗,which implies
(10)limdfμn,fμ∗≤limαdμn,μ∗,dμ∗,fμ∗,dμn,fμ∗+dfμn,μ∗2s+βdμn,μn+22s+γdfμn,μ∗.
So, we get that(11)dlimμn+1,fμ∗≤limαmaxdlimμn,μ∗,dlimμn,limμn+1,dμ∗,fμ∗,dlimμn,fμ∗+dlimμn+1,μ∗2s+βdμn,μn+22s+γdlimμn+1,μ∗.
Hence,(12)dμ∗,fμ∗≤αmaxdμ∗,μ∗,dμ∗,μ∗,dμ∗,fμ∗,dμ∗,fμ∗+dμ∗,μ∗2s+βdμ∗,μ∗2s+γdμ∗,μ∗=αdμ∗,fμ∗.
It follows thatμ∗=fμ∗ because α∈0,1. Finally, suppose that there are two fixed points of mapping f, i.e., fμ∗=μ∗, fν∗=ν∗. Then, we get
(13)dμ∗,ν∗=dfμ∗,fν∗≤αdμ∗,ν∗,dμ∗,μ∗,dν∗,ν∗,dμ∗,ν∗+dμ∗,ν∗2s+βdμ∗,ν∗2s+γdμ∗,ν∗≤α+β+γdμ∗,ν∗.
which implies thatμ∗=ν∗.Corollary 11.
LetX,d,s be a left complete quasi b-metric space and a mapping f:X⟶X. If there exist α∈0,1 such that
(14)dfμ,fν≤αdμ,ν,
for anyμ,ν∈X, then for any μ0∈X sequence of Picard iterates μn defined by mapping f at μ0 is left Cauchy sequence. Moreover, if f is sequentially continuous or d is sequentially continuous, then, f has unique fixed point μ∗∈X and μn⟶μ∗ as n⟶+∞.Example 12.
LetX=0,1 and mapping f:X⟶X defined by fμ=μ/2,μ∈X. Let d:X×X⟶0,+∞ defined by
(15)dμ,ν=μ−ν2,μ>ν,μ−ν4,μ<ν,0,μ=ν.
Since,a+b2≤2a2+b2 and a+b4≤8a4+b4, for all a,b∈ℝ, we obtain that for d holds
(16)dμ,ξ≤8dμ,ν+dν,ξ,for all μ,ν,ξ∈X. Also, dμ,ν=0 if and only if μ=ν. So, X,d,8 is a quasi b-metric space. Note that dμ,ν=dν,μ does not hold in the general case. In this case, all the conditions of Corollary 11 are valid, and we conclude that the mapping f has a fixed point.


## 4. A Common Fixed Point Theorem inb-Metric Spaces

Now we obtain a common fixed point result for mappings defined onb-metric spaces. Our result improves the classical results presented by Ćirić [4] and Wong [10].Theorem 13.
LetX,d,s be a complete b-metric space and the mappings f,g:X⟶X. If there exist α,β∈0,1 such that α+2β<1 and
(17)dfμ,gν≤αmaxdμ,ν,dμ,fμ,dν,gν,dμ,gν+dfμ,ν2s+βdμ,gνs+βdfμ,νs,
for anyμ,ν∈X, then for any μ0∈X sequence of Picard iterates μn defined by g∘f at μ0 is left Cauchy sequence. If f and g are sequentially continuous or d is sequentially continuous then f and g has unique fixed point which is unique limit of all Picard sequences defined by g∘f.Proof.
Letμ0∈X be arbitrary and μn sequence defined by μ2n+1=fμ2n and μ2n+2=gμ2n+1. Then
(18)dμ2n+1,μ2n+2=dfμ2n,gμ2n+1≤αmaxdμ2n,μ2n+1,dμ2n,fμ2n,dμ2n+1,gμ2n+1,dμ2n,gμ2n+1+dμ2n,μ2n+12s+βdμ2n,fμn+1s+βdfμ2n,μ2n+1s≤αmaxdμ2n,μ2n+1,dμ2n+1,μ2n+2,dμ2n,μ2n+1+dμ2n+1,μ2n+22+βdμ2n,μn+2s≤αmaxdμ2n,μ2n+1,dμ2n+1,μ2n+2+βdμ2n,μ2n+1+βdμ2n+1,μ2n+2.
