# Fixed Point Results for Single and Multivalued Maps on Partial Extendedb-Metric Spaces

Authors: Hassen Aydi; Samina Batul; Muhammad Aslam; Dur-e-Shehwar Sagheer; Eskandar Ameer
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This article is based on the concept of partial extendedb-metric spaces, which is inspired by the notions of new extended b-metric spaces and partial metric spaces. Fixed point results for single and multivalued mappings on such spaces are also presented. Few examples are also provided to elaborate the concepts.


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## Body

## 1. Introduction and Preliminaries

Many problems in science and engineering defined by nonlinear equations can be solved by reducing them to an equivalent fixed point problem. Banach contraction principal provided a platform for researchers to prove the existence of fixed point of mappings on different abstract spaces. Fixed point theorems are developed for single-valued or set-valued mappings of abstract metric spaces. One can see a lot of literature on metric fixed point theory [1–9]. Recently, Shukla and Panicker [10] investigated some fixed point results on generalized enriched nonexpansive mappings defined on Banach spaces. These results are generalization of many existing results.A partial metric space was introduced by Matthews [11, 12]. It is the generalization of a metric space, in which dx,x=0 may be nonzero and dx,x≤dx,y for every x,y∈X. Matthews proved a fixed point theorem for contractive mappings in that space and also discussed its topological properties. After this development, many researchers worked in this direction (see [4, 9, 13–15]). Aydi et al. [16] initiated the idea of a partial Hausdorff metric space, and they proved fixed point theorems in partial metric spaces for multivalued mappings. A beautiful piece of work in this aspect can be seen in [17]. Authors used partial metric spaces by incorporating a strictly increasing F-mapping to introduce multivalued F-contractions. This paper extends some results in partial metric spaces. Shukla [18] presented the notion of a partial b-metric space and proved some related fixed point results. A lot of research can be seen in literature on b-metric spaces (see [1, 2, 7, 8, 19–23]), which is a stimulation towards the concept of extended b-metric spaces introduced by Kamran et al. [6] who proved the Banach contraction theorem in the setting of extended b-metric spaces. Afterwards, many authors have focused on the subject and generalized different results of metric spaces in extended b-metric spaces (see [3, 24–28]). Very recently, Aydi et al. [5] have introduced the concept of new extended b-metric spaces by replacing the triangle inequality of extended b-metric spaces with a modified functional inequality. In this setting, they proved few fixed point theorems for nonlinear contractive mappings. Motivated by the above-mentioned research, we initiate the idea of partial extended b-metric spaces and we extend the result of Kamran et al. [6] in setting of the partial extended b-metric spaces. We have also presented few examples to elaborate the idea.To reach the goal of proving some fixed point results in the new notion, first we give some definitions.Definition 1 [21].
Consider a nonempty setX with a real number b≥1. The function d:X×X⟶0,∞ is called a b-metric if it satisfies the following properties for each x,y,z∈X:
(b1) dx,y=0⇔x=y
(b2) dx,y=dy,x
(b3) dx,y≤bdx,z+dz,y
Here, the pairX,d is called a b-metric space.
In the following, we give the definition of a new extendedb-metric space (in the sense of Aydi et al. [5]).Definition 2.
LetX be a nonempty set and given θ:X×X×X⟶1,∞. The function dθ:X×X⟶0,∞ is called a new extended b-metric if it satisfies the following conditions for all x,y,z∈X:
(1)
dθx,y=0⇔x=y(2)
dθx,y=dθy,x(3)
dθx,z≤θx,y,zdθx,y+dθy,z
The pairX,dθ is called a new extended b-metric space.Definition 3 [11].
LetM be a nonempty set. The function p:M×M⟶ℝ+ is said to be a partial metric on M if p satisfies the conditions listed below:
(p1) w1=w2⇔pw1,w1=pw2,w2=pw1,w2
(p2) pw1,w1≤pw1,w2
(p3) pw1,w2=pw2,w1
(p4) pw1,w2≤pw1,w3+pw2,w3−pw3,w3
The pairM,p is called a partial metric space.
Shukla [18] has presented a partial b-metric space as follows.Definition 4 [18].
LetM be a nonempty set and b≥1 be a given real number. The function pb:M×M⟶0,∞ is called a partial b-metric with coefficient b if for all w1,w2,w3∈M, the following conditions are satisfied:
(PB1):pbw1,w1=pbw2,w2=pbw1,w2 if and only if w1=w2
(PB2):pbw1,w1≤pbw1,w2
(PB3):pbw1,w2=pbw2,w1
(PB4):pbw1,w2≤bpbw1,w3+pbw3,w2−pbw3,w3
TheM,pb is called a partial b-metric space.


