# Admissible HybridZ-Contractions in Extended b-Metric Spaces

Authors: Quanita Kiran; Mamuna Azhar; Hassen Aydi; Yaé Ulrich Gaba
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we define the notation of admissible hybridZ-contractions in the setting of extended b-metric spaces, which unifies and generalizes previously existing results in literature. Furthermore, as an application, we discuss Ulam-Hyers stability and well-posedness of a fixed point problem.


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## Body

## 1. Introduction

The source of metric fixed point theory is considered to be the Banach Contraction Principle which is a very important mechanism for finding the existence and solutions for many problems including differential and integral equations. Afterwards, numerous papers on generalizations and extensions of the Banach’s theorem for both single-valued and multivalued mappings have been published, either by changing the contraction conditions or by changing the structure of metric space to more generalized form, e.g., see [1] and all the references therein.The concept of ab-metric space was accomplished by the works of Bourbaki [2], Bakhtin [3], and Czerwik [4]. Subsequently, several articles have appeared in literature which dealt with the fixed point theorems by taking into account more general forms, of a metric spaces, i.e., b-metric space, see [5], and the applications of relaxed triangular inequalities, like NEM (nonlinear elastic matching), ice floes, etc., were also utilized in various directions [6, 7]. Following the idea of b-metric spaces, a number of authors have presented several results in this direction, see [8, 9]. To have some insight about miscellaneous generalizations of a metric, we refer the readers to [10–15] for some works on b-metric spaces.In 2017, Kamran et al.[16] generalized the structure of a b-metric space and referred it as an extended b-metric space. He weakened the triangle inequality of a metric and established fixed point results for a class of contractions. Thereafter, many researchers have studied and generalized fixed point results for single and multivalued mappings. Proving extensions of the Banach contraction principle from metric spaces to b-metric spaces and hence to extended b-metric spaces is useful to prove existence and uniqueness theorems for different types of integral and differential equations. Keeping the length of paper concise, we refer to [17–35] and to references mentioned therein. For more topological properties of extended b-metric spaces, see [20].The main purpose of this paper is to merge different linear and nonlinear results existing in literature in setup of an extendedb-metric space, which is a real generalization of a b-metric and a standard metric space. We express our results in a more refined form by combining the notations, like admissible mappings, simulation functions, and hybrid contractions. We will prove fixed point results involving a certain type of mappings. The obtained results generalize [36]. Moreover, we prove Ulam-Hyers stability [37–39] and well-posedness of fixed point problems as well.


## 2. Preliminaries

In this section, we recollect some definitions and results from literature along with some examples.Definition 1 [40].
LetX be a nonempty set and θ:X×X⟶1,∞. A function dθ:X×X⟶0,∞ is called an extended b-metric, if it satisfies the following properties for allμ,γ,ν∈X:
(dθ1) dθμ,ν=0⇐μ=ν
(dθ2) dθμ,ν=dθν,μ
(dθ3) dθμ,γ≤θμ,γdθμ,ν+dθν,γ
The pairX,dθ is called an extended b-metric space.Example 1.
LetX=0,1 and θ:X×X⟶1,∞ defined by θμ,ν=1+μ+ν/μ+ν. Define dθ:X×X⟶0,∞as
dθμ,ν=1/μν for μ,ν∈0,1,μ≠ν
dθμ,ν=0 for μ,ν∈0,1,μ=ν
dθμ,0=dθ0,μ=1/μν for μ∈0,1
Note thatX,dθ is an extended b-metric space.Example 2.
LetX=1,2,3, θ:X×X⟶1,∞, and dθ:X×X⟶0,∞as
θμ,ν=1+μ+ν such that
dθ1,1= dθ2,2=dθ3,3=0
dθ1,2 = dθ2,1=70
dθ1,3 = dθ3,1=90
dθ2,3 = dθ3,2=20
Here,dθ is an extended b-metric on X.Note that the extendedb-metric space becomes a b-metric space, whenever θμ,ν=δ, where δ≥1 and a standard metric space forδ=1.Definition 2 [16].
LetX,dθ be an extended b-metric space. The sequence μn in X is termed as follows:(i)
Cauchy if and only ifdθμn,μm⟶0 as n,m⟶∞(ii)
Convergent if and only if there existsμ∈X such that dθμn,μ⟶0 asn⟶∞ and we write limn⟶∞μn=μ
Note that the extendedb-metric space X,dθ is complete if every Cauchy sequence is convergent.Theb-metric is not continuous in general and so the same for an extended b-metric. We define the concept of f-orbital continuity (in case of an extended b-metric space) as used in [41].Definition 3 [42].
