# New Contributions in GeneralizationS-Metric Spaces to S∗p-Partial Metric Spaces with Some Results in Common Fixed Point Theorems

Authors: Asma Al Rwaily; A. M. Zidan
Journal: Complexity
 (2021)

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## Abstract

In this paper, we introduce the notion ofS∗p-partial metric spaces which is a generalization of S-metric spaces and partial-metric spaces. Also, we give some of the topological properties that are important in knowing the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorems in this spaces.


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## Body

## 1. Introduction

There are a large number of generalizations of the Banach contraction principle with different use forms of contractual terms in a variety of generalized metric spaces. Some of the circulares are obtained through contractual terms expressed in reasonable terms. Shahkoohi and Razani [1] introduced a new class of rational Geraghty contractive mappings in the setup of b-metric spaces and proved some fixed point for such mappings in ordered b-metric spaces which are investigated. One such generalization is a partial-metric space which was introduced by Matthews [2] in 1994, since, in this space, a self-distance of an arbitrary point need not be equal zero. Also, in 2007, Sedghi et al. [3] introduced D∗-metric spaces which are generalization of the notion of D-metric spaces introduced by Dhage [4]. Also, Sedghi et al. proved some basic properties in D∗-metric spaces. In 2012, Sedghi et al. [5] introduced S-metric spaces and gave some properties and fixed point theorem for a complete S-metric space.Recently, in 2019, Z. Mustafa et al. [6] introduced the structure of Sb-metric spaces which generalization of S-metric space and gave some properties and fixed point theorem for these spaces.


## 2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.Definition 1.
(see [2]). Let ϒ be a set. A function p:ϒ×ϒ⟶0,∞ is said to be a partial metric spaces on a nonempty set ϒ; for all w,t,r∈ϒ, the following conditions are held:P10≤Pw,w≥Pw,t (nonnegativity and small self-distances)P2 If Pw,w=Pw,t=Pt,t, then w=t (indistancy implies equality)P3Pw,t=Pt,w (symmetry)P4Pw,t≤Pw,r+Pr,t−Pr,r (triangularity)
Hence, the functionP is called an P-metric on ϒ and the pair ϒ,P is called a partial metric space.Example 1.
(see [7, 8]). Let ϒ=0,∞ and p:ϒ×ϒ⟶0,∞ given by pw,t=maxw,t, for all w,t∈Υ; then, ϒ,p is partial metric space.Definition 2.
Letϒ,p be a partial metric space; then, one has the following:(i)
A sequencewn in ϒ converges to a point w∈ϒ if and only if limn⟶∞pwn,w=pw,w(ii)
A sequencewn in ϒ is Cauchy if limn,m⟶∞pwn,wm exists and is finite(iii)
ϒ is complete if every Cauchy sequence wn in ϒ converges to a point w∈ϒ, that is, limn,m⟶∞pwn,wm=pw,wLemma 1 (see [2]).
Letϒ,p be a partial metric space. Then, ϒ is complete if and only if the metric space ϒ,ps is complete. Furthermore, limn⟶∞pswn,r=0 if and only if(1)pr,rlimn⟶∞pswn,r=limn,m⟶∞pswn,wm.In [9], Romaguera introduced the definitions of 0-complete partial metric spaces and 0-Cauchy sequence. Also, he obtained a characterization of completeness for partial metric space using the notion of 0-completeness.Mustafa and Sims [10] extended Banach principle and introduced the notation of generalized metric spaces, so-called G-metric spaces, as follows.Definition 3.
Letϒ be a set. A function G:ϒ×ϒ×ϒ⟶0,∞ is said to be a generalized metric spaces on a nonempty set ϒ, for all w,t,r,η∈ϒ, and the following conditions hold:G1Gw,t,r=0⇔w=t=rG2Gw,t,r>0 and w≠tG3Gw,w,t≤Gw,t,r, and t≠rG4Gw,t,r=Gw,r,t=Gt,r,w=⋯(symmetry in all three variables)G5Gw,t,r≤Gw,η,η+Gη,r,w
Hence, the functionG is called an G-metric on ϒ and the pair ϒ,G is called an generalized metric space.To find some basic properties and examples ofG-metric spaces in Mustafa and Sims [10].After that, Zand and Nezhad [11] introduced definition of Gp-metric space by generalization and unification of both partial metric space and G-metric space as follows.Definition 4.
