# On Fixed Point Results in Partialb-Metric Spaces

Authors: Haitham Qawaqneh; Mohd Salmi Md Noorani; Hassen Aydi; Amjed Zraiqat; Arslan Hojat Ansari
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Partialb-metric spaces are characterised by a modified triangular inequality and that the self-distance of any point of space may not be zero and the symmetry is preserved. The spaces with a symmetric property have interesting topological properties. This manuscript paper deals with the existence and uniqueness of fixed point points for contraction mappings using triangular weak α-admissibility with regard to η and C-class functions in the class of partial b-metric spaces. We also introduce an example to demonstrate the obtained results.


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## Body

## 1. Introduction and Preliminaries

The contribution of fixed point theorems has been appeared as a really effective and successful strategy for understanding many mathematical issues, which rise from actual applications. Fixed point theory is an essential tool to solve several equations in diverse fields, such as integral and differential (fractional) equations appeared in economics, game theory, engineering, physics, and chemistry. For instance, see [1–8]. As the significance and accessibility of Banach contraction mapping (BCM), various authors have amplified and generalized this contraction and set up various interesting supplements and alterations. In this continuation, Czerwik [9] and Bakhtin [10] originated the idea of a b-metric space as a suitable generalization of metric spaces. They demonstrated the truth of new weak contraction mapping standards in b-metric spaces that generalize the celebrated Banach contraction. Subsequently, several authors have dealt with related fixed point results in this setting (see [11–17]). Furthermore, Matthews [18] presented the idea of a partial metric space by considering the distance between any point to itself that requires not be equal to zero. For more related works in this setting, see [19–21]. Recently, Shukla [22] studied the connection, as topological properties, between a partial metric and a b-metric space to introduce an unification of the concept of a partial b-metric space.Definition 1 ([22]).
Given a nonempty setW. A function ρpb:W×W⟶0,∞ is denominated as a b-partial metric if for all υ,κ,z∈W(1)ρpb1υ=κifandonlyifρpbυ,υ=ρpbυ,κ=ρpbbκ,κ,ρpb2ρpbυ,υ≤ρpbυ,κ,ρpb3ρpbυ,κ=ρpbκ,υ,ρpb4thereisarealnumbers≥1suchthatρpbυ,κ≤sρpbυ,z+ρpbz,κ−ρpbz,z.Ifρpbυ,κ=0, then υ=κ. The converse does not hold in general. Now, consider W=0,∞ and p>1. Take ρpb:W×W⟶ℝ+ as
(2)ρpbυ,κ=maxυ,κp−υ−κp,for allυ,κ∈W. Then, X,ρpb is a partial b-metric space (Here, s=2p>1). It is neither a partial metric, nor a b-metric space. Let ρp be a partial metric and ρb be a b−metric (with s>1) on a nonempty set W. Then, ρpb:W×W⟶0,∞ defined by ρpbυ,κ=ρpυ,κ+ρbυ,κ for all υ,κ∈W, is a partial b-metric on W.Definition 2 ([22, 23]).
LetW,ρpb be a partial b-metric space with s≥1. Let υn be a sequence in W and υ∈W. Then,
(i)
υn⊆W converges to υ∈W if limn⟶∞ρpbυn,υ=ρpbυ,υ(ii)
υn⊆W is Cauchy in W,ρpb if limn,m⟶∞ρpbυn,υm exists and is finite(iii)
W,ρpb is complete if for every Cauchy sequence υn⊆W, there is υ∈W so that(3)limn,m⟶∞ρpbυn,υm=limn⟶∞ρpbυn,υ=ρpbυ,υ.Given a nonempty setW. Given f:W⟶W and α:W×W⟶0,∞. Then, f is called α-admissible [24] if for all υ,κ∈W with αυ,κ≥1, we have αfυ,fκ≥1. The conception of triangular α-admissibility was initiated in [25]. This concept has been weakened as follows.Definition 3 ([26]).
