# New Fixed Point Theorems for Admissible Hybrid Maps

Authors: Maha Noorwali; Seher Sultan Yeşilkaya
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The aim of this work is to investigate the concept of a new hybrid Suzuki contractive by using the Rus-Reich-Ćirić-type interpolative mappings inb-metric spaces. We seek the presence and uniqueness of a fixed point of such new contraction type mappings and prove some related results. We further give an application of Ulam-Hyers-type stability to show the well-posedness of our results.


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## Body

## 1. Introduction and Preliminaries

Fixed point hypothesis has been a considerable area of research for mathematics and other sciences for the last century. It is the basis of functional analysis in mathematics, which is one of the critical topics of mathematics. The first concept of fixed point theory is knowing to appear in the work of Liouville in 1837 and Picard in 1890. But the main fixed point theorem was introduced by Banach [1]. The theorem is named after Banach. There are many generalizations of Banach theorem in the literature. In 1968, one of the most famous generalizations due to know, Kannan [2] introduced a new and useful contraction using Banach’s theorem. Suzuki [3] introduced important extensions of Banach’s main theorem, which we refer to [4–6]. In one of these studies [7], the researchers investigated a new extensive result by using simulation function. On the other hand, in [8], by using other auxiliary functions, called the Wardowski functions, they observed a contraction that combines both linear and nonlinear contractions. We also mention that in [9], the author obtained a fixed point theorem without the Picard operator. For more interesting results, see, e.g., [10–19]. In addition, Banach’s fixed point theorem is a significant mean in the theory of metric spaces. The metric concept has been generalized from different angles. One of the significant generalizes is defined b-metric which was defined as follows.Definition 1 (see [20, 21]).
LetL be a (nonempty) set and s≥1 a real number. A function b:L×L⟶0,∞ is a b-metric space on L if following conditions are satisfied:
(i)
br,v=0, if r=v(ii)
br,v=bv,r(iii)
br,v≤sbr,q+bq,v, for every r,v,q∈L
In this case, the pairL,b,s is called a b-metric space.We recollect some basic notions that are used in our study.A mapφ:0,∞⟶0,∞ is defined as a comparison function if it is increasing and φqz⟶0,q⟶∞, for any z∈0,∞. We state by Φ the class of all the comparison functions φ:0,∞⟶0,∞, see, e.g., [22–24]. Defined by Ψ=ψ:0,∞⟶0,∞ψistheb‐comparisonfunction}.Lemma 2 (see [22, 23]).
For a comparison function,φ:0,∞⟶0,∞ satisfying the below statements take
(1)
every iterateφl of φ,l≥1 is a comparison function(2)
φ is continuous(3)
φz<z, for any z>0Lemma 3 (see [25]).
Ifφ:0,∞⟶0,∞ is a b-comparison function, then,
(1)
the series∑l=0∞slφlz converges for any z∈0,∞(2)
the functionbs:0,∞⟶0,∞ defined by bsz=∑l=0∞slφlz,z∈0,∞ is increasing and continuous at 0.
We state that anyb-comparison function is a comparison function because of Lemma 2.3, and thus, in Lemma 2.2 any b-comparison function ψ satisfies ψz<z.Karapinar [26] introduced interpolation Kannan-type contraction generalized from the famous Kannan fixed point theorem by using interpolative operator. In the following, the common fixed point theorem for this contraction was obtained [27]. In [28], the authors extended the results in [26] by introducing the interpolative Reich-Rus-Ćirić contractive in a general framework, in the setting of partial metric space. In addition, the interpolative Hardy-Rogers-type contractive was defined and discussed in [28]. The contraction, defined in [29], was generalized in [30] by involving the admissibility into the contraction inequality. Furthermore, in [31], hybrid contractions were considered. Indeed, the notion of hybrid contraction here refers to combination of interpolative (nonlinear) contraction and linear contraction. For more interesting papers, see [32–34].In 2019, inspired by interpolative contraction, researchers [35] obtained and published a hybrid contractive that integrates Reich-Rus-Ćirić-type contractive and interpolative-type mappings. In particular, this approach was applied for Pata-Suzuki-type contraction in [36]. On the other hand, by using hybrid contraction, a solution for a Volterra fractional integral equation was proposed in [37]. Furthermore, the hybrid contractions were discussed in a distinct abstract space, namely, Branciari-type distance space, in [38]. Another advance was recorded in [39] where the authors investigated the Ulam-type stability of this consideration. In addition, new hybrid contractions were developed in b-metric spaces [40]. As a result, as can be seen in the literature review, many papers were published on the subject of interpolative contraction and hybrid contraction inspired by it. The contractions are a current study topic for fixed point theory. Therefore, the results of the study contribute to the existing literature.Now we give the idea ofα-admissibility defined by Samet et al. [41] and Karapnar and Samet [42].Definition 4.
A mappingM:L⟶L is defined α-admissible if for each r,v∈L we have
(1)αr,v≥1⇒αMr,Mv≥1,where α:L×L⟶0,∞ is a given function.The mapping ofw-orbital admissibility was presented by Popescu [43] as a modification of α-admissibility as follows:Definition 5.
Letw:L×L⟶0,∞ be a mapping and L≠∅. A map M:L⟶L is defined w-orbital admissible if for every r∈L, we get
(2)wr,Mr≥1⇒wMr,M2r≥1.The following condition has often been considered on account of refraining from the continuity of the concerned contractive mappings.Definition 6.
A spaceL,b,s is defined w-regular, if rq is a sequence in L such that αrq,rq+1≥1 for all q∈ℕ and rq⟶r∈L as q⟶∞; then, there exists a subsequence rqp of rq such that wrqp,r≥1 for all p.The framework of this study is organized into four sections. After the first introduction section, in Section2, we introduced the definitions, theorems, and some results on the Ćirić-Rus-Reich-Suzuki-type hybrid. In Section 3, we give an application Ulam-Hyers-type stability to show the well-posedness for our fixed point theorem. Finally, in the last section, the conclusions are drawn.


