# S∗p‐b-Partial Metric Spaces with some Results in Common Fixed Point Theorems

Authors: A. M. Zidan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we introduce the notion ofS∗P‐b-partial metric spaces which is a generalization each of S‐b-metric spaces and partial-metric space. Also, we study and prove some topological properties, to know the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorem in these 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. For some of these, the circulares are obtained through contractual terms expressed in reasonable terms. Latif et al. [1] introduced the notion of G-rational Geraghty contractive mappings in the setup of ordered generalized b-metric spaces and investigated the existence of fixed points for such mappings. They also provided an example to illustrate the presented results and show that they are more general than some existing ones.One such generalization is a partial-metric space introduced by Matthews [2] in 1994. Since in this space, a self-distance of an arbitrary point need not to be equal zero. Shahkoohi and Razani [3] introduced new classes of rational Geraghty contractive mappings in the setup of b-metric spaces and the existence of some fixed point for such mappings in ordered b-metric spaces. Bakhtin [4] and Bourbaki [5] introduced a notion of b-metric spaces. Later on, Czerwik [6] generalized the Banach contraction mapping theorem by formally defining a b-metric space and giving a postulate which was weaker than the condition of triangular inequality. After that, Fagin and Stockmeyer [7] defined some kind of relaxation in triangular inequality and called this new distance measure as the nonlinear elastic mathing. All these notions and applications pushed us to introduce the concept of extended b-metric space by S-metric space and partial metric space.In 2007, Sedghi et al. [8] introduced D∗-metric spaces which are the generalization of the notion of D-metric spaces introduced by Dhage [9]. Also, Sedghi et al. proved some of the basic properties in D∗-metric spaces. In 2012, Sedghi et al. [10] introduced S-metric spaces and give some properties and fixed point theorem for a complete S-metric space. Soliman and Zidan [11] introduced a new coupled fixed point theorem in a generalized metric space, and they utilized the same to study the stability for a system of set-valued functional equations. In 2015, Gupta and Deep [12] proved fixed point theorems for nonlinear contractive mappings in S-metric spaces. On the other way, Chauhan and Gupta [13] introduced the notion of fuzzy cone b-metric space and defined fuzzy cone b-contractive mapping and proved Banach contraction theorem for a single mapping in the setting of fuzzy cone b-metric space.Recently, in 2019, Mustafa et al. [14] introduced the structure of Sb-metric spaces which is a generalization of S-metric space and gave some properties and fixed point theorem for this spaces.


## 2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.Definition 1.
LetY be a nonempty set and ζ≥1 be a given real number. A function db:Y×Y⟶0,∞ is called b-metric, if it satisfies the following properties for each w,t,r∈Y:
(b1) dbw,t=0⇔w=t
(b2) dbw,t=dbt,w
(b3) dbw,t=≤ζdbw,r+dbr,t
The pairY,db is called a b-metric space.Example 2.
LetY=l℧ℝ with 0<℧<1 where l℧ℝ:wn⊂ℝ:Σn=1∞wn℧<∞. Define db:Y×Y⟶0,∞ as
(1)dbw,t=∑n=1∞wn−tn℧1/℧,where w=wn, t=tn. Then, db is a b-metric space.Definition 3 (see [2]).
LetY be a set. A function P:Y×Y⟶0,∞ is said to be a partial metric spaces on a nonempty set Y, if for all w,t,r∈Y, the following conditions hold:
(P1) 0≤Pw,w≥Pw,t (nonnegativity and small self-distances)
(P2) If Pw,w=Pw,t=Pt,t then w=t (indistancy implies equality)
(P3) If Pw,t=Pt,w (symmetry)
(P4) Pw,t≤Pw,r+Pr,t−Pr,r (triangularity)
Hence, the functionP is called a P-metric on Y and the pair Y,P is called a partial-metric space.
Mustafa and Sims [15] extended the Banach principle by introducing the notation of generalized metric spaces the so-called G-metric spaces as follows.Definition 4. LetY be a set.