Given a nonempty setW. Given α,η:W×W→0,∞. S:W⟶W is triangular weak α-admissible with regard to η if
(1)
αυ,Snυ≥ηυ,Snυ, implies that αSnυ,Sn+1υ≥ηSnυ,Sn+1υ, for all n∈ℕ(2)
αυ,z≥ηυ,z and αz,κ≥ηz,κ, imply that αυ,κ≥ηυ,κ.LetW=0,∞. Given S:W⟶W as Sυ=υ2. Take α,η:W×W→0,∞ as αυ,κ=eυ+κ and ηυ,κ=eκ−υ. Here, S is triangular weak α-admissible with regard to η.Lemma 4 ([26]).
Givenα,η:W×W⟶ℝ. Let S:W⟶W be triangular weak α-admissible with regard to η. Assume that there is υ0∈W such that αυ0,Sυ0≥ηυ0,Sυ0. Define υn in W by Sυn=υn+1. Then, for all integers n<m(4)αυn,υm≥ηυn,υm.LetW,σ be a metric space. Given α,η:W×W⟶0,∞. Following [27], the mapping T:W⟶W is said to be α-η-continuous if every sequence υn in W with αυn,υn+1≥ηυn,υn+1 for all n∈ℕ and limn⟶∞συn,υ=0, we have limn⟶∞σTυn,Tυ=0. This concept could be extended to b-partial metric spaces as follows.Definition 5.
LetW,ρpb be a b-partial metric space. Given α,η:W×W⟶0,∞. The mapping T:W⟶W is said to be α-η-continuous if every sequence υn in W with αυn,υn+1≥ηυn,υn+1 for all n∈ℕ and limn⟶∞ρpbυn,υ=ρpbυ,υ, we have limn⟶∞dTυn,Tυ=ρpbTυ,Tυ.The conception ofC-class functions is posed as follows.Definition 6 ([28]).
We say thatF:0,∞×0,∞⟶ℝ is a C-class function if it is continuous and for υ,κ∈0,∞, we have
(i)
Fυ,κ≤υ(ii)
Fυ,κ=υ yields that either υ=0, or κ=0
Denote byC the set of C-class functions. The following functions are elements in C (for all υ,κ∈0,∞):
(1)
Fυ,κ=υ−κ(2)
Fυ,κ=μυ (with 0<μ<1)(3)
Fυ,κ=Θυ (with Θ:0,∞⟶0,1 is continuous)
Denote byΨ the set of functions [29] ψ:0,∞⟶0,∞ verifying that:
(i)
ψ is monotone and nondecreasing(ii)
ψυ=0⟺υ=0
LetΦ be the set of functions φ:0,∞⟶0,∞ so that
(i)
φ is continuous(ii)
φυ=0⟺υ=0
In this paper, we initiate the concept of generalizedα−η−ψ−φ−F-contraction self-mappings via C-class functions and the connotation of triangular weak α-admissibility with regard to η. The goal is to prove some related fixed point results in the context of partial b-metric spaces.


## 2. Main Results

We start with the following.Definition 7.
LetW,ρpb be a complete b-partial metric space with coefficient s≥1 and S:W⟶W. If there areα,η:W×W⟶0,∞, F∈C, ψ∈Ψ, and φ∈Φ, so that for all υ,κ∈W with αυ,κ≥ηυ,κ,
(5)ψsρpbSυ,Sκ≤FψPυ,κ,φPυ,κifρpbυ,Sκ+ρpbSυ,κ≠0,0otherwiseρpbυ,Sκ+ρpbSυ,κ=0,where either
(6)Pυ,κ=maxρpbυ,κ,ρpbυ,Sυ,ρpbκ,Sκ,ρpbυ,Sκ+ρpbκ,Sυ2s,or
(7)Pυ,κ=ρpbυ,Sυ+ρpbκ,Sκ2,then S is called a generalized α−η−ψ−φ−F-contraction.Theorem 8.
LetW,ρpb be a complete b-partial metric space (with s≥1) and S:W⟶W. Assume that
(a)
S is a generalized α−η−ψ−φ−F-contraction(b)
S is triangular weak α-admissible(c)
There isυ0∈W such that αυ0,Sυ0≥ηυ0,Sυ0(d)
S is α−η−continuousThen,S possesses a unique fixed point.Proof.