Given a mappingf:D⊂X⟶X. Suppose that there exists some μ0∈D such that Oμ0=μ0,fμ0,f2μ0,⋯⊂D. The set Oμ0 is called the orbit of μ0∈D. A self-mapping f:X⟶X is called orbitally continuous if limn⟶∞fnη=η for some η∈X implies that(1)limn⟶∞ffnη=fη.
Moreover, if every Cauchy sequence of the formfnη as n⟶∞, η∈X converges in X,dθ, then an extended b-metric space X,dθ is called f-orbitally complete.Definition 4 [40].
LetX,dθ be an extended b-metric space. A function ϕ:ℝ+⟶ℝ+ is called an extended b-comparison function if it is increasing and also there exists a mapping f:D⊂X⟶X such that for some μ0∈D, Oμ0⊂D, ∑n=0∞ϕnυ∏i=0nθμi,μm converges for each υ∈ℝ+ and for every m∈ℕ. Here, μn=fnμ0 for n=0,1,2,⋯. We say that ϕ is an extended b-comparison function for f at μ0 and denotes the collection of all extended b-comparison functions byΨs.Example 3 [40].
LetX,dθ be an extended b-metric space and f be a self mapping on X. Assume that limn,m⟶∞θμn,μm exists. Define ϕ:0,∞⟶0,∞ such that ϕυ=λυ, with limn,m⟶∞θμn,μm<1/λ. Then, the series, ∑n=0∞ϕnυ∏i=0nθμi,μm converges by ratio test.
Here,λ∈0,1 and μn=fnμ0 forn=1,2,⋯.The notation ofα-admissible mappings also played a vital role in fixed point theory, see [43, 44].Definition 5 [36].
Letα:X×X⟶0,∞ be a mapping. A function f:X⟶X is α-orbital admissible ifαμ,fμ≥1αfμ,f2μ≥1.
Anα-orbital admissible, mapping f is called triangular α-orbital admissible, if αμ,ν≥1 and αν,fν≥1αμ,fν≥1 for allμ,ν∈X.Example 4.
LetX=0,1,2,3 and f:X⟶X such that f0=0 and f1=f2=f3=1. Consider α:X×X⟶0,∞ given as α1,2=α2,1=α1,3=α3,1=α1,1=1 and 0 otherwise. Clearly, f is α-orbital admissible.Definition 6 [45].
A mappingζ:0,∞×0,∞⟶ℝ satisfying the following conditions
(ζ1) ζt,s<s−t for allt,s>0
(ζ2) If tnn∈ℕ, snn∈ℕ are the sequences in 0,∞ such that
limn⟶∞tn=limn⟶∞sn>0. Then, limsupn⟶∞ζtn,sn<0 is termed as a simulation function
LetZ represents the set of all simulation functions defined above.The main idea of the simulation function is very useful and effective. For a self-mappingT on a metric space, the contraction dTμ,Tν≤κdμ,ν can be represented as 0≤κdμ,ν−dTμ,Tν=ζdμ,ν,dTμ,Tν, where κ∈0,1 and ζ:0,∞×0,∞⟶ℝ. By letting dμ,ν=s and dTμ,Tν=t, the corresponding simulation function for Banach’s fixed point theorem is ζt,s=κs−t. It is clear that for many other well-known results (Geraghty, Boyd-Wong, etc.), one can find a corresponding simulation function, see [36, 46–51]. In other words, a simulation function can be considered as a generator of different contraction type inequalities.Definition 7.
A self-mappingf, defined on a metric space X,d, is called a Z-contraction with respect to ζ∈Z, if it satisfies(2)ζdfμ,fν,dμ,ν≥0forallμ,ν∈X.EveryZ-contraction defined on a complete metric space has a 0 fixed point, as described in [46]. A Z-contraction generalizes the Banach contraction principle by assuming γ∈0,1 and ζt,s=γs−t for all s,t∈0,∞. It also unifies several known type of contractions. Many authors have extended their work on Z-contractions in order to prove a more generalized version (see [47]).The notion of admissible hybrid contractions is introduced [36, 46, 48–50] in order to generalize and unify the several existing fixed point results in the literature. The main goal of this paper is to investigate the existence and uniqueness of a fixed point of admissible hybrid Z-contractions in the context of an extended b-metric space. We shall also list some existing results in the literature as corollaries and consequences of our main results. Consequently, the results in the class of b-metric spaces and standard metric spaces become a special case of our obtained results.


## 3. Main Results

Definition 8.