Ifdμ2n,μ2n+1<dμ2n+1,μn+2 then
(19)1−α−βdμ2n+1,μ2n+2<βdμ2n,μ2n+1.
So, we get that(20)dμ2n+1,μ2n+2≤β1−α−βdμ2n,μ2n+1<dμ2n,μ2n+1,therefore, dμ2n,μ2n+1≥dμ2n+1,μ2n+2 which implies
(21)1−βdμn+1,μn+2≤α+βdμ2n,μ2n+1.
Hence, we get that(22)dμ2n+1,μ2n+2≤α+β1−βdμ2n,μ2n+1.
So, we obtained(23)dμ2n+1,μ2n+2≤λdμ2n,μ2n+1,where λ=α+β/1−β<1. Further, we have
(24)dμ2n,μ2n+1=dgμ2n−1,fμ2n=dfμ2n,gμ2n−1≤αmaxdμ2n−1,μ2n,dμ2n,fμ2n,dμ2n−1,gμ2n−1,dμ2n−1,fμ2n+dμ2n,gμ2n−12s+βdμ2n,gμ2n−1s+βdfμ2n,μ2n−1s≤αmaxdμ2n−1,μ2n,dμ2n,μ2n+1,dμ2n,μ2n+1+dμ2n−1,μ2n2+βdμ2n−1,μ2n+1s≤αmaxdμ2n,μ2n+1,dμ2n+1,μ2n+2+βdμ2n−1,μ2n+βdμ2n,μ2n+1.
Ifdμ2n−1,μ2n<dμn,μn+1 then
(25)1−α−βdμ2n,μ2n+1<βdμ2n−1,μ2n,which implies
(26)dμ2n,μ2n+1≤β1−α−βdμ2n,μ2n−1<dμ2n−1,μ2n,therefore, dμ2n−1,μ2n≥dμ2n,μ2n+1 which implies that
(27)dμ2n,μ2n+1≤αdμ2n−1,μ2n+βdμ2n−1,μ2n+βdμ2n,μ2n+1.
Therefore, we obtain(28)1−βdμ2n,μ2n+1≤α+βdμ2n−1,μ2n.
It follows(29)dμ2n,μ2n+1≤α+β1−βdμ2n−1,μ2n.
So we obtain,(30)dμ2n,μ2n+1≤λdμ2n−1,μ2n,where λ=α+β/1−β<1. Hence,
(31)dμn,μn+1≤λdμn−1,μn,for each positive integer n. So, by Lemma 7, we obtain that μn is a Cauchy sequence. It is convergent because X,d,s is complete. Therefore, there exists ξ∈X such that ξ=limμn.☐Case 14.
Letf and g be sequentially continuous functions. Then, we have
(32)ξ=limμn=limfμn=fξ=limμn=limgμn=gξ.Case 15.
Letd be a sequentially continuous. Then,
(33)dfξ,gμ2n+1≤αmaxdξ,μ2n+1,dξ,fξ,dμ2n+1,gμ2n+1,dμ2n+1,fξ+dgμ2n+1,ξ2s+βdμ2n+1,fξ2s+βdgμ2n+1,ξs,which implies
(34)limdfξ,gμ2n+1≤limαmaxdξ,μ2n+1,dξ,fξ,dμ2n+1,gμ2n+1,dμ2n+1,fξ+dgμ2n+1,ξ2s+βdμ2n+1,fξ2s+βdgμ2n+1,ξs.
So, we get that(35)dlimgμ2n+1,fξ≤αmaxdlimμ2n+1,ξdξ,fξ,dlimμ2n+1,limfμ2n+1,dlimμ2n+1,fξ+dlimgμ2n+1,ξ2s+βdlimμ2n+1,fξ2s+βdlimgμ2n+1,ξs.