Letυ0∈W be in order that αυ0,Sυ0≥ηυ0,Sυ0. Realize a sequence υn in W in order that υn+1=Sυn for all n∈ℕ. If for some n, υn=υn+1, the υn is a fixed point. Now, suppose that υn≠υn+1 for all n∈ℕ. By using assumption (b), the triangular weak α-admissibility and Lemma 4, we get that for all integers m>n,
(8)αυn,υm≥ηυn,υm.In particular,αυn,υn+1≥ηυn,υn+1 for all n∈ℕ.Case 1.
Here, we considerPυ,κ=maxρpbυ,κ,ρpbυ,Sυ,ρpbκ,Sκ,ρpbυ,Sκ+ρpbκ,Sυ/2s. We need three steps.Step 1.
In view of assumption (a), one writes(9)ψρpbυn+1,υn=ψρpbSυn,Sυn−1≤ψsρpbSυn,Sυn−1≤FψPυn,υn−1,φPυn,υn−1≤ψPυn,υn−1,for all n∈ℕ, where
(10)Pυn,υn−1=maxρpbυn,υn−1,ρpbυn,Sυn,ρpbυn−1,Sυn−1,ρpbυn,Sυn−1+ρpbυn−1,Sυn2s=maxρpbυn,υn−1,ρpbυn,υn+1,ρpbυn−1,υn,ρpbυn,υn+ρpbυn−1,υn+12s=maxρpbυn,υn−1,ρpbυn,υn+1.IfPυn,υn−1=ρpbυn,υn+1 for some n, then we get
(11)ψρpbυn+1,υn≤FψPυn,υn−1,φPυn,υn−1=Fψρpbυn+1,υn,φρpbυn+1,υn≤ψρpbυn+1,υn.This implicates that eitherψρpbυn+1,υn=0 or φρpbυn+1,υn=0, so ρpbυn+1,υn=0; hence, υn+1=υn. It is a contradiction. Thus. Pυn,υn−1=ρpbυn,υn−1 for all n≥1. By (5), we have
(12)ψρpbυn+1,υn≤ψρpbυn,υn−1.Sinceψ is nondecreasing, we get ρpbυn+1,υn≤ρpbυn,υn−1 for all n≥1. Thus, ρpbυn,υn+1 is nonincreasing. Thus, there is ω≥0 in order that limn⟶∞ρpbυn,υn+1=ω. We claim that ω=0. We have
(13)ψρpbυn+1,υn+2≤ψsρpbSυn,Sυn+1≤Fψρpbυn,υn+1,φρpbυn,υn+1≤ψρpbυn,υn+1.At the limit, we acquire that(14)ψω≤Fψω,φω≤ψω.This infers that eitherψω=0, or φω=0. That is, ω=0, i.e.,
(15)limn⟶∞ρpbυn,υn+1=0.To prove thatυn is a Cauchy sequence in the b-partial metric space W,ρpb, we argue by contradiction. For this, suppose that there exist ε>0 and subsequences mll=1∞ and nll=1∞ of positive integers with nl>ml>l such that
(16)ρpbυnl,υml≥εandρpbυml,υnl−1<ε.Usingρpb4 and applying (15) and (16), one easily gets
(17)εs2≤limsupl⟶∞ρpbυnl−1,υml−1≤εs,(18)εs≤limsupl⟶∞ρpbυnl−1,υml≤ε,(19)εs≤limsupl⟶∞ρpbυnl,υml−1≤εs2.In view of assumption (a), we have(20)ψsρpbυnl,υml=ψsρpbSυnl−1,Sυml−1≤FψPυnl−1,υml−1,φPυnl−1,υml−1≤ψPυnl−1,υml−1,where
(21)Pυnl−1,υml−1=maxρpbυnl−1,υml−1,ρpbυnl−1,Sυnl−1,ρpbυml−1,Sυml−1,ρpbυnl−1,Sυml−1+ρpbυml−1,Sυnl−12s=maxρpbυnl−1,υml−1,ρpbυnl−1,υnl,ρpbυml−1,υml,ρpbυnl−1,υml+ρpbυml−1,υnl2s.By (15)–(19), one gets
(22)limsupl⟶∞Pυnl−1,υml−1≤εs.Lettingl⟶∞ and having in mind properties of ψ, φ and F, we deduce that
(23)ψsε≤Fψsε,φsε≤ψsε.That is,(24)Fψsε,φsε=ψsε,so either ψsε=0 or φsε=0. Thus, ε=0, which is, a contradiction. Hence, υn is a Cauchy sequence.Step 2.