LetX,dθ be an extended b-metric space. A self-mapping f is called an admissible hybrid contraction if there exist an extended b-comparison function ψ:0,∞⟶0,∞∈Ψs and α:X×X⟶0,∞ such that(3)αμ,νdθfμ,fν≤ψPfrμ,ν,
wherer≥0 and λi≥0, i=1,2,3,4,5 with ∑i=15λi=1, and(4)Pfrμ,ν=Qμ,ν1/r,forr>0andμ,ν∈X,Rμ,ν,forr=0andμ,ν∈X,
where(5)Qμ,ν≔λ1dθrμ,ν+λ2dθrμ,fμ+λ3dθrν,fν+λ4dθν,fν1+dθμ,fμ1+dθμ,νr+λ5dθν,fμ1+dθμ,fν1+dθμ,νr,and(6)Rμ,ν≔dθλ1μ,ν·dθλ2μ,fμ·dθλ3ν,fν·dθν,fν1+dθμ,fμ1+dθμ,νλ4·dθμ,fν+dθν,fμ2θμ,fνλ5.Definition 9.
LetX,dθ be an extended b-metric space. A self-mapping f is said to be an admissible hybrid Z-contraction, if there exist an extended b-comparison function ψ∈Ψs, α:X×X⟶0,∞, and ζ∈Z such that(7)ζαμ,νdθfμ,fν,ψPfrμ,ν≥0,∀μ,ν∈X.Further, we discuss the existence and uniqueness of a fixed point of an admissible hybridZ-contraction mapping.Note that we assume thatdθ is continuous and limm,n⟶∞θμm,μn<∞, throughout Section 3 and Section 4.Theorem 10.
LetX,dθ be an extended b-metric space. Let f:X⟶X be an admissible hybrid Z-contraction, which satisfies the following axioms:(i)
The functionf is triangular α-orbital admissible(ii)
There existsμ0∈X such thatαμ0,fμ0≥1(iii)
Eitherf is continuous or(iv)
f2 is continuous and αμ,fμ≥1 for any μ∈Fixf2X
Then,f possesses a fixed point.Proof.
Letμ0∈X be any arbitrary point. We start from μ0 and iteratively, we construct a sequence μnn∈ℕ such that μn=fnμ0 for all n∈ℕ. Suppose that there exists some m∈ℕ such that fμm=μm+1=μm, we find that μm is a fixed point of f, and in this way, the proof is completed. Thus, we can assume that μn≠μn−1 for any n∈ℕ. By condition (i), as f is an admissible hybrid Z-contraction, so by assuming μ=μn−1 and ν=μn in equation (3), we have(8)0≤ζαμn−1,μndθfμn−1,fμn,ψPfrμn−1,μn<ψPfrμn−1,μn−αμn−1,μndθfμn−1,fμn,which gives,(9)αμn−1,μndθfμn−1,fμn≤ψPfrμn−1,μn.
By condition (ii), asf is triangular α-orbital admissible and αμn−1,μn≥1, so(10)dθμn,μn+1≤αμn−1,μndθfμn−1,fμn≤ψPfrμn−1,μn.☐Case 1.
Considerr>0 so that(11)Pfrμn-1,μn=λ1dθrμn-1,μn+λ2dθrμn-1,fμn-1+λ3dθrμn,fμn+λ4dθμn,fμn1+dθμn-1,fμn-11+dθμn-1,μnr+λ5dθμn,fμn-11+dθμn-1,fμn1+dθμn-1,μnr1/r=λ1dθrμn-1,μn+λ2dθrμn-1,μn+λ3dθrμn,μn+1+λ4dθμn,μn+11+dθμn-1,μn1+dθμn-1,μnr+λ5dθμn,μn1+dθμn-1,μn+11+dθμn-1,μnrλ5dθμn,μn1+dθμn-1,μn+11+dθμn-1,μn=λ1dθrμn-1,μn+λ2dθrμn-1,μn+λ3dθrμn,μn+1+λ4dθμn,μn+1r1/r=λ1+λ2dθrμn-1,μn+λ3+λ4dθrμn,μn+11/r.
From equation (10),(12)dθμn,μn+1≤αμn−1,μndθfμn−1,fμn≤ψPfrμn−1,μn=ψλ1+λ2dθrμn−1,μn+λ3+λ4dθrμn,μn+11/r.
Supposedθμn−1,μn≤dθμn,μn+1. As ψ is an increasing function, so above inequality can be written as(13)dθμn+1,μn≤αμn−1,μndθfμn−1,fμn≤ψλ1+λ2dθrμn−1,μn+λ3+λ4dθrμn,μn+11/r≤ψλ1+λ2+λ3+λ4dθrμn,μn+11/r=ψλ1+λ2+λ3+λ41/rdθμn,μn+1.