## 2. Partial Extendedb-Metric Space

In this section, we first elaborate the idea of a partial extendedb-metric space (in the sense of Aydi et al. [5]), and then with the help of an example, we explain our new definition.Definition 5.
LetM be nonempty set and θ:M×M×M⟶1,+∞ be a function. The function pθ:M×M⟶0,+∞ is called a partial extended b-metric if for all w1,w2,w3∈M, it satisfies the following:
(PE1):pθw1,w1=pθw2,w2=pθw1,w2⇔w1=w2
(PE2):pθw1,w1≤pθw1,w2
(PE3):pθw1,w2=pθw2,w1
(PE4):pθw1,w3≤θw1,w2,w3pθw1,w2+pθw2,w3−pθw2,w2The partial extendedb-metric space pθ on M generates a T0 topology τpθ with a base of the family of open pθ-balls Bm,ε:m∈M,ε>0, where Bm,ε=m1∈M:pθm,m1<pθm,m+ε, for all m∈M and ε>0. A sequence hn in a partial new extended b-metric space pθ converges with respect to τpθ to a point h∈M if and only if
(1)limn⟶+∞pθh,hn=pθh,h.Remark 6.
(1)
Ifθw1,w2,w3=b, then the above definition coincides with a partial b-metric space(2)
Ifθw1,w2,w3=1, then the partial new extended b-metric becomes a partial metric
The following is an example of a partial extendedb-metric space, which shows that the notion of a partial extended b-metric space is a generalization of a partial b-metric space.Example 7.
ConsiderM=ℕ and define pθ:M×M⟶ℝ by
(2)pθw1,w2=maxw1,w2ifw1=w2,1w1+maxw1,w2ifw1isevenandw2isodd,1w2+maxw1,w2ifw2isevenandw1isodd,1+maxw1,w2otherwise.
Then,pθ is a partial extended b-metric on ℕ, where θ:M×M×M⟶1,∞ is defined by
(3)θw1,w2,w3=1,ifw1=w3andw2isoddoreven,5ptw1w3w1+w3,ifw1≠w3,w1andw3areevenandw2odd,5ptw22,ifw1≠w3,w1andw3areoddandw2iseven,5pt32,ifw1≠w3,w1,w2,andw3allareevenorallareodd,5ptw1+w21+w1w11+w2,ifw1,w2evenandw3odd,5ptw3+w2w3+1w3w2+1,ifw1odd,w3andw2areeven,5pt2+w31+w3,ifw1w2areoddandw3iseven,5ptw1+1w1,ifw1iseven,w2andw3areodd.Example 8.
Consider the setM=1,2,3, and define the function θ:M×M×M⟶1,∞ by
(4)θw1,w2,w3=2+w1+w2+w2.
Definepθ:M×M⟶ℝ+ in the following manner:
(5)pθ1,1=1=pθ2,2=2=pθ3,3=3,pθ1,2=pθ2,1=82,pθ1,3=pθ3,1=1000,pθ2,3=pθ3,2=600.
Then,pθ is a partial extended b-metric on M. In fact, the first three axioms of Definition 5 hold, so there is only need to check the triangular inequality. To do this, proceed as follows:
(6)pθ1,2=82<θ1,2,3pθ1,3+pθ3,2−pθ3,3=81000+600−3=8×1600−3=12797.
Similarly, the remaining pairs can be verified as above; hence, for allw1,w2,w3∈M,
(7)pθw1,w3≤θw1,w2,w3pθw1,w2+pθw2,w3−pθw2,w2holds. Hence, M,pθ is a partial extended b-metric space.Remark 9.