We claim thatS has a fixed point. The sequence υn is Cauchy in the complete b-partial metric space, so there is υ∗∈W in order that
(25)limn,m⟶∞ρpbυn,υ∗=limn,m⟶∞ρpbυn,υm=ρpbυ∗,υ∗=0.For eachn∈ℕ, we have
(26)ρpbυ∗,Sυ∗≤sρpbυ∗,υn+1+ρpbυn+1,Sυ∗−ρpbυn+1,υn+1≤sρpbυ∗,υn+1+ρpbSυn,Sυ∗≤sρpbυ∗,υn+1+sρpbSυn,Sυ∗.By assumption (d) and Definition5, ρpbSυn,Sυ∗⟶ρpbSυ∗,Sυ∗ as n⟶∞. Using (25), we deduce that ρpbυ∗,Sυ∗≤sρpbSυ∗,Sυ∗. We will show that υ∗=Sυ∗. Suppose that υ∗≠Sυ∗, then we obtain that ρpbυ∗,Sυ∗≠0. This implies that
(27)ψρpbSυ∗,Sυ∗≤ψsρpbSυ∗,Sυ∗≤FψPυ∗,υ∗,φPυ∗,υ∗≤ψPυ∗,υ∗,where
(28)Pυ∗,υ∗=maxρpbυ∗,υ∗,ρpbυ∗,Sυ∗,ρpbυ∗,Sυ∗,ρpbυ∗,Sυ∗+ρpbυ∗,Sυ∗2s=ρpbυ∗,Sυ∗.So,(29)ψρpbSυ∗,Sυ∗≤ψρpbυ∗,Sυ∗≤ψρpbSυ∗,Sυ∗.Hence, ψρpbSυ∗,Sυ∗=ψρpbSυ∗,υ∗. We can write
(30)ψρpbSυ∗,υ∗≤FψρpbSυ∗,υ∗,φρpbSυ∗,υ∗≤ψρpbSυ∗,υ∗.Then,(31)FψρpbSυ∗,υ∗,φρpbSυ∗,υ∗=ψρpbSυ∗,υ∗.Thus,ψρpbSυ∗,υ∗=0 or φρpbSυ∗,υ∗=0. Hence, υ∗=Sυ∗. It is a contradiction. Consequently, S has a fixed point.Step 3.
We demonstrate that the fixed point ofS is unique.
Givenu≠u^∈W so that Su=u and Su^=u^. From (5),
(32)ψρpbu,u^≤ψsρpbu,u^=ψsρpbSu,Su^≤Fψmaxρpbu,u^,ρpbu,Su,ρpbu^,Su^,ρpbu,Su^+ρpbu^,Su2s,φmaxρpbu,u^,ρpbu,Su,ρpbu^,Su^,ρpbu,Su^+ρpbu^,Su2s=Fψmaxρpbu,u^,ρpbu,u,ρpbu^,u^,ρpbu,u^+ρpbu^,u2s,φmaxρpbu,u^,ρpbu,u,ρpbu^,u^,ρpbu,u^+ρpbu^,u2s=Fψρpbu,u^,φρpbu,u^≤ψρpbu,u^.This infers that eitherψρpbu,u^=0 or φρpbu,u^=0. This implies that ρpbu,u^=0, so u=u^.Case 1.
We considerPυ,κ=ρpbυ,Sυ+ρpbκ,Sκ/2.