## 2. Main Results

We begin with the definition of the Ćirić-Rus-Reich-Suzuki-type hybrid contraction:Definition 7.
LetL,b,s be a b-metric space and w:L×L⟶0,∞ be a function. A map M:L⟶L is a Ćirić-Rus-Reich-Suzuki-type hybrid contraction (CRRS-type hybrid contraction) if there exist ψ∈Ψ such that
(3)12sbr,Mr≤br,v⇒wr,vbMr,Mv≤ψχMar,v,for each r,v∈L, where a≥0 and ρi≥0,i=1,2,3, such that ρ1+ρ2+ρ3=1,
(4)χMar,v=ϱ1br,va+ϱ2br,Mra+ϱ3bv,Mva1/a,fora>0,r≠vbr,vρ1br,Mrρ2bv,Mvρ3,fora=0,r,v∈L\FixM.Theorem 8.
LetL,b,s be a complete b-metric space and w-orbital admissible map also wr0,Mr0≥1 for some r0∈L. Given that M:L⟶L be a CRRS-type hybrid contraction satisfying one of the following conditions:
(h1) L,b,s is w-regular
(h2) M is continuous
(h3) M2 is continuous and wr,Mr≥1, where r∈FixM2.
Thereupon,M admits a fixed point in L.Proof.
We install an iterative sequencerq of points such that Mqr0=rq for q=0,1,2,⋯ and r0∈L with wr0,Mr0≥1. If rq0=rq0+1 for some integers q0, then rq0=Mrq0. Thus, suppose that rq≠rq+1, as M is w-orbital admissible, then wr0,Mr0=wr0,r1≥1 implies that wr1,Mr1=wr1,r2≥1. Continuing this process, we get
(5)wrq,rq+1≥1.
Condition 1:a>0, by taking χMar,v choosing r=rq−1 and v=Mrq−1=rq in (3) we get
(6)12sbrq−1,Mrq−1=12sbrq−1,rq≤brq−1,rq⇒,(7)wrq−1,rqbMrq−1,Mrq≤ψχMarq−1,rq,where
(8)χMarq−1,Mrq−1=ϱ1brq−1,Mrq−1a+ϱ2brq−1,Mrq−1a+ϱ3bMrq−1,M2rq−1a1/a=ϱ1brq−1,rqa+ϱ2brq−1,rqa+ϱ3brq,rq+1a1/a.
Whereupon, we deduce that(9)brq,rq+1≤wrq−1,rqbMrq−1,Mrq≤ψχMarq−1,rq=ψϱ1brq−1,rqa+ϱ2brq−1,rqa+ϱ3brq,rq+1a1/a=ψϱ1+ϱ2brq−1,rqa+ϱ3brq,rq+1a1/a.
If we have given thatbrq,rq+1≥brq−1,rq, then, accompanying that ψ is nondecreasing with (9), we get
(10)brq,rq+1≤ψϱ1+ϱ2brq−1,rqa+ϱ3brq,rq+1a1/a≤ψϱ1+ϱ2brq,rq+1a+ϱ3brq,rq+1a1/a=ψϱ1+ϱ2+ϱ3brq,rq+1a1/a=ψbrq,rq+1a1/a=ψbrq,rq+1<brq,rq+1,which is a contradiction. Thus, we obtain
(11)brq,rq+1<brq−1,rq.
As a result, from (9), we will turn into(12)brq,rq+1≤ψbrq−1,rq<brq−1,rq,and by similarly this process, we obtain that
(13)brq,rq+1≤ψqbr0,r1.for any q∈ℕ.
We argue thatrq is a fundamental sequence in L,b,s. Then, let q,l∈ℕ such that l>q and using the triangle inequality with (13), we take
