# On New Type ofF-Contractive Mapping for Quasipartial b-Metric Spaces and Some Results of Fixed-Point Theorem and Application

Authors: A. M. Zidan; Asma Al Rwaily
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we introduce the concept of new type ofF-contractive type for quasipartial b-metric spaces and some definitions and lemmas. Also, we will prove a new fixed-point theorem in quasipartial b-metric spaces for F-contractive type mappings. In addition, we give an application which illustrates a situation when Banach’s fixed-point theorem for complete quasipartial b-metric spaces cannot be applied, while the conditions of our theorem are satisfying.


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## Body

## 1. Introduction

In 1922, Banach [1] introduced the most significant outcome in spaces identified with metric spaces. This principle has been used in proving fixed-point theorems in different spaces and has been used successfully in proving the existence of the solutions of some nonlinear differential equations, integral equations, nonlinear integral differential equations, etc. A great deal of extensions of Banach’s principle have been done, for the most part by generalizing the contraction operator and also by expanding the necessity of completeness or even both (see, for example, [2–4]). The concept of partial metric was introduced by Matthews [5] in 1994.Furthermore, Matthews introduced the concept of metric spaces via putting self-distances which are not necessarily equal to zero. In the year 1994, Czerwik [6] introduced the notion of a b-metric space as another generalization of metric space. Karapinar et al. [7] defined the concept of quasipartial metric space. This notion is a unification of both quasi metric spaces and partial metric spaces under the same substructure. In the following, we present some results that are related to our paper and show the motivation of our concern.Wardowski [8] in 2012 Introduced the definition of contraction called F-contraction. After that, in 2017, Górnicki [9] called F-contraction to be Wardowski contraction and also extended fixed-point theorems established by Ran and Reurings [10] and Nieto and Rodríguez-López [11] to CJM contractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Recently, in 2019, Goswami et al. [12] defined F-contractive type mappings in b-metric space; they proved fixed-point theorems with suitable examples. After that, they had given F-expanding type mappings and a fixed-point result was obtained. In this paper, we introduce the concept of new type of F-contractive type for quasipartial b-metric space and some definitions and lemmas. Also, we state and prove some fixed-point results in new concept and generalize a notion of F-contractive type mappings in quasipartial b-metric spaces with application.