Letϒ be a set. A function Gp:ϒ×ϒ×ϒ⟶0,∞ is said to be a generalized partial metric spaces on a nonempty set ϒ, for all w,t,r,η∈ϒ, and the following conditions hold:G1pw=t=r if Gpw,w,w=Gpt,t,t=Gpr,r,rG2p0≤Gpw,w,w≤Gpw,w,t≤Gpw,t,yG3pGpw,t,r=Gpw,r,tGpt,r,w=⋯(symmetry in all three variables)G4pGpw,t,r≤Gpw,η,η+Gpη,r,w−Gpη,η,η
Hence, the functionGp is called an Gp-metric on ϒ and the pair ϒ,Gp is called an generalized partial metric space.Example 2.
(see [11]). Let ϒ=0,∞ and Gpw,t,r=maxw,t,r for all w,t,r∈Υ, then ϒ,Gp is a Gp-metric space. Also, ϒ,Gp is not a G-metric space.Proposition 1.
Letϒ,Gp is a Gp-metric space, then, for any w,t,r∈ϒ and η∈ϒ, it follows that(i)
Gpw,t,r≤Gpw,w,t+Gpw,w,r−Gpw,w,w(ii)
Gpw,t,t≤2Gpw,w,t−Gpw,w,w(iii)
Gpw,t,r≤Gpw,η,η+Gpt,η,η+Gpr,η,η−2Gpη,η,ηShaban Sedghi et al. [3] introduced the notion of D∗-metric which is a modification of the definition of D-metric introduced by Dhage [4, 12], and they proved some basic properties in D∗-metric spaces.Definition 5.
(see [3]). Let ϒ be a set. A function D∗:ϒ×ϒ×ϒ⟶0,∞ is said to be a D∗-metric spaces on a nonempty set ϒ, for all w,t,r∈ϒ, and the following conditions hold:D1∗D∗w,t,r≥0D2∗D∗w,t,r⇔w=t=wD3∗D∗w,t,r=D∗ℵw,t,r (where ℵ is a permutation function)
Hence, the functionD∗ is called an D∗-metric on ϒ, and the pair ϒ,D∗ is called an generalized partial metric space.Example 3.
(see [3]). Let ϒ=ℝ. Denote D∗w,t,r=w−t+t−r+r−w, for all w,t,r∈ℝ. Since(2)BD∗w,λ=t∈Υ:D∗w,t,t<λ.
Hence,(3)BD∗1,2=t∈ℝ:D∗1,t,t<2=t∈ℝ:t−1+t−1<2=t∈ℝ:t−1<1=0,2.On other way, Sedghi et al. introducedS-metric spaces as follows.Definition 6.
(see [5]). Let ϒ be a set. A function S:ϒ3⟶0,∞ is said to be a S-metric spaces on a nonempty set ϒ, for all w,t,r,η∈ϒ, and the following conditions hold:S1Sw,t,r>0S2Sw,t,r⇔w=t=rS3Sw,t,r≤Sw,w,η+St,t,η+Sr,r,η for all w,t,r,η∈ϒ (rectangle inequality)
Hence, the functionS is called an S-metric on ϒ and the pair ϒ,S is called an S-partial metric space.Example 4.
(see [5]). Let ϒ=ℝn and ⋅ a norm on ϒ; then,(4)Sw,t,r=t+r−2w+t−ris a S-metric on ϒ.Remark 1.
EveryD∗-metric is S-metric, but in general, the converse is not true, see the following example.Example 5.
Letϒ=ℝn and ⋅ a norm on ϒ; then, Sw,t,r=t+r−2w+t−r is S-metric on ϒ, but it is not D∗-metric because it is not symmetric.Example 6.
Letϒ=ℝn and d is an ordinary metric on ϒ. Therefore, Sw,t,r=dw,t+dw,r+dt,r is an S-metric on ϒ. Then, if the points w,t,andr are connected by a line, hence, we have a triangle and if we choose a point a mediating this triangle, then the inequality(5)Sw,t,r≤sw,w,η+st,t,η+r,r,η,for all w,t,r,η∈ϒ holds.Definition 7.