(14)brq,rl≤sbrq,rq+1+s2brq+1,rq+2+⋯+sl−qbrl−1,rl≤sψqbr0,r1+s2ψq+1br0,r1+⋯+sl−qψl−1br0,r1=1sq−1sqψqbr0,r1+sq+1ψq+1br0,r1+⋯+sl−1ψl−1br0,r1=1sq−1∑q=ql−1sqψqbr1,r0.
By using Lemma3, the series ∑q=0∞sqψqbr1,r0 is convergent where Ht=∑q=0tsqψqbr0,r1, the above inequality finds
(15)brq,rl≤1sq−1Hl−1−Hq−1and q,l⟶∞, we obtain
(16)brq,rl⟶0.
Thus,rq is a fundamental sequence. Accompanying this together with the fact that the space L,b,s is complete, it will imply that there exists p∈L such that
(17)limq⟶∞brq,p=0.
We argue thatp is a fixed point of M.
If the supposeh1 takes, we get wrq,p≥1, and we assert that
(18)either12sbrq,Mrq≤brq,por12sbMrq,MMrq≤bMrq,p,for every q∈ℕ. Since, if we have given that
(19)12sbrq,Mrq>brq,pand12sbMrq,MMrq>bMrq,p,then, by using conditions of b-metric spaces L,b,s, since the sequence brq,rq+1 is decreasing, we write that
(20)brq,rq+1=brq,Mrq≤sbrq,p+bp,Mrq<12brq,Mrq+12bMrq,MMrq=12brq,rq+1+12brq+1,rq+2<12brq,rq+1+12brq,rq+1=brq,rq+1a contradiction. Therefore, for all q∈ℕ, either
(21)12sbrq,Mrq≤brq,p,or
(22)12sbMrq,MMrq≤bMrq,pprovides. In the condition that (21) takes, then by (3), we get
(23)brq+1,Mp≤wrq,pbMrq,Mp≤ψϱ1brq,pa+ϱ2brq,Mrqa+ϱ3bp,Mpa1/a<ϱ1brq,pa+ϱ2brq,Mrqa+ϱ3bp,Mpa1/a=ϱ1brq,pa+ϱ2brq,rq+1a+ϱ3bp,Mpa1/a.
If the second condition, (22) holds, we obtain(24)brq+2,Mp≤wrq+1,pbM2rq,Mp≤ψϱ1bMrq,pa+ϱ2bMrq,M2rqa+ϱ3bp,Mpa1/a<ϱ1bMrq,pa+ϱ2bMrq,M2rqa+ϱ3bp,Mpa1/a=ϱ1brq+1,pa+ϱ2brq+1,rq+2a+ϱ3bp,Mpa1/a.
Thereupon, takingq⟶∞ in (23) and (24),
(25)bp,Mp<ϱ31/abp,Mp≤bp,Mpwhich is contraction. Therefore, we get that bp,Mp=0 that is p=Mp.
If the presumeh2 is correct, and the map M is continuous, we get
(26)Mp=limq⟶∞Mrq=limq⟶∞rq+1=p.
In case that last supposition,h3 holds, from above, we write M2p=limq⟶∞M2rq=limq⟶∞rq+2=p, we want to show that Mp=p. Let us pretend otherwise, that is, p≠Mp from
(27)12sbMp,M2p=12sbMp,p≤bMp,pusing (3) we obtain that
(28)bp,Mp=bM2p,Mp≤wMp,pbM2p,Mp≤ψϱ1bMp,pa+ϱ2bMp,M2pa+ϱ3bp,Mpa1/a<ϱ1bMp,pa+ϱ2bMp,M2pa+ϱ3bp,Mpa1/a=ϱ1+ϱ2+ϱ3bp,Mpa1/a=bp,Mpa contradiction. Eventually, p=Mp.
Condition 2: ifa=0, in the equation χMar,v taking r=rq−1 and v=Mrq−1=rq in (3) we write
(29)12sbrq−1,Mrq−1=12sbrq−1,rq≤brq−1,rq⇒,(30)brq,rq+1≤wrq−1,rqbMrq−1,Mrq≤ψχMarq−1,rq=ψbrq−1,rqρ1⋅brq−1,Mrq−1ρ2⋅brq,Mrqρ3<brq−1,rqρ1⋅brq−1,rqρ2⋅brq,rq+1ρ3.