Asψt<t, so(14)dθμn+1,μn<λ1+λ2+λ3+λ41/rdθμn,μn+1.
Sinceλ1+λ2+λ3+λ4≤1, we get(15)dθμn+1,μn<λ1+λ2+λ3+λ41/rdθμn,μn+1≤dθμn,μn+1,which is a contradiction. Thus, for every n∈ℕ, dθμn,μn+1<dθμn−1,μn, and thus equation (10) becomes(16)dθμn,μn+1≤ψλ1+λ2dθrμn−1,μn+λ3+λ4dθrμn,μn+11/r<ψλ1+λ2+λ3+λ41/rdθμn−1,μn≤ψdθμn−1,μnψ2dθμn−2,μn−1≤⋯≤ψndθμ0,μ1.
Letm>n, by triangular inequality, we have(17)dθμn,μm≤θμn,μmdθμn,μn+1+θμn,μmθμn+1,μmdθμn+1,μn+2+⋯+θμn,μmθμn+1,μmθμn+2,μm⋯θμm−2,μmθμm−1,μmdθμm−1,μm≤θμn,μmψndθμ0,μ1+θμn,μmθμn+1,μmψn+1dθμ0,μ1+⋯+θμn,μmθμn+1,μmθμn+2,μm⋯θμm−2,μmθμm−1,μmψm−1dθμ0,μ1≤θμ1,μmθμ2,μm⋯θμn−1,μmθμn,μmψndθμ0,μ1+θμ1,μmθμ2,μm⋯θμn,μmθμn+1,μmψn+1dθμ0,μ1+⋯+θμ1,μmθμ2,μm⋯θμn,μmθμn+1,μm⋯θμm−2,μmθμm−1,μmψm−1dθμ0,μ1.
Sinceψ is an extended b-comparison function, the series(18)∑n=1∞ψndθμ0,μ1∏x=1nθμx,μm,is convergent for every m∈ℕ.
DenoteS=∑n=1∞ψndθμ0,μ1∏x=1nθμx,μm and Sn=∑j=1nψjdθμ0,μ1∏x=1jθμx,μm.
Thus, form>n, the above inequality becomes(19)dθμn,μm≤Sm−1−Sn.
Lettingn,m⟶∞, we get(20)dθμn,μm⟶0,which implies that μnn∈ℕ is a Cauchy sequence in a complete extended b-metric space. Therefore, it is convergent, so there exists μ′∈ℕ such that(21)limn⟶∞dθμn,μ′=0.
Now, we prove thatμ′ is a fixed point of f. By condition (iii), if f is continuous, we have(22)dθμ′,fμ′=limn⟶∞dθμn,fμn=limn⟶∞dθμn,μn+1=0,μ′=fμ′.
Therefore,μ′ is a fixed point of f. Now, consider that f2 is continuous. It follows that f2μ′=limn⟶∞f2μn=μ′. We shall now prove that fμ′=μ′. Contrarily, suppose that, fμ′≠μ′, from equation (3)(23)0≤ζαfμ′,μ′dθf2μ′,fμ′,ψPfrfμ′,μ′=ψPfrfμ′,μ′−αfμ′,μ′dθf2μ′,fμ′.
It implies that(24)dθμ′,fμ′=dθf2μ′,fμ′≤αfμ′,μ′dθfμ′,μ′.
Asψt<t, so(25)ψPfrfμ′,μ′<Pfrfμ′,μ′,Pfrfμ′,μ′=λ1dθrfμ′,μ′+λ2dθrμ′,fμ′+λ3dθrfμ′,f2μ′+λ4dθμ′,fμ′1+dθfμ′,f2μ′1+dθμ′,fμ′r+λ5dθfμ′,fμ′1+dθμ′,f2μ′1+dθμ′,fμ′r1/r=λ1dθrfμ′,μ′+λ2dθrμ′,fμ′+λ3dθrfμ′,μ′+λ4dθμ′,fμ′1+dθfμ′,μ′1+dθμ′,fμ′r+λ5dθfμ′,fμ′1+dθμ′,μ′1+dθμ′,fμ′r1/r≤λ1+λ2+λ3+λ4dθμ′,fμ′1/r=λ1+λ2+λ3+λ41/rdθrμ′,fμ′≤dθμ′,fμ′,which leads to a contradiction. Thus, fμ′=μ′.Case 2.