(see [5]). Let ϒ,S be a S-metric space. Then, for w0∈Υ and λ>0, the S-open ball and S-closed ball of radius λ with centered at w0 are(6)BSw0,λ=t∈ϒ:Sw0,w0,t<λ,B¯Sw0,λ=t∈ϒ:Sw0,w0,t≤λ.Proposition 2 (see [5]).
Letϒ,S be a S-metric space and A⊂ϒ:(1)
If, for everyw0∈A, there exists λ>0 such that BSw0,λ, then the subset A is called an open subset of ϒ.(2)
A subsetA of ϒ is said to be S-bounded if there exists λ>0 such that Sw,w,t<λ, for all w,t∈A.(3)
A sequencewn in ϒ converges to w if and only if Swn,wn,w⟶0 as n⟶∞. That is, for each λ>0, there exists n0∈N such that, for all n,n0, Swn,wn,w<λ and we denote this by limn⟶∞wn=w.Lemma 2 (see [5]).
Letϒ,S be an S-metric space; then, the following satisfy:(1)
Ifλ>0 and w∈ϒ, then the ball BSw0,λ is an open subset of ϒ(2)
If the sequencewn in ϒ converges to w, then w is unique(3)
If the sequencewn in ϒ converges to w, then wn is a Cauchy sequenceDefinition 8.
(see [10]). Two classes of the following mappings are(1)
Π=π:π:0,∞⟶0,∞ is nondecreasing, continuous, and π−10=0(2)
Γ=γ:γ:0,∞⟶0,∞ is nondecreasing, lower semicontinuous, and π−10=0Definition 9.
Letϒ,≺¯ be a partially ordered set. Two maps f,g:ϒ×ϒ are said to be weakly increasing if fw≺¯gfw and gw≺¯fgw, for all w∈ϒ.Barakat and Zidan [13] proved a common fixed point theorem for weak contractive maps by using the concept of Gp-metric spaces.Theorem 1.
Letϒ,≺¯ be a partially ordered set with f and g as weakly increasing self-mapping on a complete Gp-partial metric space. Suppose that there exist π∈Π and γ∈Γ such that(7)πGpfw,fw,gt≤πMw,w,t−γMt,t,w,for all w,t∈ϒ, where(8)Mw,w,t=ζ1Gpw,w,t+ζ2Gpw,w,fw+ζ3Gpt,t,gt+ζ4Gpw,w,gt+Gpt,t,fw,where ζ1>0, for i=1,2,3,4, with ζ1+ζ2+ζ3+ζ4≥1.Assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈ϒ implies wn≺¯r for all n∈ℕ(b)
f or g is continuousTherefore, the mapsf or g have a common fixed point.


## 3. S∗p-Partial Metric Spaces and Some Properties

We first introduce the concept of aS∗p-partial metric space or S∗P.Definition 10.
Letϒ be a set. A function S∗P:ϒ3⟶0,∞ is said to be a S∗p-partial metric space on a nonempty set ϒ, for all w,t,r,η∈ϒ; the following conditions hold:S1∗PS∗Pw,t,r=S∗Pw,w,w=S∗Pt,t,t=S∗Pr,r,r; then, w=t=rS2∗PS∗Pw,t,r≤S∗Pw,w,η+S∗Pt,t,η+S∗Pr,r,η−S∗Pη,η,η, for all w,t,r,η∈ϒ (rectangle inequality)
Hence, the functionS∗P is called an S∗p-partial metric on ϒ and the pair ϒ,S∗P is called an S∗-partial metric space.Example 7.
Letϒ=ℝ and ⋅ a norm on ϒ; then, we have(9)S∗Pw,t,r=maxw,t,r+w−r+t−w,which is a S-partial metric on ϒ.Remark 2.
From Example7, we get every S-metric is S∗P-metric, but the converse is not true at all, see the following example.Definition 11.