## 2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.Definition 1.
(see [7]). Let ξ be a set. A function qp:ξ×ξ⟶0,∞ is said to be a quasipartial metric on a nonempty set ξ, such that for all w,t,r∈ξ, the following conditions hold:(1)qp1Ifqpw,w=qpw,t=qpt,t,thenw=t,qp2qpw,w≤qpw,t,qp3qpw,t=qpt,w,qp4qpw,t+qpr,r≤qpw,r+qpr,tfor allw,t,r∈ξ.
A quasipartial metric space is a pairξ,qp such that ξ is a nonempty set and qp is a quasipartial metric on ξ. Then, dqpw,t=qpw,t+qpt,w−qpw,w−qpt,t is a metric on ξ.Lemma 1.
For a quasipartial metricqp on ξ,(2)pqpw,t=12qpw,t+qpt,e,∀w,t∈ξ,and then pqp is a partial metric on ξ.In 2015, Gupta and Gautam [13] introduced the concept of quasipartial b-metric space, lemma, example, and some other definitions.Definition 2.
Letξ be a set. A function qbp:ξ×ξ⟶0,∞ is said to be a quasipartial b-metric on a nonempty set ξ, such that for all w,t,r∈ξ, the following conditions hold:(3)qbp1Ifqbpw,w=qbpw,t=qbpt,t⟹w=t,qbp2qbpw,w≤qbpw,t,qbp3qbpw,t=qbpt,w,qbp4qbpw,t+qbpr,r≤ηqbpw,r+qbpr,t,for allw,t,r∈ξ.
A quasipartialb-metric space is a pair ξ,qbp such that ξ is a nonempty set and ξ,qbp is a quasipartial b-metric space on ξ. The number η is called the coefficient of ξ,qbp. Further, qbp is a quasipartial b-metric space on the set ξ. Then,(4)dqbpw,t=qpw,t+qpt,w−qpw,w−qpt,tis a metric on ξ.Lemma 2.
Everyξ,qp is a ξ,qbp. But the opposite does not have to be true.Example 1.
Letξ=0,1; then, qbpw,t=w−t+w. Hence, ξ,qbp is quasipartial b-metric space with η>0.Definition 3.
Letξ,qbp be a quasipartial b-metric space which satisfies the following:(i)
A sequencewn⊂ξ converges to G2w∈ξ iff(5)qbpw,w=limn⟶∞qbpw,wn=limn⟶∞qbpwn,w.(ii)
A sequencewn⊂ξ is said to be a Cauchy sequence iff(6)limn,m⟶∞qbpwn,wm,limn,m⟶∞qbpwm,wn.exist and are finite.(iii)
ξ,qbp is called complete if every Cauchy sequence wn⊂ξ converges with respect to τqbp to a point w∈ξ so that(7)qbpw,w=limn,m⟶∞qbpwn,wm=limn,m⟶∞qbpwm,wn.(iv)
A mappingΛ:ξ⟶ξ is called continuous at w0⊂ξ if, for every ε>0, there is δ>0, so that ΛBw−0,δ⊂BΛw−0,ε.Remark 1.
Letξ,qbp be a quasipartial b-metric space. Then, the following holds:(i)
Ifqbpw,t=0⟹w=t.(ii)
Ifw≠t⟹qbpw,t>0 and qbpt,w>0.Definition 4.
(see [8]). Let F:0,∞⟶R be a map satisfying the following conditions:F1 F is strictly increasing.F2limn⟶∞σn=0⟺limn⟶∞Fσn=−∞, for each sequence σnn∈N of positive numbers.F3 If there exists l∈0,1, then limσ⟶0+σlFσ=0.Letξ,d be a metric space. A mapping Λ:ξ⟶ξ is called a Wardowski F-contraction if there exists τ>0 such that(8)dΛw,Λt>0⟹τ+FdΛw,Λt≤Fdw,t,for allw,t∈ξ.After that, in 2015, Cosentino et al. [14] extended the notion of quasipartial b-metric space with the following condition added to these concepts.F4 Let η≥1 be a real number. For each sequence σnn∈N of positive numbers,(9)τ+Fησn≥Fσn−1⟹τ+Fηnσn≥Fηn−1σn−1,for all n∈N and some τ>0. Alsulami et al. in 2015, [15] defined a generalized F-Suzuki type in a b-metric space ξ,d,η as a mapping Λ:ξ⟶ξ; then, there exists τ>0 with w=t, such that(10)2ηdw,Λw<dw,t⟹τ+FdΛw,Λt≤σFdw,t+δFdw,Λw+ρFdt,Λt,for all w,t∈ξ, since σ,δ∈0,1 and ρ∈0,1 under condition σ+δ+ρ=1 with σ and F satisfying conditions F1 and F2.Lemma 3 (see [16]).
Letξ,d,η be a b-metric space and wn be a convergent sequence in ξ with limn⟶∞wn=wl. Therefore, t∈ξ, and we have(11)η−1dw,t≤limn⟶∞infdwn,t≤limn⟶∞supdwn,t≤ηdw,t.The present paper aims to establish a similar type of result forF-contractive type mappings in a quasipartial b-metric space. In our results, we consider that a quasipartial b-metric is not a continuous functional.