A functionG:Y×Y×Y⟶0,∞ is said to be generalized metric spaces on a nonempty set Y, if for all w,t,r,η∈Y, the following conditions hold:
(G1) Gw,t,r=0⇔w=t=r
(G2) Gw,t,r>0 and w≠t
(G3) Gw,w,t≤Gw,t,r and t≠r
(G4) Gw,t,r=Gw,r,t=Gt,r,w=⋯, (symmetry in all three variables)
(G5) Gw,t,r≤Gw,η,η+Gη,r,w
Hence, the functionG is called a G-metric on Y and the pair Y,G is called a generalized metric space.
Mustafa and Sims [15] found some basic properties and examples of G-metric spaces.
After that, Zand and Nezhad [16] introduced the definition of Gp-metric space by generalization and unification of both the partial-metric space and G-metric space as follows:Definition 5. LetY be a set.
A functionGp:Y×Y×Y⟶0,∞ is said to be a generalized partial metric spaces on a nonempty set Y, if for all w,t,r,η∈Y, the following conditions hold:
(G1p) w=t=r if Gpw,w,w=Gpt,t,t=Gpr,r,r
(G2p) 0≤Gpw,w,w≤Gpw,w,t≤Gpw,t,y
(G3p) Gpw,t,r=Gpw,r,t=Gpt,r,w=⋯, (symmetry in all three variables)
(G4p) Gpw,t,r≤Gpw,η,η+Gpη,r,w−Gpη,η,η
Hence, the functionGp is called a Gp-metric on Y and the pair Y,Gp is called a generalized partial metric space.Example 6 (see [16]).
LetY=0,∞ and Gpw,t,r=maxw,t,r for all w,t,r∈Y, then Y,Gp is a Gp-metric space. Also, Y,Gp is not a G-metric space.Proposition 7.
LetY,Gp is a Gp-metric space, then for any w,t,r∈Y and η∈Y, 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,η
Sedghi et al. [8] introduced the notion of D∗-metric which is a modification of the definition of D-metric introduced by Dhage [9, 17], and they proved some basic properties in D∗-metric spaces.Definition 8 (see [8]).
LetY be a set. A function D∗:Y×Y×Y⟶0,∞ is said to be a D∗-metric spaces on a nonempty set Y, if for all w,t,r∈Y, the following conditions hold:
(D1∗) D∗w,t,r≥0
(D2∗) D∗w,t,r⇔w=t=w
(D3∗) D∗w,t,r=D∗ℵw,t,r, (where ℵ is a permutation function)
Hence, the functionD∗ is called a D∗-metric on Y and the pair Y,D∗ is called a generalized partial metric space.Example 9 (see [8]).
LetY=ℝ. Denote D∗w,t,r=∣w−t∣+∣t−r∣+∣r−w∣, for all w,t,r∈ℝ. Since
(2)BD∗w,λ=t∈Y: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 the other way, Sedghi et al. introducedS-metric spaces as follows.Definition 10 (see [10]).
LetY be a set. A function S:Y3⟶0,∞ is said to be S-metric spaces on a nonempty set Y, if for all w,t,r,η∈Y the following conditions hold:
(S1) Sw,t,r>0
(S2) Sw,t,r⇔w=t=r
(S3) Sw,t,r≤Sw,w,η+St,t,η+Sr,r,η for all w,t,r,η∈Y (rectangle inequality)
Hence, the functionS is called a S-metric on Y and the pair Y,S is called an S-metric space.Example 11 (see [10]).
LetY=ℝn and ∥·∥ a norm on Y, then
(4)Sw,t,r=∥t+r−2w∥+∥t−r∥,is a S-metric on Y.Remark 12.
EveryD∗-metric isS-metric, but in general, the converse is not true, see the following example.Example 1.
LetY=ℝn and ∥·∥ a norm on Y, then Sw,t,r=∥t+r−2w∥+∥t−r∥isS-metric onY, but it is not D∗-metric because it is not symmetric.Example 2.
LetY=ℝ2 and d is an ordinary metric on Y. Therefore, Sw,t,r=dw,t+dw,r+dt,r is an S-metric on Y. Then, if the points w,t,r connected by a line. Hence, the triangle when choose a point a mediating this triangle then the inequality
(5)Sw,t,r≤sw,w,η+st,t,η+r,r,η,
for allw,t,r,η∈Y holds.Definition 13 (see [10]).