Considerr=0. Let μ=μn−1 and ν=μn. One writes(26)Pfrμn−1,μn=dθλ1μn−1,μn·dθλ2μn−1,fμn−1·dθλ3μn,fμn·dθμn,fμn1+dθμn−1,fμn−11+dθμn−1,μnλ4·dθμn−1,fμn+dθμn,fμn−12θμn−1,fμnλ5=dθλ1μn−1,μn·dθλ2μn−1,μn·dθλ3μn,μn+1·dθμn,μn+11+dθμn−1,μn1+dθμn−1,μnλ4·dθμn−1,μn+1+dθμn,μn2θμn−1,μn+1λ5=dθλ1μn−1,μn·dθλ2μn−1,μn·dθλ3μn,μn+1·dθλ4μn,μn+1·dθμn−1,μn+12θμn−1,μn+1λ5.
Using triangular inequality, we obtain(27)Pfrμn−1,μn≤dθλ1μn−1,μn·dθλ2μn−1,μn·dθλ3μn,μn+1·dθλ4μn,μn+1·θμn−1,μn+1dθμn−1,μn+dθμn,μn+12θμn−1,μn+1λ5≤dθλ1μn−1,μn·dθλ2μn−1,μn·dθλ3μn,μn+1·dθλ4μn,μn+1·dθμn−1,μn+dθμn,μn+12λ5.
By using the following inequality, one gets(28)x+y2c≤xc+yc2,∀x,y,c>0.
So, we have(29)Pfrμn−1,μn≤dθλ1μn−1,μn·dθλ2μn−1,μn·dθλ3μn,μn+1·dθλ4μn,μn+1·dθμn−1,μnλ5+dθμn,μn+1λ52,and by equation (7),(30)0≤ζαμn−1,μndθfμn−1,fμn,ψPfrμn−1,μn<ψPfrμn−1,μn−αμn−1,μndθfμn−1,fμn,which gives,(31)dθμn,μn+1≤αμn−1,μndθfμn−1,fμn≤ψPfrμn−1,μn.
Suppose thatdθμn−1,μn≤dθμn,μn+1. As ψ is an increasing function, so(32)dθμn,μn+1≤dθλ1+λ2+λ3+λ4+λ5μn,μn+1=dθμn,μn+1,dθμn,μn+1≤dθμn,μn+1,so, we have two cases here that dθμn,μn+1=dθμn,μn+1 or dθμn,μn+1<dθμn,μn+1 which is a contradiction. Thus, we obtain(33)dθμn,μn+1≤ψPfrμn−1,μn<ψndθμ0,μ1.
Utilizing the same arguments as of Case1, we obtain that μnn∈ℕ is a Cauchy sequence in an extended b-metric space. Therefore, it is convergent, so there exists μ′∈ℕ such that(34)limn⟶∞μn=μ′.
Now, further we claim thatμ′ is a fixed point of f. By condition (iii), if f is continuous, then(35)dθμ′,fμ′=limn⟶∞dθμn,fμn=limn⟶∞dθμn,μn+1=0.
Thus,(36)μ′=fμ′.
Hence,μ′ is a fixed point of f. Now, consider f2 is continuous. It follows that f2μ′=μ′. Now, we shall validate that fμ′=μ′. Contrarily, suppose that fμ′≠μ′, then(37)0≤ζαfμ′,μ′dθf2μ′,fμ′,ψPfrf2μ′,fμ′=ψPfrf2μ′,fμ′−αfμ′,μ′dθf2μ′,fμ′.
It implies that(38)dθμ′,fμ′=dθf2μ′,fμ′≤αfμ′,μ′dθf2μ′,fμ′≤ψPfrf2μ′,fμ′=ψPfrμ′,fμ′,where(39)Pfrμ′,fμ′=dθλ1+λ2+λ3μ′,fμ′·dθμ′,fμ′1+dθμ′,fμ′1+dθμ′,fμ′λ4·dθμ′,f2μ′+dθfμ′,fμ′2θμ′,fμ′λ5=dθλ1+λ2+λ3+λ4μ′,fμ′<dθμ′,fμ′.
Therefore, we have(40)dθμ′,fμ′≤ψPfrμ′,fμ′<ψdθμ′,fμ′<dθμ′,fμ′,which is a contradiction. So, fμ′=μ′.Theorem 11.
Assume that all the postulates of Theorem10 hold. Additionally, suppose that αμ′,ν′≥1, for any μ′,ν′∈FixfX. Then, f has a unique fixed point.Proof.
Assume that the fixed point of the mappingf is not unique. Let ν′∈X be another fixed point of f, where μ′≠ν′. For case r>0, by equation (7), we get(41)0≤ζαμ′,ν′dθfμ′,fν′,ψPfrμ′,ν′<ψPfrμ′,ν′−αμ′,ν′dθfμ′,fν′.