## 3. Main Convergent Results

Lemma 4.
Letξ,qbp,η be a quasipartial b-metric space and wn be a convergent sequence in ξ with limn⟶∞wn=wl, for all t∈ξ; we have(12)1ηqbpw,t≤limn⟶∞infqbpwn,t≤limn⟶∞supqbpwn,t≤ηqbpw,t,(13)1ηqbpt,w≤limn⟶∞infqbpt,wn≤limn⟶∞supqbpt,wn≤ηqbpt,w.Proof.
We will apply twice the relaxed triangle inequalityqbp4; then, for every n∈N, we obtain(14)1ηqbpw,t≤qbpw,wn+qbpwn,t−qbpwn,wn,≤qbpw,wn+qbpwn,t,and then we have(15)1ηqbpw,t−qbpw,wn≤qbpwn,w+qbpw,t−qbpw,w,≤qbpwn,w+qbpw,t.
Therefore, we take on the left-hand side inequality and on the right-hand side inequality, and we obtain equation (12); in the same way, we get equation (13).Definition 5.
Letξ,qbp,η be a quasipartial b-metric space, and a mapping Λ:ξ⟶ξ is said to be an F-contractive type mapping if there exists τ>0 such that qbpw,Λw,qbpt,Λt≠0 and qbpΛw,w,qbpΛt,t≠0 imply(16)τ+FηqbpΛw,Λt≤13Fqbpw,t+Fqbpw,ΛW+Fqbpt,Λt,(17)τ+FηqbpΛt,Λw≤13Fqbpt,w+Fqbpt,Λt+Fqbpw,Λw.
Also, ifqbpw,Λw,qbpt,Λt=0 and qbpΛw,w,qbpΛt,t=0, we have the same equations ((16) and (17)) for all w,t∈ξ.Lemma 5.
Letξ,qbp,η be a quasipartial b-metric space and let ξ,dqbp,η be the corresponding b-metric space. Then, the following statements are equivalent:(i)
ξ,qbp,η is complete.(ii)
ξ,dqbp,η is complete.Moreover,(18)limn⟶∞dqbpz,xn=0⟺qbpz,z=limn⟶∞qbpz,xn=limn⟶∞qbpxn,xm,limn⟶∞dqbpxn,z=0⟺qbpz,z=limn⟶∞qbpxn,z=limn⟶∞qbpxm,xn,since z,xn,xn∈ξ, for all m,n∈N and m>n.Definition 6.
Letξ,qbp,η be a quasipartial b-metric space and Λ:ξ⟶ξ be a mapping; then, there exists τ>0 with w=t, for all w,t∈ξ, such that(19)2ηqbpw,Λw<dw,t⟹τ+FqbpΛw,Λt≤σFqbpw,t+δFqbpw,Λw+ρFqbpt,Λt,2ηqbpΛw,w<qbpt,w⟹τ+FqbpΛt,Λw≤σFqbpt,w+δFqbpΛw,w+ρFqbpΛt,t,qbpw,Λwqbpt,Λy=qbpΛw,wqbpΛt,t=0.
Hence, we have(20)τ+FqbpΛw,Λt≤σFqbpw,t+δFqbpw,Λw+ρFqbpt,Λt,⟹τ+FqbpΛt,Λw≤σFqbpt,w+δFqbpΛw,w+ρFqbpΛt,t.
Sinceσ,δ∈0,1 and ρ∈0,1 under conditionsince σ+δ+ρ=1 with σ, F satisfies each conditions between F−F4.
Now, we establish a new type of result forF-contractive type mappings in a quasipartial b-metric space by appealing to Lemma 4.Theorem 1.
Letξ,qbp,η be a complete quasipartial b-metric space and Λ:ξ⟶ξ be an F-contractive type mapping. Then, Λ satisfies a Picard operator.Proof.
We start our proof by supposing thatw0∈ξ is arbitrary (but fixed) and considering the sequence wn; since wn=Λwn, n∈N. Suppose that λn>0 for all n∈N and denote qbpwn,wn+1, qbpwn+1,wn by λn; then, we have(21)Fηqbpwn,wn+1=Fηλn(22)≤13Fqbpwn−1,wn+Fqbpwn−1,wn+Fqbpwn,wn+1−τ,Fηqbpwn+1,wn=Fηλn≤13Fqbpwn,wn−1+Fqbpwn,wn+1+Fqbpwn−1,wn−τ.
From (21), we can write(23)Fηλn≤Fηλn−1−32τ.
By conditionF4, we have(24)Fηnλn≤Fηn−1λn−1−32τ.
Then, by induction, we get(25)Fηnλn≤Fηn−1λn−1−32τ≤⋯≤Fλ0−32n.
By taking a limit asn⟶∞, we have(26)limn⟶∞Fηnλn=−∞⟹limn⟶∞ηnλn=0.
Therefore, from conditionF3, there exists l∈0,1 such that(27)limn⟶∞ηnλnlFηnλn=0.
Multiplication of (25) with ηnλnl yields(28)0≤ηnλnlFηnλn+32nηnλnlτ≤ηnλnlFλ0.
By taking a limit asn⟶∞, we have(29)limn⟶∞nηnλnl=0.
Now, we can show thatwn is a Cauchy sequence. Since ξ,qbp,η is a complete quasipartial b-metric space, there exists r∈ξ such that(30)limn⟶∞wn=r.
By applying Lemma4, we have(31)limn⟶∞qbpr,wn=limn⟶∞supqbpr,wn≤ηqbpr,r≥0.
On the other hand, letm and n be natural numbers with m>n; then, using qp4, we get(32)qbpwn,wn+2≤ηqbpwn,wn+1+qbpwn+1,wn+2−qbpwn+1,wn+1,≤ηqbpwn,wn+1+qbpwn+1,wn+2,qbpwn,wn+3≤ηqbpwn,wn+2+qbpwn+2,wn+3−qbpwn+2,wn+2,≤ηηqbpwn,wn+1+qbpwn+1,wn+2+ηqbpwn+2,wn+3,≤η2qbpwn,wn+1+η2qbpwn+1,wn+2+ηqbpwn+2,wn+3.
On generalization, we have(33)qbpwn,wm≤ηm−n−1qbpwn,wn+1+qbpwn+1,wn+2+ηm−n−2qbpwn+2,wn+3+⋯+ηqbpwm−1,wm,=∑i=n+1m−1ηm−iqbpwi,wi+1+ηm−n−1qbpwn,wn+1,=∑i=nm−1ηm−iqbpwi,wi+1−ηm−nqbpwn,wn+11−1η,≤∑i=nm−1ηm−iqbpwi,wi+1,≤∑i=nm−1ηm−i·liqbpw0,w1,≤∑i=nm−1lηiηmqbpw0,w1.
By taking limit asn⟶∞ in (33), since m holding fixed, we have(34)limn⟶∞qbpwn,wm≤0.
But, we know(35)limn⟶∞qbpwn,wm≥0.
Thus,(36)limn⟶∞qbpwn,wm=0.
Similarly, we can show that(37)limn⟶∞qbpwm,wn=0.
Due to (18) in Lemma 5 and (36), we get(38)limn⟶∞dqbpr,wn=0⟹qbpr,r=limn⟶∞supqbpr,wn=limn⟶∞supqbpwn,wm=0.
Also, using (16), we have for all n⟶N,(39)τ+FηqbpΛr,Λwn≤13Fqbpr,wn+Fqbpr,Λr+Fqbpwn,Λwn,since. Then, by taking a limit as n⟶∞ in last equation, we get(40)τ+limn⟶∞FηqbpΛr,Λwn≤−∞.
This implies(41)limn⟶∞qbpΛr,wn+1=limn⟶∞qbpΛr,Λwn=0.
Since the sequencewn converges to both r and Λr, we have Λr=r. Finally to show the uniqueness of the fixed point, let x be another fixed point of Λ with r≠x; then, from (16),(42)τ+FηqbpΛr,Λx≤13Fqbpr,x+Fqbpr,Λr+Fqbpx,Λx,or(43)FηqbpΛr,Λx<FqbpΛr,Λx.
This is a contradiction. Hence, we proved the theorem.