LetY,S be a S-metric space. Then, for w0∈Y and λ>0, the S-open ball and S-closed ball of radius λ with centered at w0 is
(6)BSw0,λ=t∈Y:Sw0,w0,t<λ,B¯Sw0,λ=t∈Y:Sw0,w0,t≤λ.Proposition 14 (see [10]).
LetY,S be a S-metric space and A⊂Y:
(1)
If for everyw0∈A there exists λ>0 such that BSw0,λ, then the subset A is called an open subset of Y(2)
A subsetA of Y 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 Y 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=wLemma 15 (see [10]).
LetY,S be an S-metric space, then the following is satisfied:
(1)
Ifλ>0 and w∈Y, then the ball BSw0,λ is an open subset of Y(2)
If the sequencewn in Y converges to w, then w is unique(3)
If the sequencewn in Y converges to w, then wn is a Cauchy sequenceDefinition 16 (see [15]).
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 semi-continuous and π−10=0Definition 17. LetY,≼ be a partially ordered set.
Two mapsf,g:Y×Y are said to be weak increasing if fw≼gfw and gw°fgw for all w∈Y.
Barakat and Zidan [18] proved a common fixed point theorem for weak contractive maps by using the concept of Gp-metric spaces.Theorem 18.
LetY,≼ be a partially ordered set wit f and g be 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 allw,t∈Y, where
(8)Mw,w,t=ζ1Gpw,w,t+ζ2Gpw,w,fw+ζ3Gpt,t,gt+ζ4Gpw,w,gt+Gpt,t,fw,
whereζ1>0 for i=1,2,3,4 with ζ1+ζ2+ζ3+ζ4≥1.
Then, of the following two cases, assume that one of the following cases is satisfied:(a)
If a nondecreasing sequencewn converges to r∈Y implies wn°r for all n∈ℕ(b)
f or g is continuous
Therefore, the mapsf or g have a common fixed point.The present paper is aimed at introducing the notion ofSb∗-partial metric spaces which is a generalization each of S-metric spaces and partial-metric space. 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 fixed point theory in this spaces.


## 3.Sb∗-Partial Metric Spaces and some Properties

We first introduce the concept of aSb∗P-partial metric space or Sb∗P.Definition 19. LetY be a set and ζ≥1.
A functionSb∗P:Y3⟶0,∞ is said to be a Sb∗p-partial metric spaces on a nonempty set Y, if for all w,t,r,η∈Y, the following conditions hold:
Sb∗P1Sb∗Pw,t,r=Sb∗Pw,w,w=Sb∗Pt,t,t=Sb∗Pr,r,r then, w=t=r
S2∗PSb∗Pw,t,r≤ζSb∗Pw,w,η+Sb∗Pt,t,η+Sb∗Pr,r,η−Sb∗Pη,η,η for all w,t,r,η∈Y (rectangle inequality)
Hence, the functionSb∗P is called an Sb∗-partial metric on Y and the pair Y,Sb∗P is called an Sb∗-partial metric space.Example 20.
LetY=ℝ and ∥·∥ a norm on Y, then we have
(9)Sb∗Pw,t,r=maxw,t,r+∥w−r∥+∥t−w∥,is a Sb∗-partial metric on Y.Remark 21.
From Example20, we get every S-metric is Sb∗P-metric, but the converse is not true at all.Definition 22. LetY,Sb∗P be a Sb∗-partial metric space.
Then, forw0∈Y and λ>0, the Sb∗P-open ball and Sb∗P-open closed of radiusλ with centered at w0is(10)BSb∗Pw0,λ=t∈Y:Sb∗Pw0,w0,t<λ+Sb∗Pw0,w0,w0−Sb∗Pt,t,w0,B¯Sb∗Pw0,λ=t∈Y:Sb∗Pw0,w0,t≤λ+Sb∗Pw0,w0,w0−Sb∗Pt,t,w0.Proposition 23. LetY,Sb∗P be a Sb∗P-partial metric space.
Then forw0∈Y and λ>0, the following statements are satisfying:
(1)
IfSb∗Pw0,w0,w≤λ+Sb∗Pw0,w0,w0−Sb∗Pw,w,,w0 and Sb∗Pw0,w0,t≤λ+Sb∗Pw0,w0,w0−Sb∗Pt,t,w0, then w,t∈BSb∗Pw0,λ(2)
Ift∈BSb∗Pw0,λ, then their exist ρ>0, such that BSb∗Pt,ρ⊆BSb∗Pw0,λProof.