Letϒ,S∗P be a S∗p-partial metric space. Then, for w0∈ϒ and λ>0, the S∗P-open ball and S∗P-open closed of radius λ with centered at w0 are(10)BS∗Pw0,λ=t∈ϒ:S∗Pw0,w0,t<λ+S∗Pw0,w0,w0−S∗Pt,t,w0,B¯S∗Pw0,λ=t∈ϒ:S∗Pw0,w0,t≤λ+S∗Pw0,w0,w0−S∗Pt,t,w0.Proposition 3.
Letϒ,S∗P be a S∗p-partial metric space. Then, for w0∈ϒ and λ>0, the following statements are satisfied:(1)
IfS∗Pw0,w0,w≤λ+S∗Pw0,w0,w0−S∗Pw,w,w0 and S∗Pw0,w0,t≤λ+S∗Pw0,w0,w0−S∗Pt,t,w0, then w,t∈BS∗Pw0,λ(2)
Ift∈BS∗Pw0,λ, then their exist ρ>0, such that BS∗Pt,ρ⊆BS∗Pw0,λProof.
(1)
The proof is straightforward.(2)
Lett∈BS∗Pw0,λ; then, we have(11)S∗Pw0,w0,t≤λ+S∗Pw0,w0,w0−S∗Pt,t,w0.
Also, we suppose(12)ρ=λ+S∗Pw0,w0,w0−S∗Pr,r,w0−2S∗Pw0,w0,t, and r∈BS∗Pt,ρ; then, we obtain(13)S∗Pt,t,r≤ρ+S∗Pt,t,t−S∗Pr,r,t.
Therefore,t∈BS∗Pw0,λ, and so (2) holds.(14)S∗Pw0,w0,r≤S∗Pw0,w0,t+S∗Pw0,w0,t+S∗Pr,r,t−S∗Pt,t,t≤2S∗Pw0,w0,t+S∗Pr,r,t+λ+S∗Pw0,w0,w0−S∗Pr,r,w0−2S∗Pw0,w0,t−S∗Pt,t,t≤λ+S∗Pw0,w0,w0−S∗Pr,r,t.
Hence,r∈BS∗Pw0,λ.Definition 12.
Letϒ,S∗P be a S∗p-partial metric space and wn a sequence in ϒ. A point w∈ϒ is said to be the limit of the sequence if(15)limn,m⟶∞S∗Pw,wn,wm=S∗Pw,w,w.
Hence, the sequencewn is S∗P-convergent to w.
Therefore, ifwn⟶w in a S-partial metric space ϒ,S∗P, then, for any ℏ>O, there exists ℑ∈N such that, for all n,m∈ℑ, we have(16)S∗Pw,wn,wm−S∗Pw,w,w<ℏ.Definition 13.
AS∗p-partial metric space ϒ,S∗P is called a S∗p-partial asymmetric space if(17)S∗Pw,w,t=S∗Pt,t,w.Lemma 3.
ϒ,S∗P be a S∗P-partial metric space. If the sequence wn in ϒ converges to w. Therefore, we get w is unique.Proof.
Letwn converges to w and w. Therefore, for each λ>0, there exist n1,n2∈N; then, we have(18)n≥n1⇒S∗Pwn,wn,w<λ3,n≥n2⇒S∗Pwn,wn,t<λ3.
If setn0=maxn1,n2, then, for every n≥n0, we have a third condition of S∗P-partial metric:(19)S∗Pw,w,t≤S∗Pw,w,wn+S∗Pw,w,wn+S∗Pt,t,wn−S∗Pwn,wn,wn≤2S∗Pw,w,wn+S∗Pt,t,wn−S∗Pwn,wn,wn≤2S∗Pw,w,wn+S∗Pt,t,wn<2λ3+λ3=λ.
Hence,S∗Pw,w,t=0⇒w=t, but the converse is not necessarily true.Definition 14.
Letϒ,S∗P be a S∗p-partial metric space. Then, for a sequence wn⊆ϒ and a point w∈Υ, the following are equivalent:(1)
wn is a S∗P convergent to w(2)
limn,m⟶∞S∗Pwn,wn,w=S∗Pw,w,wProposition 4.
Letϒ,S∗P be a S∗p-partial metric space. If we get a sequence wn∈ϒ which is called a Cauchy sequence if, for each λ>0, there exists n0∈N such that(20)S∗Pwn,wn,wm<λ+S∗Pw,w,w,Proof.
for eachn,m≥0, if the sequence wninϒ converges to w∈ϒ.Definition 15.