Note that in the above examples, the functionθ depends on all the three variables w1, w2, and w3. So from this fact, it is clear that a partial extended b-metric needs not to be a partial extended b-metric.
In partial extendedb-metric spaces, the notion of Cauchyness and convergence are defined as follows.Definition 10.
LetM,pθ be a partial extended b-metric space.
(1)
A sequencewn is said to be a Cauchy sequence if limn,m⟶∞pθwm,wn exists and is finite(2)
A sequencewn converges to w∈M if and only if(8)limm⟶+∞pθw,wm=pθw,w.
It is worth to mention that ifpθ is a partial extended b-metric on M, the dθ:M×M⟶ℝ+ defined by
(9)dθm,n=2pθm,n−pθm,m−pθn,nis an extended b-metric on M.Definition 11.
A partial extendedb-metric space M,pθ is complete if for each Cauchy sequence wm in M, there is w∈M such that
(10)limm⟶+∞pθw,wm=pθw,w=limn,m⟶∞pθwm,wn.Theorem 12.
LetM,pθ be a complete partial extended b-metric space and pθ be a continuous functional. Consider a self map T:M⟶M such that
(11)pθTw,Tv≤kpθw,v∀w,v∈M,
wherek∈0,1. Also assume that there exists w0∈M such that
(12)limn⟶∞θwn,wn+1,wm<1k,forallm>n,
wherewn=Tnw0. Then, T has a unique fixed point.Proof.
For an arbitraryw0∈M, take the iterative sequence wn defined by wn=Twn−1=Tnw0∀n∈ℕ, which satisfies (12). Inequality (11) implies
(13)pθTw1,Tw2≤kpθw1,w2≤kpθTw0,Tw1≤k2pθw0,w1.
Thus, by applying (11) successively, we obtain
(14)pθwn,wn+1≤knpθw0,w1∀n∈ℕ.By use of triangular inequality, we have (form>n)
(15)pθwn,wm≤θwn,wn+1,wmpθwn,wn+1+pθwn+1,wm−pθwn+1,wn+1=θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmpθwn+1,wm−pθwn+1,wn+1≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmpθwn+1,wm≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+pθwn+2,wm−pθwn+2,wn+2≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+pθwn+2,wm.Continuing this process, we get(16)pθwn,wm≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+θwn,wn+1,wmθwn+1,wn+2,wmθwn+2,wn+3,wmpθwn+2,wn+3+⋯+θwn,wn+1,wmθwn+1,wn+2,wm⋯θwm−2,wm−1,wmpθwm−1,wm.By using (14), one writes
(17)pθwn,wm≤θwn,wn+1,wmknpθw0,w1+θwn,wn+1,wmθwn+1,wn+2,wmkn+1pθw0,w1+⋯+θwn+1,wn+2,wmθwn+2,wn+3,wm⋯θwm−2,wm−1,wmkm−1pθw0,w1.Therefore,(18)pθwn,wm≤pθw0,w1θwn,wn+1,wmkn+θwn,wn+1,wmθwn+1,wn+2,wmkn+1+⋯+θwn,wn+1,wmθwn+1,wn+2,wmθwn+2,wn+3,wm⋯θwm−2,wm−1,wmkm−1≤pθw0,w1θw1,w2,wmθw2,w3,wm⋯θwn−1,wn,wmθwn,wn+1,wmkn⋯+θw1,w2,wmθw2,w3,wm⋯θwn−1,wn,wmθwn,wn+1,wmθwn+1,wn+2,wmkn+1⋯+θw1,w2,wmθw2,w3,wm⋯θwn−1,wn,wmθwn,wn+1,wmθwn+1,wn+2,wm⋯θwn+2,wn+3,wm⋯θwm−2,wm−1,wmkm−1.Set(19)Sn=pθw0,w1∑j=1nkj∏i=1jθwi,wi+1,wm.Form>n, we conclude that