(1)
The proof is straightforward.(2)
Lett∈BSb∗Pw0,λ, then we have(11)Sb∗Pw0,w0,t≤λ+Sb∗Pw0,w0,w0−Sb∗Pt,t,w0
Also, we suppose(12)ρ=λ+Sb∗Pw0,w0,w0−Sb∗Pr,r,w0−2Sb∗Pw0,w0,t,and r∈BSb∗Pt,ρ, then we get
(13)Sb∗Pt,t,r≤ρ+Sb∗Pt,t,t−Sb∗Pr,r,t.
Therefore,t∈BSb∗Pw0,λ, and so (2) holds.
(14)Sb∗Pw0,w0,r≤ζSb∗Pw0,w0,t+Sb∗Pw0,w0,t+Sb∗Pr,r,t−S∗Pt,t,t≤ζ2Sb∗Pw0,w0,t+Sb∗Pr,r,t+λ+Sb∗Pw0,w0,w0−Sb∗Pr,r,w0−2Sb∗Pw0,w0,t−Sb∗Pt,t,t≤ζλ+Sb∗Pw0,w0,w0−Sb∗Pr,r,t≤λ+Sb∗Pw0,w0,w0−Sb∗Pr,r,t.
Hence,r∈BSb∗Pw0,λ.Definition 24.
LetY,Sb∗P be a Sb∗P-partial metric space and wn a sequence in Y. A point w∈Y is said to be the limit of the sequence if
(15)limn,m⟶∞Sb∗Pw,wn,wm=Sb∗Pw,w,w.
Hence, the sequencewn is Sb∗P-convergent to w.
Therefore, ifwn⟶w in a Sb∗P-partial metric space Y,Sb∗P. Then, for any ℏ>0, there exists I∈N such that, for all n,m>I, we have
(16)∣Sb∗Pw,wn,wm−Sb∗Pw,w,w∣<ℏ.Definition 25.
ASb∗P-partial metric spaceY,Sb∗P is called a Sb∗P-partial asymmetric space if(17)Sb∗Pw,w,t=Sb∗Pt,t,w.Lemma 26.Y,Sb∗P be a Sb∗-partial metric space.
If the sequencewninY converges to w. Therefore, we get w is unique.Proof.
Letwn converges to w and t. Therefore, for each λ>0, there exist n1,n2∈N, then we have
(18)n≥n1⇒Sb∗Pwn,wn,w<λ3,n≥n2⇒Sb∗Pwn,wn,t<λ3.
If setn0=maxn1,n2, then for every n≥n0. Also, we have a third condition of Sb∗P-partial metric
(19)Sb∗Pw,w,t≤ζSb∗Pw,w,wn+Sb∗Pw,w,wn+Sb∗Pt,t,wn−Sb∗Pwn,wn,wn≤ζ2Sb∗Pw,w,wn+Sb∗Pt,t,wn−Sb∗Pwn,wn,wn≤ζ2Sb∗Pw,w,wn+Sb∗Pt,t,wn≤2Sb∗Pw,w,wn+Sb∗Pt,t,wn<2λ3+λ3=λ.
Hence,Sb∗Pw,w,t=0⇒w=t, but the converse is not necessarily true.Definition 27. LetY,Sb∗P be a Sb∗P-partial metric space.
Then, for as′sequencewn⊆Y and a point w∈Y, the following are equivalent:
(1)
wn is a Sb∗P convergent to w(2)
limn,m⟶∞Sb∗Pwn,wn,w=Sb∗Pw,w,wProposition 28. LetY,Sb∗P be a Sb∗P-partial metric space.
If then, we get(1)
A sequencewn∈Y is called a Cauchy sequence if for each λ>0, there exists n0∈N such that(20)Sb∗Pwn,wn,wm<λ+Sb∗Pw,w,w,for each n,m≥0.Definition 29. LetY,Sb∗P be a Sb∗P-partial metric space.
Then,Y,Sb∗P is said to be complete if every Cauchy sequence is convergent.Definition 30. LetY,Sb∗P and let Y2,Sb2∗P be two Sb∗P-partial metric space.