Hence,(36)dξ,fξ≤αmaxdξ,ξ,dξ,fξ,dξ,ξ,dξ,fξ+dξ,ξ2s+βdξ,fξ2s+βdξ,ξ<α+βdξ,fξ.
It follows thatξ=fξ because α+β∈0,1. Further, we have
(37)dfμ2n,gξ≤αmaxdμ2n,ξ,dμ2n,fμ2n,dξ,gξ,dμ2n,gξ+dfμ2n,ξ2s+βdμ2n,gξ2s+βdfμ2n,ξs,which implies
(38)limdfμ2n,gξ≤limαmaxdμ2n,ξ,dμ2n,fμ2n,dξ,gξ,dμ2n,gξ+dfμ2n,ξ2s+βdμ2n,gξ2s+βdfμ2n,ξs.
So, we get that(39)dlimfμ2n,gξ≤αmaxdlimμ2n,ξ,dlimμ2n,limfμ2n,dξ,gξ,dlimμ2n,gξ+dlimfμ2n,ξ2s+βdlimμ2n,gξ2s+βdlimfμ2n,ξs.
Hence,(40)dξ,gξ≤αmaxdξ,ξ,dξ,ξ,dξ,gξ,dξ,gξ+dξ,ξ2s+βdξ,gξ2s+βdξ,ξ<α+βdξ,gξ.
It follows thatξ=gξ because α+β∈0,1.
Now, we prove that the fixed point is unique. Suppose that there areξ and ξ′, i.e., gξ=fξ=ξ and gξ′=fξ′=ξ′. Then, we obtain
(41)dξ,ξ′=dfξ,gξ′≤αdξ,ξ′,dξ,ξdξ′,ξ′,dξ,ξ′+dξ,ξ′2s+βdξ,ξ′2s+βdξ,ξ′≤α+2βdξ,ξ′.
which implies thatξ=ξ′.Corollary 16.
LetX,d,s be a complete b-metric space and mapping f:X⟶X. If there exist α,β,γ∈0,1 such that α+β+γ<1 and
(42)dfμ,fν≤αmaxdμ,ν,dμ,fμ,dν,fν,dμ,fν+dfμ,ν2s+βdμ,fνs+γdfμ,νs,
for anyμ,ν∈X, then for any μ0∈X sequence of Picard iterates μn defined by f at μ0 is Cauchy sequence. If f is sequentially continuous or d is sequentially continuous, then, f has unique fixed point which is unique limit of all Picard sequences defined by f.Proof.
From(43)dfμ,fν≤αmaxdμ,ν,dμ,fμ,dν,fν,dμ,fν+dfμ,ν2s+βdμ,fνs+γdfμ,νs,and
(44)dfν,fμ≤αmaxdν,μ,dν,fν,dμ,fμ,dν,fμ+dfν,μ2s+βdν,fμs+γdfν,μs,it follows
(45)dfμ,fν≤αmaxdμ,ν,dμ,fμ,dν,fν,dμ,fν+dfμ,ν2s+δdμ,fνs+δdfμ,νs,
whereδ=β+γ/2.☐Example 17.
LetX=0,4 and dμ,ν=μ−ν2, for each μ,ν∈X. Then X,d,2 is a b-metric space. Define a mapping f:X⟶X by
(46)ft=t3,t∈0,3,t6,t∈3,4,for any t∈X. For μ,ν∈0,3, we have
(47)dfμ,fν=19μ−ν2.
Forμ,ν∈3,4, we have
(48)dfμ,fν=136dμ,ν.
Forμ∈0,3 and ν∈3,4, we have
(49)dfμ,fν=μ3−ν62≤49<12dν,fν,because dν,fν=5ν/62>25·9/36.
Forν∈0,3 and μ∈3,4, we have
(50)dfμ,fν=μ3−ν62≤169<34dμ,fμ,because dμ,fμ=5μ/62>25·9/36.Since conditions of Corollary16 is satisfied for α=3/4 and β=γ=0. So, f has unique fixed point which is unique limit of all Picard sequences defined by f, because d is sequentially continuous.



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