Letϒ,S∗P be a S∗p-partial metric space. Then, ϒ,S∗P is said to be complete if every Cauchy sequence is convergent.Definition 16.
Letϒ,S∗P, ϒ1,S1∗P, and ϒ2,S2∗P be two S∗-partial metric space. Also, we suppose a function(21)F:ϒ1,S1∗P⟶ϒ2,S2∗P,and then, F is said to be S∗P-continuous at a point; x∈ϒ1 if and only if, for given ℏ>O, there exists λ>0 such that w,t∈ϒ1 and the inequality(22)S1∗Px,w,t<ℏ+S1∗Pw,w,w.
This is an indication:(23)S2∗PFx,Fw,Ft<λ+S2∗PFw,Fw,Fw.
Hence, a functionF is S∗P-continuous on ϒ1 if and only if it is S∗P-continuous at all x∈ϒ1.


## 4. A Generalization of Common Point Theorems inS∗P-Partial Metric Spaces

Theorem 2.
Letϒ,≺¯ be a partially ordered set with f and g as weakly increasing self-mapping on a complete S∗P-partial metric space. Suppose that there exist π∈Π and γ∈Γ such that(24)πGpfw,fw,gt≤πMw,w,t−γMt,t,w,for all w,t∈ϒ, where(25)Mw,w,t=maxS∗Pw,w,t,S∗Pw,w,fw,S∗Pt,t,gt,S∗Pw,w,gt+S∗Pt,t,fw.Assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈ϒ implies wn≺¯r for all n∈ℕ(b)
f or g is continuousTherefore, the mapsf or g have a common fixed point.Proof.
Suppose thatℏ is a fixed point of f and S∗Pℏ,ℏ,gℏ>0. From (24), with w=t=ℏ, we have(26)πS∗Pℏ,ℏ,gℏ=πS∗Pfℏ,fℏ,gℏ≤πMℏ,ℏ,ℏ−γMℏ,ℏ,ℏ,where(27)Mℏ,ℏ,ℏ=maxS∗Pℏ,ℏ,ℏ,S∗Pℏ,ℏ,fℏ,S∗Pℏ,ℏ,gℏ,S∗Pℏ,ℏ,gℏ+S∗Pℏ,ℏ,fℏ=maxS∗Pℏ,ℏ,ℏ,S∗Pℏ,ℏ,ℏ,S∗Pℏ,ℏ,gℏ,S∗Pℏ,ℏ,gℏ+S∗Pℏ,ℏ,ℏ=maxS∗Pℏ,ℏ,ℏ,S∗Pℏ,ℏ,gℏ=S∗Pℏ,ℏ,gℏ.
Hence, we have(28)πS∗Pℏ,ℏ,gℏ=πS∗Pfℏ,fℏ,gℏ≤πS∗Pfℏ,fℏ,gℏ−γS∗Pfℏ,fℏ,gℏ⇒γS∗Pℏ,ℏ,gℏ≤0,which is a contradiction. Therefore, S∗Pfℏ,fℏ,gℏ=0. So, ℏ is common fixed point of f and g. Similarly, if ℏ is a fixed point of ℏ, then one can deduce that ℏ is also fixed point of f.
If we letw0 be an arbitrary point of ϒ with fw0=w0, then the proof is finished, so we assume that fw0≠w0.
Now, one can construct a sequencewnϒ as follows:(29)w1=fw0≺¯gfw0=gw1=w2w2=fw1≺¯gfw1=gw1=w3⋮wn≺¯wn+1.