Also, we suppose a function(21)F:Y1,Sb1∗P⟶Y2,Sb2∗P,then F is said to be Sb∗P- continuous at a point a x∈Y1 if and only if, for given ℏ>0, there exists λ>0 such that w,t∈Y1 and the inequality
(22)Sb1∗Px,w,t<ℏ+Sb1∗Pw,w,w.
This is an indication(23)Sb2∗PFx,Fw,Ft<λ+Sb2∗PFw,Fw,Fw.
Hence, a functionF is Sb∗P- continuous on Y1 if and only if it Sb∗P-continuous at all x∈Y1.Definition 31.
The two classes of following mappings are definedΠ=π;π:0,∞→0,∞ is continuous, nondecreasing, and π−10=0. Γ=γ;γ:0,∞→0,∞ is lower semicontinuous, nondecreasing, and γ−10=0.


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

Theorem 32.
LetY,≼ be a partially ordered set, f and g be weakly increasing self-mapping on a complete Sb∗P-metric space with ζ≥1. Suppose that there exist π∈Π and γ∈Γ such that
(24)πSb∗Pfw,fw,gt≤πMw,w,t−γMt,t,w,for all w,t∈Y, where
(25)Mw,w,t=maxSb∗Pw,w,t,Sb∗Pw,w,fw,Sb∗Pt,t,gt,Sb∗Pw,w,gt+Sb∗Pt,t,fw.
Assume that one of the following cases is satisfied:(a)
if a nondecreasing sequencewn converges to r∈Y implies wn°r for all n∈ℕ(b)
f or g is continuous
Therefore, 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 4.1 with w=t=ℏ, we have
(26)πSb∗Pℏ,ℏ,gℏ=πSb∗Pfℏ,fℏ,gℏ≤πMℏ,ℏ,ℏ−γMℏ,ℏ,ℏ,where
(27)Mℏ,ℏ,ℏ=maxSb∗Pℏ,ℏ,ℏ,Sb∗Pℏ,ℏ,fℏ,Sb∗Pℏ,ℏ,gℏ,Sb∗Pℏ,ℏ,gℏ+Sb∗Pℏ,ℏ,fℏ=maxSb∗Pℏ,ℏ,ℏ,Sb∗Pℏ,ℏ,ℏ,Sb∗Pℏ,ℏ,gℏ,Sb∗Pℏ,ℏ,gℏ+Sb∗Pℏ,ℏ,ℏ=maxSb∗Pℏ,ℏ,ℏ,Sb∗Pℏ,ℏ,gℏ=Sb∗Pℏ,ℏ,gℏ.
Hence, we have(28)πSb∗Pℏ,ℏ,gℏ=πSb∗Pfℏ,fℏ,gℏ≤πSb∗Pfℏ,fℏ,gℏ−γSb∗Pfℏ,fℏ,gℏ⇒γSb∗Pℏ,ℏ,gℏ≤0.A contradiction. Therefore,Sb∗Pfℏ,fℏ,gℏ=0. So, ℏ is common fixed point of f and g. Similarly, if ℏ is a fixed point of g, then one can deduce that ℏ is also fixed point of f.If, we letw0 be an arbitrary point of Y with fw0=w0, then the proof is finished, so we assume that fw0≠w0.Now, one can construct a sequencewn∈Y as follows:
(29)w1=fw0≼gfw0=gw1=w2,w2=fw1≼gfw1=gw1=w3,⋮wn≼wn+1.Sincew2n and w2n+1 are comparable, then we may assume that Sb∗Pw2n,w2n,w2n+1>0, for every n∈N. If not, then w2n=w2n+1 for some n. For all those n, using (24), we obtain