Again, in view of assumption (a), one writes(33)ψρpbυn+1,υn≤ψsρpbυn+1,υn=ψsρpbSυn,Sυn−1≤FψPυn,υn−1,φPυn,υn−1≤ψPυn,υn−1,for all n≥1, where
(34)Pυn,υn−1=ρpbυn,Sυn+ρpbυn−1,Sυn−12=ρpbυn,υn+1+ρpbυn−1,υn2.Then,(35)ψρpbυn+1,υn≤FψPυn,υn−1,φPυn,υn−1=Fψρpbυn,υn+1+ρpbυn−1,υn2,φρpbυn,υn+1+ρpbυn−1,υn2≤ψρpbυn,υn+1+ρpbυn−1,υn2.We deduce that(36)ρpbυn+1,υn≤ρpbυn,υn+1+ρpbυn−1,υn2.That is,(37)ρpbυn+1,υn≤ρpbυn−1,υn,n≥1.Letρpbυn+1,υn=θn. Then, θn<θn−1.This emphasizes thatθn is a bounded decreasing sequence and bounded below, so it must be a convergent sequence.For eachn∈ℕ, we must have,
(38)θn<θn−1<⋯<θ1=Mboundedvalue.Again, for alln,m∈ℕ, we get
(39)ρpbυn,υm≤12θn−1+θm−1<M.Therefore,υn is a bounded sequence in W. By Bolzano Weierstrass theorem that every bounded sequence of real numbers has a convergent subsequence, υn must have a convergent subsequence, like υnk which converges to some z∈W. So,
(40)limk⟶∞ρpbυnk,υnk+1=ρpblimk⟶∞υnk,limk⟶∞υnk+1=ρpbz,z=0.This confirms that the subsequenceθnk itself converges to 0. So, the sequence θn must converge to 0. Hence, for all n,m∈ℕ,
(41)ρpbυn,υm≤12ρpbυn−1,υn+ρpbυm−1,υm⟶0,asn,m⟶∞.This concludes thatυn is a Cauchy sequence. As the subsequence υnk of υn converges to z, the limit of υn must be z. Also, we have
(42)ρpbz,Sz≤ρpbz,Sυn+1+ρpbSυn+1,Sz<ρpbz,Sυn+1+12ρpbυn,Sυn+1+ρpbz,Sz⇒12ρpbz,Sz≤ρpbz,Sυn+1+12ρpbυn,Sυn+1⟶0asn⟶∞.This infers thatz=Sz. Hence, z is a fixed point of S.Next, we assay the uniqueness ofz. Debating by contradiction, let ż≠z be another fixed point of S, then ρpbz,ż>0. We have
(43)ψρpbz,ż≤ψsρpbSz,Sż≤FψPz,ż,φPz,ż=Fψρpbz,Sz+ρpbż,Sż2,φρpbz,Sz+ρpbż,Sż2≤ψρpbz,z+ρpbż,ż2≤ψρpbz,ż.Thus,ρpbz,ż=0, which is a contradiction. Then, z=ż.Now, the continuity ofS in Theorem 8 has been dropped.Theorem 9.
LetW,ρpb be a complete b-partial metric space (with s≥1) and S:W⟶W. Assume that:
(1)
S is a generalized α−η−ψ−φ−F-contraction(2)
S is triangular weak α-admissible(3)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(4)
Ifυn is a sequence in W in order that αυn,υn+1≥ηυn,υn+1 and υn⟶υ∗∈W as n⟶∞, then there is υnk in order that αυnk,υ∗≥ηυnk,υ∗Then,S possesses a unique fixed point.Proof.
As in the proof of Theorem8, we build the sequence υn by υn+1=Sυn for all n∈ℕ, which converges to υ∗∈W in order that αυn,υn+1≥ηυn,υn+1 for all n∈ℕ. By condition (4), there is a subsequence υnk of υn in order that αυnk,υ∗≥ηυnk,υ∗ for all k∈ℕ. Consequently,
(44)ψρpbυnk+1,Sυ∗≤ψsρpbSυnk,Sυ∗=ψsρpbSυnk,Sυ∗,≤FψPυnk,υ∗,φPυnk,υ∗,≤FψPυnk,υ∗,for all n∈ℕ, where
Case 1.(45)Pυnk,υ∗=maxρpbυn,υ∗,ρpbυnk,Sυnk,ρpbυ∗,Sυ∗,ρpbυnk,Sυ∗+ρpbυ∗,Sυnk2s,=maxρpbυnk,υ∗,ρpbυnk,υnk+1,ρpbυ∗,υ∗,ρpbυnk,υ∗+ρpbυ∗,Sυnk2s,≤maxρpbυnk,υ∗,ρpbυnk,υnk+1,ρpbυ∗,Sυnk.