(20)pθwn,wm≤pθw0,w1Sm−1−Sn−1.Consider the series(21)pθw0,w1∑n=1∞kn∏i=1nθwi,wi+1,wm.Take(22)vn=kn∏i=1nθwi,wi+1,wm.Thus, we have(23)limn⟶∞vn+1vn=kθwn+1,wn+2,wm.By using the condition(24)kθwn+1,wn+2,wm<1by ratio test, the series is convergent. So we conclude that limm,n⟶∞Sm−1−Sn−1=0. Hence, by (20), wn is a Cauchy sequence in M. Since M is complete, we have wn⟶η∈M such that limn⟶∞pθwn,η=pθη,η=limm,n⟶∞pθwn,wm=0. Now, we prove that η is a fixed point of T. Consider
(25)pθTwn,Tη≤kpθwn,η⇒pθwn+1,Tη≤kpθwn,η.Taking limit asn⟶∞, we get pθTη,η=0 due to the fact that pθ is continuous. We also conclude pθη,η=0 and pθTη,Tη=0. Hence, η is a fixed point of T. To verify its uniqueness, assume that there exists another fixed point γ of T, that is, pθγ,γ=pθTγ,Tγ=pθTγ,γ. By (11), we have
(26)pθη,γ=pθTη,Tγ≤kpθη,γ.Sincek<1, pθη,γ=0. Also, we have pθγ,γ=0 and pθη,η=0. Thus, η=γ; that is, the fixed point of T is unique.Definition 13.
LetT:M⟶M be a self map and w0∈M. The set Ow0=w0,Tw0,T2w0⋯ is called the orbit of w0. The function H:M⟶ℝ is said to be T-orbitally lower semicontinuous at w∈M if wn⊂Ow0 and wn⟶w, which implies
(27)Hw≤liminfn⟶∞Hwn.Theorem 14.
LetM,pθ be a complete partial extended b-metric space and pθ represent a continuous functional. Consider a self map T:M⟶M and w0∈M such that
(28)pθTw,T2w≤kpθw,Tw,foreachw∈Ow0,
wherek∈0,1. Also for such w0∈M,
(29)limn⟶∞θwn,wn+1,wm<1k,form>n,where wn=Tnw0, n=1,2,3⋯. Then, Tnw0⟶η∈M. Further, η is a fixed point of T if Hw=pθw,Tw is T-orbitally lower semi continuous at η.Proof.
For a givenw0∈M, define an orbit Ow0 with w1=Tw0, w2=Tw1=TTw0=T2w0,⋯, wn=Tnw0,⋯. By the use of successive iteration of the inequality (28), we get
(30)pθTnw0,Tn+1w0=pθwn,wn+1≤kpθwn−1,wn≤k2pθwn−2,wn−1⋯≤knpθw0,w1.
Proceeding as in Theorem12, we can prove wn is a Cauchy sequence. As M is complete, then wn=Tnw0⟶η∈M. Since H is T-orbitally lower semi continuous at η∈M,
(31)pθη,Tη≤liminfn⟶∞pθTnw0,Tn+1w0,≤liminfn⟶∞knpθw0,w1=0.
Thus,pθη,Tη=0. Hence, η is a fixed point of T.Example 15.
TakeM=0,∞. Define pθ:M×M⟶ℝ+ by
(32)pθw1,w2=maxw1,w2
andθ:M×M×M⟶1,∞ by
(33)θw1,w2,w3=w1+w2+w3+1.
Then,pθ represents a complete partial extended b-metric on M.
LetT:M⟶M be defined by
(34)Tw=w2.
We have(35)pθTw,T2w=pθw2,w4=w2=kpθw,Tw.
For eachw∈M and Tnw=w/2n, we get
(36)limm,n⟶∞θTnw,Tn+1w,Tmw=limm,n⟶∞w2m+w2n+w2n+1+1<2.
Since all the conditions of Theorem14 are satisfied, the mapping T has a fixed point.