From (30), we find(31)brq,rq+11−ϱ3<brq−1,rqϱ1+ϱ2and from ϱ1+ϱ2+ϱ3=1, we attain that brq,rq+1<brq−1,rq for every q∈ℕ. Using (30), we take
(32)brq,rq+1≤ψbrq−1,rqand as in condition 1, we can prove that
(33)brq,rq+1≤ψqbr0,r1.
Since the equal methods as in the case ofa>0, we clearly prove that rq is a fundamental sequence in a complete b-metric space. Additionally, for p∈L so, limq⟶∞brq,p=0 also we assert that p=Mp. In the meanwhile, L,b,s is w-regular; thus, as rq confirm (5), and wrq,rq+1≥1 for each q∈ℕ, we obtain wrq,p≥1. Moreover, as in the proof of condition 1, we know that either
(34)12sbrq,Mrq≤brq,p,or
(35)12sbMrq,MMrq≤bMrq,pholds, for each q∈ℕ. If (34) is taken, we conclude that
(36)brq+1,Mp≤wrq,pbMrq,Mp≤ψbrq,pϱ1⋅brq,Mrqϱ2⋅bp,Mpϱ3,=ψbrq,pϱ1⋅brq,rq+1ϱ2⋅bp,Mpϱ3,<brq,pϱ1⋅brq,rq+1ϱ2⋅bp,Mpϱ3,
Let us assume that inequality (35) is satisfied, then(37)brq+2,Mp≤wrq+1,pbM2rq,Mp≤ψbMrq,pϱ1⋅bMrq,M2rqϱ2⋅bp,Mpϱ3,=ψbrq+1,pϱ1⋅brq+1,rq+2ϱ2⋅bp,Mpϱ3,<brq+1,pϱ1⋅brq+1,rq+2ϱ2⋅bp,Mpϱ3,
Then, getting to the limit, we conclude thatbp,Mp=0, and p=Mp. Now, the continuity of M implies p=Mp (from condition 1). Therefore, supposition h3 lead to M2p=limq⟶∞M2rq=limq⟶∞rq+2=p. We will prove that Mp=p. Let’s presume otherwise, that is, p≠Mp(38)12sbMp,M2p=12sbMp,p≤bMp,pusing (3) we find that
(39)bp,Mp=bM2p,Mp≤wMp,pbM2p,Mp≤ψbMp,pϱ1⋅bMp,M2pϱ2⋅bp,Mpϱ3<bMp,pϱ1⋅bMp,pϱ2⋅bp,Mpϱ3=bMp,p,a contradiction. Consequently, p=Mp. Thus, the proof of the Theorem is completed.Theorem 9.
Addingwp,p∗≥1 for any p,p∗∈FixML and if supplying to all the hypothesis of Theorem 8, we prove the uniqueness of fixed point.Proof.
Supposing that differentp∗ is fixed point of M, that is Mp∗=p∗ with p≠p∗.In the case that a>0, then, from (3) we have
(40)12sbp,Mp=0≤bp,p∗implies(41)bp,p∗≤wp,p∗bMp,Mp∗≤ψχMap,p∗<χMap,p∗=ϱ1dp,p∗a+ϱ2bp,Mpa+ϱ3bp∗,Mp∗a1/a.
Thus,(42)bp,p∗<ϱ11/abp,p∗≤bp,p∗which is contradiction. In the case that a=0, then, from (4) we get that
(43)0<bp,p∗<0,a contradiction. Eventually, p=p∗, so p is a unique fixed point of M.Example 1.
Letb:L×L⟶0,+∞, br,v=r−v2 for every r,v∈L with s=2 and
(44)wr,v=4,ifr,v∈0,11,ifr=0,v=20,otherwisealso, the function ψ∈Ψ with ψt=t/4. Define a mapping M:L⟶L as