(30)πSb∗Pw2n+1,w2n+1,w2n+2=πSb∗Pfw2n,fw2n,gw2n+1≤πMw2n,w2n,w2n+1−γMw2n,w2n,gw2n+1,Mw2n,w2n,w2n+1=maxSb∗Pw2n,w2n,w2n+1,Sb∗Pw2n,w2n,fw2n,Sb∗Pw2n+1,w2n+1,gw2n+1,Sb∗Pw2n,w2n,gw2n+1,Sb∗Pw2n+1,w2n+1,fw2n2=maxSb∗Pw2n,w2n,w2n+1,Sb∗Pw2n,w2n,w2n+1,Sb∗Pw2n+1,w2n+1,w2n+2,Sb∗Pw2n,w2n,w2n+2,Sb∗Pw2n+1,w2n+1,w2n+12≤maxSb∗Pw2n+1,w2n+1,w2n+2,12ζSb∗Pw2n,w2n,w2n+1+Sb∗Pw2n,w2n,w2n+1+Sb∗Pw2n+1,w2n+1,w2n+1−Sb∗Pw2n+1,w2n+1,w2n+1+Sb∗Pw2n+1,w2n+1,w2n+1=Sb∗Pw2n+1,w2n+1,w2n+2.Therefore(31)πSb∗Pw2n+1,w2n+1,w2n+2≤πSb∗Pw2n+1,w2n+1,w2n+2−γSb∗Pw2n+1,w2n+1,w2n+2.It means thatγSb∗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 g.By takingSb∗Pw2n,w2n,w2n+1>0 for n=1,2,3,⋯, now, we consider
(32)πSb∗Pw2n+1,w2n+1,w2n+2=πSb∗Pfw2n,fw2n,gw2n+1≤πMw2n,w2n,w2n+1−γMw2n,w2n,w2n+1.Mw2n,w2n,w2n+1=maxSb∗Pw2n,w2n,w2n+1,Sb∗Pw2n+1,w2n+1,w2n+2.Now ifSb∗Pw2n+1,w2n+1,w2n+2≥Sb∗Pw2n,w2n,w2n+1 for some n=0,1,2,⋯, then Mw2n,w2n,w2n+1=Sb∗Pw2n+1,w2n+1,w2n+2 and from (32), we have
(33)πSb∗Pw2n+1,w2n+1,w2n+2≤πSb∗Pw2n+1,w2n+1,w2n+2,−γSb∗Pw2n+1,w2n+1,w2n+2,implying thatγSb∗Pw2n+1,w2n+1,w2n+2=0 is a contradiction. Therefore, for all n≥0,
(34)Sb∗Pw2n+1,w2n+1,w2n+2≤Sb∗Pw2n,w2n,w2n+1.Similarly, we have(35)Sb∗Pw2n,w2n,w2n+1≤Sb∗Pw2n−1,w2n−1,w2n,for alln≥0. Hence, we get
(36)Sb∗Pwn+1,wn+1,wn+2≤Sb∗Pwn,wn,wn+1.Also,Sb∗Pwn+1,wn+1,wn+2 is a nonincreasing sequence, then there exists L, such that
(37)limn⟶∞Sb∗Pwn+1,wn+1,wn+2=L.Hence, by the lower semicontinuity ofγ,
(38)γL≤liminfn⟶∞γSb∗Pwn,wn,wn+1Now, we claim thatL=0. By lower semicontinuity of γ, taking the upper limit as n⟶∞ on either side of
(39)πSb∗Pwn+1,wn+1,wn+2≤πMwn,wn,wn+1−γMwn,wn,wn+1,then, we get
(40)πL≤πL−liminfn⟶∞γMwn,wn,w2n+1≤πL−γL,this implies thatγL≤0⇒γL=0, then we have
(41)limn⟶∞Sb∗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)Sb∗Pwn,wn,wm≤ζ2Sb∗Pwn,wn,wn+1+2ζSb∗Pwn+1,wn+1,wn+2+2ζ2Sb∗Pwn+2,wn+2,wn+3+⋯+2ζmSb∗Pwm,wm,wm+1−ζ2Sb∗Pwn+1,wn+1,wn+1+⋯++2ζmSb∗Pwm+1,wm+1,wm+1.By taking the limit asn,m⟶∞ to both sides of the above inequality and from (41), we have
(43)limn,m⟶∞Sb∗Pwn,wn,wm=0.It follows thatwn is a Sb∗P-Cauchy sequence and by Sb∗P-completeness of Y, so there exist r∈Y such that wn converges to r as n⟶∞. Now, we will distinguish the cases a and b of this theorem.