## 3. Partial Hausdorff Distance via the Extendedb-Metric Space

Shukla and Panicker [10] introduced the notion of a partial Hausdorff metric. Baig and Pathak [29] has proved well-known Nadler’s fixed point theorem for multivalued mappings on weak partial metric spaces. This is further used by Kanwal et al. [30] for establishing a fixed point result on weak partial b-metric spaces. In this section, we establish a fixed point result for set-valued mappings on partial new extended b-metric spaces. We first give some requisite definitions. Let M,pθ be a partial extended b-metric space. Denote by CBpθM the collection of all nonempty bounded and closed subsets of M with respect to the partial extended b-metric pθ. For P,Q∈M,pθ and w∈M, define the following:
(37)pθw,Q=infpθw,qq∈Q,ΩpθP,Q=suppθp,Qp∈P,ΩpθQ,P=suppθq,Pq∈Q.Remark 16.
LetM,pθ be a partial extended b-metric space and B be any nonempty set in M, then
(38)w∈Q¯ifandonlyifpθw,Q=pθw,w,
whereQ¯ is the closure of Q. Further, Q is closed in M,pθ if and only if Q=Q¯.Definition 17.
LetM,pθ be a partial extended b-metric space. For P,Q∈CBpθM, define
(39)HpθP,Q=maxΩpθQ,P,ΩpθP,Q.
The mappingHpθ:CBpθM×CBpθM⟶0,∞ is called a partial Hausdorff distance function induced by pθ.Corollary 18.
IfM,pθ is partial extended b-metric space and P,Q∈CBpθM, then
(40)HpθP,Q=0⇒P=Q.Lemma 19.
LetM,pθ be a partial extended b-metric space, P,Q∈CBpθM, and h>1. Then, for any w1∈P, there exists w2∈Q such that
(41)pθw1,w2≤hHpθP,Q.Theorem 20.
LetM,pθ be a complete partial extended b-metric space so that pθ is a continuous functional. If T:M⟶CBpθM is a multivalued mapping such that for all w1,w2∈M, we have
(42)HpθTw1,Tw2≤kpθw1,w2,
wherek∈0,1. Also, there exists w0∈M such that for every wn with wn+1∈Twn, we have limn⟶∞θwn,wn+1,wm′<1/k,∀m′>n. Then, T has a fixed point.Proof.
Givenw0∈M and w1∈Tw0, recall that h=1/k>1, so by use of Lemma 19, we have w2∈Tw1 with
(43)pθw1,w2≤1kHpθTw0,Tw1.
By using (42), we get
(44)HpθTw0,Tw1≤kpθw0,w1,⇒pθw1,w2≤kpθw0,w1.
Forw2∈Tw1, by Lemma 19, we have w3∈Tw2 with
(45)pθw2,w3≤1kHpθTw1,Tw2.
Again by (42), we get
(46)HpθTw1,Tw2≤kpθw1,w2,⇒pθw2,w3≤kpθw1,w2.
Thus,(47)pθw2,w3≤k2pθw1,w2.
Continuing this process, we have a sequencewn in M such that wn∈Twn−1 and
(48)pθwn,wn+1≤kpθwn−1,wn∀n∈ℕ.
Also,(49)pθwn,wn+1≤knpθw0,w1∀n∈ℕ.
By using (PE4) of the partial extendedb-metric space, we have (for m>n)
(50)pθwn,wm≤θwn,wn+1,wmpθwn,wn+1+pθwn+1,wm−pθwn+1,wn+1=θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmpθwn+1,wm−pθwn+1,wn+1≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmpθwn+1,wm≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+pθwn+2,wm−pθwn+2,wn+2≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+pθwn+2,wm.