(45)Mr=15,ifr∈0,1r5,ifr∈1,2also, M2=r/10, we get that M2 is continuous but M is not continuous, where L=0,2.
We choosea=2 and ϱ1=ϱ2=ϱ3=1/3, then we obtain the following conditions:
(a)
: Forr,v∈0,1 we get bMr,Mv=0, then, (3) holds(b)
: Ifr=0 and v=2(46)12sb0,M0=1100<4=b0,2⇒.(47)w0,2bM0,M2=0.04≤0.3708099243547831=141322+13152+132−252=ψρ1br,v2+ρ2br,Mr2+ρ3bv,Mv2.
Other conditions are confirmed, fromwr,v=0. Consequently, the assumptions of Theorem 8, being supplied, M has a fixed point (r=1/5).Corollary 10.
LetL,b,s be a complete b-metric space and let M:L⟶L a continuous map satisfying the following inequality:
(48)12sbr,Mr≤br,vimpliesbMr,Mv≤ψχMar,v,where χMar,v is defined by (4), ψ∈Ψ and for all r,v∈L, where a≥0 and ϱi≥0,i=1,2,3 with ϱ1+ϱ2+ϱ3=1. In the case of M or M2 functions continuity, M admits a fixed point in L.Proof.
It is sufficient to getwr,v=1 for r,v∈L in Theorem 8.Corollary 11.
LetL,b,s be a complete b-metric space and let M:L⟶L a continuous map satisfying the following inequality
(49)12sbr,Mr≤br,vimpliesbMr,Mv≤ηχMar,v,where χMar,v is defined by (4), η∈0,1 and for each r,v∈L where a≥0 and ρi≥0,i=1,2,3 with ρ1+ρ2+ρ3=1. In the event of M or M2 functions continuity, M admits a fixed point in L.Proof.
It is adequate getψν=ην for any ν≥0 in Corollary 10.Corollary 12.
LetL,b,s be a complete b-metric space and M:L⟶L a continuous map. If there exist η∈0,1 such that
(50)12sbr,Mr≤br,vimplies(51)bMr,Mv≤ηbr,vbr,Mrbv,Mv3for each r,v∈L\FixM, in the case of M or M2 functions continuity, M admits a fixed point in L.Proof.
Ifa=0, using Corollary 11, getting ρ1=ρ2=ρ3=1/3.Corollary 13.
LetL,b,s be a complete b-metric space and M:L⟶L a continuous map. If there exist η∈0,1 such that
(52)12sbr,Mr≤br,vimplies(53)bMr,Mv≤η3br,v+br,Mr+bv,Mvfor each r,v∈L\FixM, in the case of M or M2 functions continuity, M admits a fixed point in L.Proof.
By using Corollary11, letting ϱ1=ϱ2=ϱ3=1/3 and a=1.Corollary 14.
LetL,b,s be a complete b-metric space and M:L⟶L a continuous map. If there exist η∈0,1 such that
(54)12sbr,Mr≤br,vimplies(55)bMr,Mv≤η3br,v2+br,Mr2+bv,Mv2for each r,v∈L\FixM, in the case of M or M2 functions continuity, M admits a fixed point in L.Proof.
By using Corollary11, taking ϱ1=ϱ2=ϱ3=1/3 and a=2.