Sincew2n and w2n+1 are comparable, then we may assume that S∗Pw2n,w2n,w2n+1, for every n∈N. If not, then w2n=w2n+1, for some n. For all those n, using (24), we obtain(30)πS∗Pw2n+1,w2n+1,w2n+2=πS∗Pfw2n,fw2n,gw2n+1≤πMw2n,w2n,w2n+1−γMw2n,w2n,gw2n+1,Mw2n,w2n,w2n+1=maxS∗Pw2n,w2n,w2n+1,S∗Pw2n,w2n,fw2n,S∗Pw2n+1,w2n+1,gw2n+1,S∗Pw2n,w2n,gw2n+1,S∗Pw2n+1,w2n+1,fw2n2=maxS∗Pw2n,w2n,w2n+1,S∗Pw2n,w2n,w2n+1,S∗Pw2n+1,w2n+1,w2n+2,S∗Pw2n,w2n,w2n+2,S∗Pw2n+1,w2n+1,w2n+12≤maxS∗Pw2n+1,w2n+1,w2n+2,12S∗Pw2n,w2n,w2n+1+S∗Pw2n,w2n,w2n+1+S∗Pw2n+1,w2n+1,w2n+1−S∗Pw2n+1,w2n+1,w2n+1+S∗Pw2n+1,w2n+1,w2n+1=S∗Pw2n+1,w2n+1,w2n+2.
Therefore,(31)πS∗Pw2n+1,w2n+1,w2n+2≤πS∗Pw2n+1,w2n+1,w2n+2−γS∗Pw2n+1,w2n+1,w2n+2.
It means thatγS∗Pw2n+1,w2n+1,w2n+2=0 and w2n+1=w2n+2. Following the similar arguments, we obtain w2n+2=w2n+3, and hence, w2n becomes a common fixed point of f and f.
By takingS∗Pw2n,w2n,w2n+1>0 for n=1,2,3,…., now, we consider(32)πS∗Pw2n+1,w2n+1,w2n+2=πS∗Pfw2n,fw2n,gw2n+1≤πMw2n,w2n,w2n+1−γMw2n,w2n,w2n+1,Mw2n,w2n,w2n+1=maxS∗Pw2n,w2n,w2n+1S∗Pw2n+1,w2n+1,w2n+2.
Now, ifS∗Pw2n+1,w2n+1,w2n+2≥S∗Pw2n,w2n,w2n+1, for some n=0,1,2,…, then Mw2n,w2n,w2n+1=S∗Pw2n+1,w2n+1,w2n+2, and from (32), we have(33)πS∗Pw2n+1,w2n+1,w2n+2≤πS∗Pw2n+1,w2n+1,w2n+2−γS∗Pw2n+1,w2n+1,w2n+2,which implies that γS∗Pw2n+1,w2n+1,w2n+2 is a contradiction. Therefore, for all n≥0,(34)S∗Pw2n+1,w2n+1,w2n+2≤S∗Pw2n,w2n,w2n+1.
Similarly, we have(35)S∗Pw2n,w2n,w2n+1≤S∗Pw2n−1,w2n−1,w2n,for all n≥0. Hence, we obtain(36)S∗Pwn+1,wn+1,wn+2≤S∗Pwn,wn,wn+1.
Also,S∗Pwn+1,wn+1,wn+2 is a nonincreasing sequence; then, there exists L such that(37)limn⟶∞S∗Pwn+1,wn+1,wn+2=L.
Hence, by the lower semicontinuity ofγ,(38)γL≤liminfn⟶∞γS∗Pwn,wn,wn+1.
Now, we claim thatL=0. By lower semicontinuity of γ, taking the upper limit as n⟶∞ on either side of(39)πS∗Pwn+1,wn+1,wn+2≤πMwn,wn,wn+1−γMwn,wn,wn+1,then we obtain(40)πL≤πL−liminfn⟶∞γMwn,wn,w2n+1≤πL−γL,and this implies that γL≤0⇒γL=0; then, we have(41)limn⟶∞S∗Pwn+1,wn+1,wn+2=0.
To show thatwn+1 is a S∗P-Cauchy sequence for each n,m, and n,m>N, we have(42)S∗Pwn,wn,wm≤2S∗Pwn,wn,wn+1+2S∗Pwn+1,wn+1,wn+2+2S∗Pwn+2,wn+2,wn+3+⋯+2S∗Pwm,wm,wm+1−+2S∗Pwn+1,wn+1,wn+1+⋯+2S∗Pwm+1,wm+1,wm+1.
By taking the limit asn,m⟶∞ on both sides of the above inequality and from (41), we have(43)limn,m⟶∞S∗Pwn,wn,wm=0.