This yields that(42)dθμ′,ν′≤αμ′,ν′dθfμ′,fν′≤ψPfrμ′,ν′<Pfrμ′,ν′=λ1dθrμ′,ν′+λ2dθrμ′,fμ′+λ3dθrν′,fν′+λ4dθν′,fν′1+dθμ′,fμ′1+dθμ′,ν′r+dθν′,fμ′1+dθμ′,fν′1+dθμ′,ν′r1/r=λ1+λ51/rdθμ′,ν′<dθμ′,ν′,which is a contradiction. Similarly, for case r=0, we obtain 0<dθμ′,ν′<0, which also gives a contradiction. Thus, μ′=ν′; that is, f has a unique fixed point.☐Example 1.
LetX=0,3 and d:X×X⟶0,∞ be such that dμ,ν=μ−ν2 and θμ,ν=μ+ν+2 for all μ,ν∈X. Consider the mapping f:X⟶X defined by(43)fμ=13,ifμ∈0,1,μ3,ifμ∈1,3,and(44)αμ,ν=3,ifμ,ν∈0,1,1,ifμ=0,ν=3,ψ:0,∞⟶0,∞definedbyψt=t3&ζt,s=s3−t,0,otherwise.
Note that(i)
X,dθ is an extended complete b-metric space with θμ,ν=μ+ν+2(ii)
f is triangular α-orbital admissible mapping(iii)
Forμ0∈0,1,fμ0=1/3∈0,1, and therefore, αμ0,fμ0=3>1(iv)
f is continuous(v)
f2 is continuous, where f2=1/3
Furthermore, forμ=1/3∈Fixf2X, we get α1/3,f1/3=3>1(vi)
ζαμ,νdθfμ,fν,ψPfrμ,ν≥0
Considerμ,ν∈0,1, then fμ=fν=1/3, and hence, dθfμ,fν=0. For all μ,ν∈0,1, we have(45)ζαμ,νdθfμ,fν,ψPfrμ,ν=ζ0,ψPfrμ,ν=13ψPfrμ,ν.
Hence,(46)ζαμ,νdθfμ,fν,ψPfrμ,ν≥1∀μ,ν∈0,1.
Now, considerμ=0,ν=3, r=2, and λ1=λ2=λ3=λ4=λ5=1/5, we have(47)ζαμ,νdθfμ,fν,ψPfrμ,ν=ζα0,3dθf0,f3,ψPfr0,3=13ψPfr0,3−α0,3dθf0,f3=13·1315dθ20,3+15dθ20,f0+15dθ23,f3+15dθ3,f31+dθ0,f01+dθ0,32+15dθ3,f01+dθ0,f31+dθ0,321/2−α0,3dθ13,1=191581+181+16+1681+642251/2−49≥0.
Hence,(48)ζα0,3dθf0,f3,ψPfr0,3≥0.
In all other cases, we haveαμ,ν=0, so(49)ζ0,ψPfrμ,ν=13ψPfrμ,ν≥0.
Hence, we acquire thatf is an admissible hybrid Z-contraction. It follows all the hypothesis of Theorem 11, and so, μ=1/3 is the fixed point of f.LetΦ be the collection of all auxiliary functions φ:0,∞⟶0,∞, which are continuous and φυ=0 if and only if υ=0.Corollary 12.
LetX,dθ be an extended b-metric space, f:X⟶X and α:X×X⟶0,∞. Suppose that there exist two functions φ1,φ2∈Φ with φ1υ<υ<φ2υ, for all υ>0, such that(50)φ2αμ,νdθfμ,fν≤φ1Pfrμ,ν,where ζ is defined as ζt,s=φ1s−φ2t. Additionally, suppose that(i)
f is triangular α-orbital admissible(ii)
There existsμ0∈X such thatαμ0,fμ0≥1(iii)
Eitherf is continuous or(iv)
f2 is continuous and αμ,fμ≥1 for anyμ∈Fixf2X(v)
Ifμ′,ν′∈FixfX, thenαμ′,ν′≥1
Then,f has a unique fixed point.Corollary 13.
LetX,dθ be an extended b-metric space. Suppose that there exists a function φ∈Φ, where Φ:0,∞⟶0,∞ such that all φ∈Φ is continuous and φυ=0 if and only if υ=0, such that(51)αμ,νdθfμ,fν≤Pfrμ,ν−φPfrμ,ν,where ζ is defined as ζt,s=s−φs−t. Furthermore, we assume that(i)
f is triangular α-orbital admissible(ii)
There existsμ0∈X such that αμ0,fμ0≥1(iii)
Eitherf is continuous or(iv)
f2 is continuous and αμ,fμ≥1 for any μ∈Fixf2X(v)
Ifμ′,ν′∈FixfX, thenαμ′,ν′≥1
Then,f has a unique fixed point.Corollary 14.