## 3. An Application: Ulam-Hyers-Type Stability

The stability of the solution is a considerable important subject of nonlinear analysis. Recently, Ulam stability [44, 45] results in fixed point theory have been investigated heavily. In what follows. we investigate the Ulam stability of our main theorem.Consider the following function:(56)ϒ:γ:0,∞⟶0,∞suchthatγiscontinuousatzerowith(57)γ0=0andincreasingAssume thatM:L⟶L is a map on a b-metric spaces L,b,s. The fixed point problem of M is to notice an r∈L such that
(58)r=Mr.Equality (58) is also known as fixed point implication. The fixed point implication is called to be general Ulam-Hyers stable if and only if there exists a functionγ∈ϒ so that for all ε>0 also for every v∗∈L which satisfies the following inequality,
(59)bv∗,Mv∗≤εthere exists u∈L providing the equation (58) such that
(60)bu,v∗≤γε.Moreover, if there exists aP>0 such that γt=Pt for all t∈ℝ+, then the fixed point equation (58) is said to be Ulam-Hyers stable. On the b-metric spaces L,b,s, fixed point problem (58) and M:L⟶L are defined to be well-known if the following suppositions are satisfy:(l1) M has a unique fixed point u∈L(l2) limq⟶∞bu,rq=0 for every sequence rq∈L such that
(61)limq⟶∞brq,Mrq=0Theorem 15.
LetL,b,s be a complete b-metric space. If we joint the condition a>0 and easaρ1<1, where ea=max1,2a−1 and saρ1+easaρi+1<1, where i=1 or i=2 or i=3, also suppositions of Theorem 9, thus the following conditions hold:
(a)
the fixed point problem (58) is Ulam-Hyers stable, ifwn,m≥1 for any n,m satisfying the condition (59)(b)
the fixed point problem (58) is well-known, ifwrq,u≥1 for any rq in L such that limq⟶∞bMrq,rq=0 and FixML=u.Proof.
(a)
Taking into account Theorem9, we consider that there is a unique u in L such that Mu=u. Assume that v∗ is a solution of (59), that is bv∗,Mv∗≤ε for ε>0. Clearly, u holds (59), then we get that wu,v∗≥1 and using triangular inequality satisfies(62)bu,v∗=bMu,v∗≤sbMu,Mv∗+bMv∗,v∗
SinceM is CRRS-type hybrid contraction, we obtain
(63)12sbu,Mu=0≤bu,v∗implies(64)bu,v∗≤sbMu,Mv∗+bMv∗,v∗≤swu,v∗bMu,Mv∗+bMv∗,v∗≤sψχMau,v∗+bMv∗,v∗<sχMau,v∗+bMv∗,v∗≤sρ1bu,v∗a+ρ2du,Mua+ρ3bMv∗,v∗a1/a+sbMv∗,v∗≤sρ1bu,v∗a+ρ3εa1/a+sε
Thus, we get(65)bu,v∗a≤easaϱ1bu,v∗a+saϱ3εa+saεathen,
(66)bu,v∗a≤1+ρ3easa1−ρ1easaεa(67)bu,v∗≤nεwhere n=1+ϱ3easa/1−ϱ1easa1/a for any a>0 and ϱ1∈0,1 such that ϱ1<1/easa..
(b)
The Picard iterations isM-stable, that is, let rq∈L such that limq⟶∞brq+1,Mrq=0 and FixML=u. From the triangular inequality, we can write(68)brq,u≤sbrq,Mrq+bMrq,Mu.
Thus,M is a CRRS contraction, we have
(69)12sbrq,Mrq≤brq,uimplies(70)brq,u≤sbrq,Mrq+bMrq,Mu≤sbrq,Mrq+wrq,ubMrq,Mu≤sψχMarq,u+brq,Mrq<sχMarq,u+brq,Mrq≤sρ1brq,ua+ρ2brq,Mrqa+ρ3bMu,ua1/a+sbrq,Mrq≤sρ1brq,ua+ρ2brq,Mrqa1/a+sbrq,Mrq.
Then, we calculate process(71)brq,ua≤easaϱ1brq,ua+saϱ2brq,Mrqa+sabrq,Mrqathen,
(72)brq,ua≤1+ϱ2easa1−ϱ1easabrq,Mrqa.
Takingq⟶∞ in the above inequality and using limq⟶∞brq,Mrq=0, we obtain limq⟶∞brq,u=0 the fixed point equation (58) is well posed.


## 4. Conclusion

In this study, we present new hybrid fixed point theorems inb-metric spaces. We obtain the extended results of the interpolative Reich-Rus-Ćirić fixed point theorem by using w-orbital admissible and Suzuki-type mapping. We also offer an example to show the availability of introduced results. Further, we obtain Ulam-Hyers-type stability of the fixed point theorem which is the application of our study.



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