Continuing this process, we get(51)pθwn,wm≤θwn,wn+1,wmpθwn,wn+1+θwn,wn+1,wmθwn+1,wn+2,wmpθwn+1,wn+2+θwn,wn+1,wmθwn+1,wn+2,wmθwn+2,wn+3,wmpθwn+2,wn+3+⋯+θwn,wn+1,wmθwn+1,wn+2,wm⋯θwm−2,wm−1,wmpθwm−1,wm.
By using (49), one writes
(52)pθwn,wm≤θwn,wn+1,wmknpθw0,w1+θwn,wn+1,wmθwn+1,wn+2,wmkn+1pθw0,w1+⋯+θwn,wn+1,wmθwn+1,wn+2,wm⋯θwm−2,wm−1,wmkn+m−1pθw0,w1.
This further gives(53)pθwn,wm≤pθw0,w1θw1,w2,wmθw2,w3,wm⋯θwn,wn+1,wmkn+θw1,w2,wmθw2,w3,wm⋯;θwn,wn+1,wmθwn+1,wn+2,wmkn+1+⋯+θw1,w2,wmθw2,w3,wm⋯θwn,wn+1,wm⋯θwm−2,wm−1,wmkm−1.
Consider(54)Sn=pθw0,w1∑j=1nkj∏i=1jθwi,wi+1,wm,
Form>n, we conclude that
(55)pθwn,wm≤pθw0,w1Sm−1−Sn−1.
Consider the series(56)pθw0,w1∑n=1∞kn∏i=1nθwi,wi+1,wm.
Substitute(57)vn=kn∏i=1nθwi,wi+1,wm.
Then, we have(58)limn⟶∞vn+1vn=kθwn+1,wn+2,wm<1.
Therefore, by ratio test, the series is convergent. So we conclude thatlimm,n⟶∞Sm−1−Sn−1=0. By using (55), wn is a Cauchy sequence in M. Since M is complete, wn⟶w∗∈M with respect to pθ, that is,
(59)limn⟶∞pθwn,w∗=pθw∗,w∗=limm,n⟶∞pθwn,wm=0.
By (42), we get HpθTwn,Tw∗≤kpθwn,w∗. Therefore, limn⟶∞HpθTwn,Tw∗=0. Now, wn+1∈Twn gives that
(60)pθwn+1,Tw∗≤ΩpθTwn,Tw∗≤HpθTwn,Tw∗.
By using continuity ofpθ, we get
(61)pθw∗,Tw∗=limn⟶∞pθwn+1,Tw∗=0.
By (59), pθw∗,w∗=0. Thus we have
(62)pθw∗,w∗=pθw∗,Tw∗=0,which implies w∗∈Tw∗.
The following corollary follows immediately from Theorem20 by setting θw1,w2,w3=b for all w1,w2,w3∈M.Corollary 21.
LetM,pb be a complete partial b-metric space and pb be a continuous functional. Suppose that T:M⟶CBpbM is a multivalued mapping such that for all w1,w2∈M, we have
(63)HpbTw1,Tw2≤kpbw1,w2,
wherek∈0,1 and b<1/k. Then, T has a fixed point.Remark 22.
By settingθw1,w2,w3=1 for all w1,w2,w3, the main result of [16] becomes a special case of Theorem 20.


## 4. Conclusion

In this article, the notion of a partial extendedb-metric space was introduced with suitable examples. Fixed point results for single-valued mappings endowed on partial extended b-metric spaces are established. These results are extensions of many existing results in literature.The idea of a partial Hausdorff distance in partial extendedb-metric spaces was presented. Furthermore, some fixed point theorems involving multivalued mappings have been proved. In the future, one can prove the above results under the platform of dislocated partial extended b-metric spaces.



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