It follows thatwn is a S∗P-Cauchy sequence and by S∗P-completeness of ϒ, so there exist r∈ϒ such that wn converges to r as n⟶∞. Now, we will distinguish the cases a and a of this theorem.(a)
Supposeg is continuous; since w2n+1⟶r, we obtain that(44)w2n+1=gw2n+1=gr.However,w2n+1⟶r, as a subsequence of wn. It follows that gr=r, and from the beginning of the prove, we get gr=r=fr.The proof, assuming that f is continuous, is similar to above.(b)
Suppose thatS∗Pr,r,gr, and for wn and a nondecreasing sequence with wn⟶r in ϒ, we indicate that w2n+1≺¯r, ∀n∈N. Therefore, from (24), we have(45)πS∗Pw2n+1,w2n+1,gr=πS∗Pfw2n,fw2n,gr≤πMw2n,w2n,r−γMw2n,w2n,r,where(46)Mw2n,w2n,r=maxS∗Pw2n,w2n,r,S∗Pw2n,w2n,fw2n,S∗Pr,r,gr,12S∗Pw2n,w2n,gr+S∗Pr,r,fw2n=maxS∗Pw2n,w2n,r,S∗Pw2n,w2n,w2n+1,S∗Pr,r,gr,12S∗Pw2n,w2n,gr+S∗Pr,r,w2n+1.
Taking limit asn⟶∞ implies(47)limn⟶∞Mw2n,w2n,r=S∗Pr,r,gz.
Hence,(48)πS∗Pr,r,gr=limn⟶∞supπS∗Pfw2n,fw2n,gr≤limn⟶∞supπMw2n,w2n,r−γMw2n,w2n,r.
This a contradiction. Thus, we have(49)S∗Pr,r,gr=0⇒r=fr=gr.Corollary 1.
Letϒ,≺¯ be a partially ordered set with f and g as weakly increasing self-mapping on a complete S∗P-partial metric space. Suppose that there exists γ∈Γ such that(50)S∗Pfw,fw,gt≤Mw,w,t−γMt,t,w,for all w,t∈ϒ, where(51)Mw,w,t=maxS∗Pw,w,t,S∗Pw,w,fw,S∗Pt,t,gt+12S∗Pw,w,gt+S∗Pt,t,fw.Assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈ϒ implies wn≺¯r for all n∈ℕ(b)
f or g is continuousTherefore, the mapsf or g have a common fixed point.Proof.
Putπt=t in Theorem 2.Corollary 2.
Letϒ,≺¯ be a partially ordered set with f and g as weakly increasing self-mapping on a complete S∗P-partial metric space. Suppose that there exist π∈Π and γ∈Γ such that(52)πS∗Pfw,fw,gt≤πMw,w,t−γMt,t,w,for all w,t∈ϒ, where(53)Mw,w,t=ζ1S∗Pw,w,t+ζ2S∗Pw,w,fw+ζ3S∗Pt,t,gt+ζ4S∗Pw,w,gt+S∗Pt,t,fw,where ζ1>0 for i=1,2,3,4 with ζ1+ζ2+ζ3+ζ4≥1.Assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈ϒ implies wn≺¯r for all n∈ℕ(b)
f or g is continuousTherefore, the mapsf or g have a common fixed point.Proof.
A corollary isS∗P-partial metric spaces version of Theorem 1.Corollary 3.
Letϒ,≺¯ be a partially ordered set with f and g as weakly increasing self-mapping on a complete S∗P-metric space. Suppose that there exist γ∈Γ such that(54)S∗Pfw,fw,gt≤Mw,w,t−γMt,t,w,for all w,t∈ϒ, where(55)Mw,w,t=ζ1S∗Pw,w,t+ζ2S∗Pw,w,fw+ζ3S∗Pt,t,gt+ζ4S∗Pw,w,gt+S∗Pt,t,fw,where ζ1>0 for i=1,2,3,4 with ζ1+ζ2+ζ3+ζ4≥1.Assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈Υ implies wn≺¯r for all n∈ℕ(b)
f or g is continuousTherefore, the mapsf or g have a common fixed point [14].Proof.
Putπt=t in Corollary 2.



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