LetX,dθ be an extended b-metric space. Suppose that there exist a function χ:0,∞⟶0,∞ such that ∫0Pfrμ,νχpdp exists and ∫0Pfrμ,νχpdp>ε, for every ε>0, with property that(52)αμ,νdθfμ,fν≤∫0Pfrμ,νχpdp,where ζ is defined as ζt,s=s−∫0tχudu. Moreover, we suppose that(i)
f is triangular α-orbital admissible(ii)
There existsμ0∈X such that αμ0,fμ0≥1(iii)
Eitherf is continuous or(iv)
f2 is continuous mapping and αμ,fμ≥1 for anyμ∈Fixf2X(v)
Ifμ′,ν′∈FixfX, thenαμ′,ν′≥1
Then,f has a unique fixed point.


## 4. Application

In this section, we explore Ulam-Hyers stability and well posedness of the fixed point problem in the setup of an extendedb-metric space (see [36] and references therein).Definition 15.
Letf:X⟶X be a self-mapping defined on an extended b-metric space. Consider the fixed point problem(53)μ=fμ.
The fixed point problem (53) is well-posed if(i)
FixfX=μ′(ii)
Ifμnn∈ℕ is a sequence such that dθμn,fμn⟶0, as n⟶∞, then μn⟶μ′, as n⟶∞Theorem 16.
LetX,dθ be an extended b-metric space. Suppose that all the assumptions of Theorem 11 hold, and r>0. Additionally, we assume that for any sequence μnn∈ℕ, dθμn,fμn⟶0, as n⟶∞, we have αμn,μ′≥1, for all n∈ℕ, where μ′∈FixfX. If λ1+λ5<1/η2r, where ηr=max1,2r−1θμ,νr∀μ,ν∈X, then the fixed point problem (53) is well-posed.Proof.
Asμ′=FixfX, by equation (7),(54)0≤ζαμn,μ′dθfμn,fμ′,ψPfrμn,μ′<ψPfrμn,μ′−αμn,μ′dθfμn,fμ′.
Consider(55)dθμn,μ′≤θμn,μ′dθμn,fμn+dθfμn,μ′=θμn,μ′dθμn,fμn+θμn,μ′dθfμn,μ′≤θμn,μ′dθμn,fμn+θμn,μ′αμn,μ′dθfμn,fμ′≤θμn,μ′dθμn,fμn+θμn,μ′ψPfrμn,μ′<θμn,μ′dθμn,fμn+θμn,μ′Pfrμn,μ′≤θμn,μ′dθμn,fμn+θμn,μ′λ1dθrμn,μ′+λ2dθrμn,fμn+λ3dθrμ′,fμ′+λ4dθμ′,fμ′1+dθμn,fμn1+dθμn,μ′r+λ5dθμ′,fμn1+dθμn,fμ′1+dθμn,μ′r1/r=θμn,μ′dθμn,fμn+θμn,μ′λ1dθrμn,μ′+λ2dθrμn,fμn+λ5dθrμ′,fμn1/r≤θμn,μ′dθμn,fμn+θμn,μ′λ1dθrμn,μ′+λ2dθrμn,fμn+θμ′,fμnrλ5dθμ′,μn+dθμn,fμnr1/r≤θμn,μ′dθμn,fμn+θμn,μ′λ1dθrμn,μ′+λ2dθrμn,fμn+2r−1θμ′,fμnrλ5dθrμ′,μn+2r−1θμ′,fμnrλ5dθrμn,fμn1/r.
In this way, we obtain(56)dθrμn,μ′≤2r−1θμn,μ′rdθrμn,fμn+2r−1θμn,μ′rλ1dθrμn,μ′+2r−1θμn,μ′rλ2dθrμn,fμn+22r−2θμn,μ′rθμ′,fμnrλ5dθrμ′,μn+22r−2θμn,μ′rθμn,fμnrλ5dθrμn,fμn,or we can write(57)1−2r−1θμn,μ′rλ1−22r−2θμn,μ′rθμ′,fμnrλ5dθrμn,μ′≤2r−1θμn,μ′r1+λ2+2r−1θμn,fμnrλ5dθrμn,fμn.
From here, we get(58)dθrμn,μ′≤2r−1θμn,μ′r1+λ2+2r−1θμn,fμnrλ51−2r−1θμn,μ′rλ1−22r−2θμn,μ′rθμ′,fμnrλ5dθrμn,fμn,(59)dθrμn,μ′≤ηr1+λ2+ηrλ51−ηrλ1−η2rλ5dθrμn,fμn.