## 4. Application in Fixed-Point Theorem

The example illustrates a situation when Banach’s fixed-point theorem for complete quasipartialb-metric spaces cannot be applied, while the conditions of Theorem 1 are satisfied.Example 2.
Letξ,qbp,η be a quasipartial b-metric space, where(44)qbpw,t=minx−y+x,2is complete with η=1. Suppose the mapping Λ:ξ⟶ξ is defined by(45)Λw=12:0≤w<1,0:w=1,12−1w:w≥2.
It is clear thatΛ is not continuous at w=1. Now, for w,t≥2 with w≠t, we have(46)qbpΛw,Λt3=min12−1w−12+1t+12−1w,23,=min−1w+1t+12−1w,23<1,qbpw,tqbpw,Λwqbpt,Λt=minw−t+w,2minw−12+1w+w,2mint−12+1t+t,2=8.
Again, ifw∈0,1 and t≥2 or conversely, we get(47)qbpΛw,Λt3=qbp12,12−1y3=min12−12+1t+12,23,=min1t+123<1,qbpw,tqbpw,Λwqbpt,Λt=minw−t+w,2minw−12+w,2mint−12+1t+t,2,=2w−12+w2=4w−12+w≥2.
Also, forw∈0,1,(48)qbpΛw,Λ13=qbp12,03=min12−0+12,23=164,qbpw,1qbpw,Λwqbp1,Λ1=minw−1+w,2minw−12+w,2min1−0+1,2≥2.
After that, forw≥2, we get(49)qbpΛw,Λ13=qbp12−1w,03=min12−1w−0+12−1w,23=164,qbpw,1qbpw,Λwqbp1,Λ1=minw−1+w,2minw−12+w,2min1−0+1,2≥2.
Thus, from the example in [12], if qbpw,Λwqbpt,Λt≠0 is satisfied for e−3τ=1/16. In addition,(50)qbpΛ1,Λ123=qbp0,123=min0−12+0,23=18,qbp1,12qbp1,Λ12qbpΛ1,12=min1−12+1,2min1−0+1,2min0−12+0,2,=32212=64=128.
Finally, forw≥2,(51)qbpΛw,Λ123=min12−1x−12+12−1x,23≤1,qbp1,12qbp1,Λ12qbpΛ1,12=min1−12+1,2min1−0+1,2min0−12+0,2,=32212=64=128.
Hence, Λ is an F-contractive type mapping and, by inspection, we see that Λ has a unique fixed point w (1/2).


## 5. Concluding Remarks and Future Works

We introduced the notion of type ofF-contractive type mapping for quasipartial b-metric space and some definitions and lemmas. Also, we proved the fixed-point theorem for F-contractive type mappings in a quasipartial b-metric space with application. In fact, applications of F-contractive type mapping are not limited just to the topic mentioned above.As future work, we are, joining with others, planning to(1)
Extend the results given in the above sections to quasipartialb-symmetric space.(2)
Study the applications of our results to the mathematical modelling of quality.Remark 2.
It will be interesting to find more applications to our current paper in other fields (see [17–31]).



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