By lettingn⟶∞, then
(46)Pυnk,υ∗≤ρpbυ∗,Sυ∗.
Case 2.(47)Pυnk,υ∗=ρpbυnk,Sυnk+ρpbυ∗,Sυ∗2,=ρpbυnk,υnk+1+ρpbυ∗,Sυ∗2.
By lettingn⟶∞, then
(48)Pυnk,υ∗≤ρpbυ∗,Sυ∗2≤ρpbυ∗,Sυ∗.
By using (44), we get
(49)ψρpbυ∗,Sυ∗≤Fψρpbυ∗,Sυ∗,ϕρpbυ∗,Sυ∗,which infers that ρpbυ∗,Sυ∗=0. Hence, Sυ∗=υ∗.Following Theorem8 and Theorem 9, we haveCorollary 10.
LetW,ρpb be a complete b-partial metric space (with s≥1). Let S be a self-mapping on W. Given α:W×W⟶0,∞. Assume that
(1)
for allυ,κ∈W with αυ,κ≥1 and ρpbυ,Sκ+ρpbSυ,κ≠0, we have ψsρpbSυ,Sκ≤Fψρpbυ,κ,φρpbυ,κ(2)
S is triangular weak α-admissible(3)
There isυ0∈W in order that αυ0,Sυ0≥1(4)
S is α−η−continuousThen,S admits a unique fixed point.Proof.
We considerη:W×W⟶ℝ as ηυ,κ=1.Corollary 11.
LetW,ρpb be a complete b-partial metric space (with s≥1). Let S be a self-mapping on W. Given α:W×W⟶0,∞. Assume that
(1)
for allυ,κ∈W with αυ,κ≥1 and ρpbυ,Sκ+ρpbSυ,κ≠0, we have ψsρpbSυ,Sκ≤λFψρpbυ,κ,φρpbυ,κ(2)
S is triangular α-admissible(3)
There isυ0∈W in order that αυ0,Sυ0≥1(4)
Ifυn is a sequence in W in order that αυn,υn+1≥1 and υn⟶υ∗∈W as n⟶∞, then there is υnk in order that αυnk,υ∗≥1Then,S possesses a unique fixed point.Proof.
Takeηυ,κ=1 in Theorem 9.Now, by takingFυ,κ=υβυ where Θ:0,∞⟶0,1 is continuous, we haveCorollary 12.
LetW,ρpb be a complete b-partial metric space (with s≥1). Assume S is a self-mapping on W. Given α,η:W×W⟶0,∞). Assume that for all υ,κ∈W with αυ,κ≥ηυ,κ and ρpbυ,Sκ+ρpbSυ,κ≠0(50)ψsρpbSυ,Sκ≤Θψρpbυ,κψρpbυ,κ,where ψ∈Ψ and Θ:0,∞⟶0,1) is continuous. If
(1)
S is triangular weak α-admissible(2)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(3)
S is α−η−continuous
thenS possesses a unique fixed point.Corollary 13.
LetW,ρpb be a complete b-partial metric space (with s≥1) and S be a self-mapping on W. Given α,η:W×W⟶0,∞). Assume that
(51)ψsρpbSυ,Sκ≤Θψρpbυ,κ,ψρpbυ,κ,where ψ∈Ψ and Θ:0,∞⟶0,1) is continuous, for all υ,κ∈W with αυ,κ≥ηυ,κ and ρpbυ,Sκ+ρpbSυ,κ≠0. If
(1)
S is triangular weak α-admissible(2)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(3)
υn is a sequence in W so that αυn,υn+1≥ηυn,υn+1 and υn⟶υ∗∈W as n⟶∞, then there is υnk of υn in order that αυnk,υ∗≥ηυnk,υ∗
thenS possesses a unique fixed point.


## 3. Consequences

LetΔ be the family of nonnegative functions δ defined on 0,∞ so that:
(1)
δ is Lebesgue-integrable on every compact of 0,∞(2)
for everyϖ>0(52)∫0ϖδιdι>0.Theorem 14.