(a)
Supposeg is continuous, since w2n+1⟶r, we obtain that(44)w2n+1=gw2n+1=gr.Butw2n+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 the above.
(b)
Suppose thatSb∗Pr,r,gr and for wn and a nondecreasing sequence with wn⟶r in Y indicate that w2n+1≼r, ∀n∈N. Therefore, from (24), we have(45)πSb∗Pw2n+1,w2n+1,gr=πSb∗Pfw2n,fw2n,gr,≤πMw2n,w2n,r−γMw2n,w2n,r,where
(46)Mw2n,w2n,r=maxSb∗Pw2n,w2n,r,Sb∗Pw2n,w2n,fw2n,Sb∗Pr,r,gr,12Sb∗Pw2n,w2n,gr+Sb∗Pr,r,fw2n=maxSb∗Pw2n,w2n,r,Sb∗Pw2n,w2n,w2n+1,Sb∗Pr,r,gr,12Sb∗Pw2n,w2n,gr+Sb∗Pr,r,w2n+1.By taking limit asn⟶∞, this implies
(47)limn⟶∞Mw2n,w2n,r=Sb∗Pr,r,gr.Hence,(48)πSb∗Pr,r,gr=limn⟶∞supπSb∗Pfw2n,fw2n,gr≤limn⟶∞supπMw2n,w2n,r−γMw2n,w2n,r.This is a contradiction. Thus, we have(49)Sb∗Pr,r,gr=0⇒r=fr=gr.Corollary 33.
LetY,≼ be a partially ordered set, f and g be weakly increasing self-mapping on a complete Sb∗P-metric space with ζ≥1. Suppose that there exist γ∈Γ such that
(50)Sb∗Pfw,fw,gt≤Mw,w,t−γMt,t,w,
for allw,t∈Y, where
(51)Mw,w,t=maxSb∗Pw,w,t,Sb∗Pw,w,fw,Sb∗Pt,t,gt+12Sb∗Pw,w,gt+Sb∗Pt,t,fw.
Assume that one of the following cases is satisfied:(a)
if a nondecreasing sequencewn converges to r∈Y implies wn≼r for all n∈ℕ(b)
f or g is continuous
Therefore, the mapsf or g have a common fixed point.Proof.
Putπt=t in Theorem 32.Corollary 34.
LetY,≼ be a partially ordered set, f and g be weakly increasing self-mapping on a complete Sb∗P-metric space. Suppose that there exist π∈Π and γ∈Γ such that
(52)πSb∗Pfw,fw,gt≤πMw,w,t−γMt,t,w,
for allw,t∈Y, where
(53)Mw,w,t=μ1Sb∗Pw,w,t+μ2Sb∗Pw,w,fw+μ3Sb∗Pt,t,gt+μ4Sb∗Pw,w,gt+Sb∗Pt,t,fw,
whereμ1>0 for i=1,2,3,4 with μ1+μ2+μ3+μ4≥1.
Then, of the following two cases, assume that one of the following cases is satisfied:(a)
if a nondecreasing sequencewn converges to r∈Y implies wn≼r for all n∈ℕ(b)
f or g is continuous
Therefore, the mapsf or g have a common fixed point.Proof.
A corollary isS∗P-partial metric spaces version of Theorem 18.Corollary 35.
LetY,≼ be a partially ordered set, f and g be weakly increasing self-mapping on a complete Sb∗P-metric space. Suppose that there exist γ∈Γ such that
(54)Sb∗Pfw,fw,gt≤Mw,w,t−γMt,t,w,
for allw,t∈Y, where
(55)Mw,w,t=μ1Sb∗Pw,w,t+μ2Sb∗Pw,w,fw+μ3Sb∗Pt,t,gt+μ4Sb∗Pw,w,gt+Sb∗Pt,t,fw,where ζ1>0 for i=1,2,3,4 with μ1+μ2+μ3+μ4≥1.
Then, of the following two cases, assume that one of the following cases is satisfied:(a)
if a nondecreasing sequencewn converges to r∈Y implies wn≼r for all n∈ℕ(b)
f or g is continuous
Therefore, the mapsf or g have a common fixed point.Proof.
Putπt=t in Corollary 34.Remark 36.
It will be interesting to find more applications to our current paper in other fields, see [11, 19–29].



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