Asn⟶∞, we have that limn⟶∞dθrμn,fμn=0. So, limn⟶∞dθrμ′,μn=0.
Thus, the fixed point problem (53) is well-posed.☐Definition 17.
The fixed point problemμ=fμ is called generalized Ulam-Hyers stable if and only if there exists ω:0,∞⟶0,∞ which is increasing, continuous at 0 with ω0=0, such that for every ξ>0 and for each ν′∈X,(60)dθν,fν≤ξ,there exists a solution μ′ of the fixed point problem such that dθν′,μ′≤ωξ.
If there existsx>0 such that ωa≔x.a, for each a∈ℝ+, then the fixed point problem is referred to be Ulam-Hyers stable.Theorem 18.
LetX,dθ be an extended b-metric space. Suppose that all the assumptions of Theorem 11 hold and r>0.
Furthermore, we assume thatαν′,μ′≥1, for all ν′∈X satisfying (60), where μ′∈FixfX. If λ1+λ5<1/η2r, where ηr=max1,2r−1θμ,νr∀μ,ν∈X, then the fixed point problem μ=fμ is Ulam-Hyers stable.Proof.
By (7),(61)0≤ζαν′,μ′dθfν′,fμ′,ψPfrν′,μ′<ψPfrν′,μ′−αν′,μ′dθfν′,fμ′.
Consider(62)dθν′,μ′=dθν′,fμ′≤θν′,μ′dθν′,fμ′+dθfν′,fμ′=θν′,μ′dθν′,fμ′+θν′,μ′dθfν′,fμ′≤θν′,μ′dθν′,fν′+θν′,μ′αν′,μ′dθfν′,fμ′≤θν′,μ′ξ+θν′,μ′ψPfrν′,μ′<θν′,μ′ξ+θν′,μ′Pfrν′,μ′≤θν′,μ′ξ+θν′,μ′λ1dθrν′,μ′+λ2dθrν′,fν′+λ3dθrμ′,fμ′+λ4dθμ′,fμ′1+dθν′,fν′1+dθν′,μ′r+λ5dθμ′,fν′1+dθν′,fμ′1+dθν′,μ′r1/r=θν′,μ′ξ+θν′,μ′λ1dθrν′,μ′+λ2ξr+λ5dθrμ′,fν′1/r≤θν′,μ′ξ+θν′,μ′λ1dθrν′,μ′+λ2ξr+θμ′,fν′rλ5dθμ′,ν′+dθν′,fμ′r1/r≤θν′,μ′ξ+θν′,μ′λ1dθrν′,μ′+λ2ξr+2r−1θμ′,fν′rλ5dθrμ′,ν′+2r−1θμ′,fν′rλ5dθrν′,fν′1/r.
Thus, we get(63)dθrν′,μ′≤2r−1θν′,μ′rξr+2r−1θν′,μ′rλ1dθrν′,μ′+2r−1θν′,μ′rλ2ξr+22r−2θν′,μ′rθμ′,fν′rλ5dθrμ′,ν′+22r−2θν′,μ′rθν′,fν′rλ5ξr,or we can write(64)1−2r−1θν′,μ′rλ1−22r−2θν′,μ′rθμ′,fν′rλ5dθrμ′,ν′≤2r−1θν′,μ′r1+λ2+2r−1θν′,fν′rλ5ξr.
Therefore, we obtain(65)dθrμ′,ν′≤2r−1θν′,μ′r1+λ2+2r−1θν′,fν′rλ51−2r−1θν′,μ′rλ1−22r−2θν′,μ′rθμ′,fν′rλ5ξr.
Thus, we get(66)dθrμ′,ν′≤ηr1+λ2+ηrλ51−ηrλ1−η2rλ5ξr.
Hence,(67)dθμ′,ν′≤xξr,where x=ηr1+λ2+ηrλ5/1−ηrλ1−η2rλ5 for all r>0 and λ1,λ5∈0,1.☐


## 5. Conclusion

In this research paper, we consolidated and refined several existing results in literature by bringing up the notation of admissible hybridZ-contractions in the setup of an extended b-metric space. Accordingly, all the exhibited results are authentic in context of complete b-metric spaces by letting θμ,ν=δ, where δ≥1, and in context of complete metric spaces by letting δ=1. Furthermore, the paper generalizes the results of [36, 45, 46, 51]. Numerous fixed point results can be concluded in standard b-metric spaces via a partial order or a cyclic contraction. Moreover, one can derive results in extended b-metric spaces using [52–55].



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