LetW,ρpb be a complete b-partial metric space (with s≥1) and S be a self-mapping on W. Suppose there are F∈C, δ1,δ2∈Δ, and α,η:W×W⟶0,∞ such for all υ,κ∈W with αυ,κ≥ηυ,κ and ρpbυ,Sκ+ρpbSυ,κ≠0, we get
(53)∫0sρpbSυ,Tκδ1zdz≤F∫0maxρpbυ,κ,ρpbυ,Sυ,ρpbSυ,Sυ,ρpbυ,Sκ+ρpbκ,Sυ/2sδ1zdz,∫0maxρpbυ,κ,ρpbυ,Sυ,ρpbSυ,Sυ,ρpbυ,Sκ+ρpbκ,Sυ/2sδ2zdz.Also, suppose that(1)
S is triangular weak α-admissible(2)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(3)
S is α−η−continuousThen,S possesses a unique fixed point.Proof.
Takeψt=∫0tδ1ι)dι and φt=∫0tδ2ι)dι. By using Theorem 8, S possesses a fixed point.Theorem 15.
LetW,ρpb be a complete b-partial metric space (with s≥1) and S:W⟶W. Suppose there are F∈C, δ1,δ2∈Δ, and α,η:W×W⟶0,∞ such for all υ,κ∈W with αυ,κ≥ηυ,κ and ρpbυ,Sκ+ρpbSυ,κ≠0, we have
(54)∫0sρpbSυ,Sκδ1zdz≤F∫0maxρpbυ,κ,ρpbυ,Sυ,ρpbSυ,Sκ,ρpbυ,Sκ+ρpbκ,Sυ/2sδ1zdz,∫0maxρpbυ,κ,ρpbυ,Sυ,ρpbSυ,Sκ,ρpbυ,Sκ+ρpbκ,Sυ/2sδ2zdz.
Assume that:(1)
S is triangular weak α-admissible(2)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(3)
Ifυn is a sequence in W in order that αυn,υn+1≥ηυn,υn+1 and υn⟶υ∗∈W as n⟶∞, then there is υnk in order that αυnk,υ∗≥ηυnk,υ∗Then,S possesses a unique fixed point.Proof.
Considerψt=∫otδ1ι)dι and φt=∫otδ2ι)dι in Theorem 9.Example 1.
GivenW=0,∞ and ρpb:W×W⟶ℝ as
(55)ρpbυ,κ=maxυ,κ2ifυ≠κ,0ifυ=κ,for all υ,κ∈W. Choose ψt=t and ϕt=13/16t. Consider Sυ=eυ/4. Given α,η:W×W⟶0,∞) as
(56)αυ,κ=2ifυ,κ∈0,1,0ifnot,ηυ,κ=1ifυ,κ∈0,1,0ifnot.PutFτ,ω=τ−ω for all τ,ω∈W. We are going to demonstrate the following:
(1)
W,ρpb is a complete partial b-metric space(2)
S is triangular weak α-admissible(3)
There isυ0∈W in order that αυ0,Sυ0≥ηυ0,Sυ0(4)
Ifυn is a sequence in W in order that αυn,υn+1≥ηυn,υn+1 and υn⟶υ∗∈W as n⟶∞Proof.
Note thatW,ρpb is a complete partial b-metric space (Here, s=3).
Ifαυ,Sυ≥ηυ,Sυ, then αSυ,S2υ≥ηSυ,S2υ, So αυ,eυ/4=2>1=ηx,eυ/4, then αeυ/4,eeυ/4/4=2≥1=ηeυ/4,eeυ/4/4. So υ≥0 and so Sυ≤0. Hence, αυ,Sυ≥ηυ,Sυ.Recall thatαυ,Sυ≥ηυ,Sυ for all υ,κ∈0,1]. In this case,
(57)ψsρpbSυ,Sκ=3maxeυ4,eκ42=316maxeυ,eκ2=316ρpbυ,κ≤316Pυ,κ=Pυ,κ−1316Pυ,κ=ψPυ,κ−ϕPυ,κ=FψPυ,κ,ϕPυ,κ.All hypotheses of Corollary11 hold, and so, S admits a unique fixed point.


## 4. Conclusion

We ensured the existence of a unique fixed point for generalized contraction type mappings involving triangular weakα-admissibility with regard to a function η and C-class functions in the class of partial b-metric spaces. Some illustrated examples have been also provided.



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