# On Some Interpolative Contractions of Suzuki Type Mappings

Authors: Andreea Fulga; Seher Sultan Yeşilkaya
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The goal of this study is to propose a new interpolative contraction mapping by using an interpolative approach in the setting of complete metric spaces. We present some fixed point theorems for interpolative contraction operators usingw-admissible maps which satisfy Suzuki type mappings. In addition, some results are given. These results generalize several new results present in the literature. Moreover, examples are provided to show the suitability of our given results.


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## Body

## 1. Introduction

In 1922, Banach [1] proved his famous remarkable fixed-point theorem; the result is known as the Banach contraction principle, which states that “Let K,d be a complete metric space and S:K⟶K be a contraction, then S has a unique fixed point.” The Banach contraction principle is one of the essential and most valuable theorems of analysis and is accepted as the main results of metric fixed-point theory. In the last century, the fixed point and its applications have been the subject of research by many authors in the literature, since it provides useful tools to solve many complex problems that have applications in different sciences like computer science, engineering, data science, physics, economics, game theory, and biosciences [2–7]. Due to several applications of “fixed point theory,” researchers were motivated to further generalize it in different directions, by generalizing the contractive conditions underlying the space concept of completeness.The background literature on the famous Banach contraction principle has been extended in various comprehensive directions by many researchers. One of the exciting generalizations was given by Kannan [8], which characterize the completeness of underlying metric spaces. Kannan introduced the following theorem.Theorem 1.
[8] Let K,d be a complete metric space. A mapping S:K⟶K is said to be a Kannan contraction if there exists λ∈0,1/2 such that(1)dSv,St≤λdv,Sv+dt,St,for all v,t∈K\FixS. Then, S posses a unique fixed point.The Kannan theorem has been generalized in different aspects by many authors; one of the crucial generalizations was given by Karapinar in [9]. Karapinar [9] introduced the notion of an interpolative Kannan contraction mapping and proved the following:A mappingS:K⟶K on K,d a complete metric space such that(2)dSv,St≤κdv,Svα·dt,St1−α,where κ∈0,1 and α∈0,1, for each v,t∈K\FixS. Then, S has a unique fixed point in K. Subsequently, Karapinar et al. [10] introduced the following notion of interpolative Ćirić-Reich-Rus contractions.Theorem 2 (see [10]).
LetK,pbe a partial metric space. The mappingS:K⟶Kis called an interpolative Ciric-Reich-Rus contraction if there existλ∈0,1and positive realsβ,α>0, withβ+α<1, such that(3)pSv,St≤λpv,tβ·pv,Svα·pt,St1−α−β,for each v,t∈K\FixS. Then, the mapping S has a fixed point in K.Afterward, this concept has been extended in different aspects, for example, [11–15].LetΨ be the set of all nondecreasing self-mappings ψ on 0,∞ such that ∑r=1∞ψrz<∞foreveryz>0. Notice that for ψ∈Ψ, we have ψ0=0 and ψz<z for all z>0 (see [16, 17]).The concept ofw-orbital admissible mappings was introduced by Popescu as a clarification of the concept of α-admissible mappings of Samet et al. [18].Definition 3 (see [19]).
LetSbe a self-map defined onKandw:K×K⟶0,∞be a function.Sis said to be anw-orbital admissible if for allv∈K, we have(4)wv,Sv≥1⇒wSv,S2v≥1.In our appointed theorems, if the continuity of the involved contractive mappings is removed, to handle this defect, it is necessary thatK,d be w-regular.R A space K,d is defined as w-regular, if vr is a sequence in K such that wvr,vr+1≥1 for each r and vr⟶ω∈K as r⟶∞, then wvr,ω≥1 for all r.Some curious results in this sense are found in the works in [20–24].Another most interesting Banach contraction principle generalization was given by Suzuki [25, 26]. He introduced a weaker notion of contraction and discussed the existence of some new fixed point theorems. Besides the famous theorem, Suzuki generalized also the results of Nemytzki [27] and Edelstein [28] for compact metric spaces. One of the recently popular topics in fixed point theory is addressing the existence of fixed points of Suzuki type mappings. As with many generalizations of the famous Banach theorems, Suzuki type generalization can be said to have many applications, such as in computer science [29], game theory [30], and biosciences [31] and in other areas of mathematical sciences such as in dynamic programming, integral equations, data dependence, and homotopy [32, 33]. Subsequently, Popescu [34] has modified the nonexpansiveness situation with the weaker C-condition presented by Suzuki. Accordingly, the existence of fixed points of maps satisfying the C-condition has been extensively studied (see [35–38]). Karapınar et al. [39] introduced the definition of a nonexpansive mapping satisfying the C-condition:Definition 4.
A mappingSon a metric spaceK,dsatisfies theC-condition if(5)12dv,Sv≤dv,t⇒dSv,St≤dv,t,for each v,t∈K.


## 2. Main Results

We start the section with the following essential definition:Definition 5.
LetK,dbe a metric space. A mappingS:K⟶Kis called anw-ψ-interpolative Kannan contraction of Suzuki type if there existψ∈Ψ, w:K×K⟶0,∞, and a real numberβ∈0,1, such that(6)12dv,Sv≤dv,t⇒wv,tdSv,St≤ψdv,Svβ·dt,St1−β,for each v,t∈K\FixS.Theorem 6.
LetK,dbe a complete metric space andS:K⟶Kbe anw-ψ-interpolative Kannan contraction of the Suzuki type. Suppose thatSis anw-orbital admissible mapping andwv0,Sv0≥1for somev0∈K. Then,Shas a fixed point inKprovided that at least one of the following conditions holds:(a)
K,d is w-regular(b)
S is continuous(c)
S2 is continuous and wv,Sv≥1 where v∈FixS2Proof.
Letv0∈K such that wv0,Sv0≥1 and vr be the sequence constructed by Srv0=vr for each positive integer r. Assuming that for some r0∈ℕ, vr0=vr0+1, we get vr0=Svr0, so vr0 is a fixed point of S. Then, vr≠vr+1 for each positive integer r. As S is w-orbital admissible, wv0,Sv0=wv0,v1≥1 implies that wv1,Sv1=wv1,v2≥1. Similarly, continuing this process, we have(7)wvr,vr+1≥1.
Thereupon, choosingv=vr−1 and t=Svr−1 in (6), we get(8)12dvr−1,Svr−1=12dvr−1,vr≤dvr−1,vr⇒dvr,vr+1≤wvr−1,vrdSvr−1,Svr≤ψdvr−1,Svr−1β·dvr,Svr1−β=ψdvr−1,vrβ·dvr,vr+11−β<dvr−1,vrβ·dvr,vr+11−β,
whence it follows that(9)dvr,vr+1β<dvr−1,vrβ,or equivalent(10)dvr,vr+1<dvr−1,vr.
Thus, on the one hand, it follows that the sequencedvr−1,vr is a nonincreasing sequence with positive terms, so there exists l≥0 such that limr⟶∞dvr−1,vr=l. On the other hand, combining (8) and (10) and keeping in mind that the function ψ is nondecreasing, we obtain(11)dvr,vr+1≤ψdvr−1,vr≤ψ2dvr−2,vr−1≤⋯≤ψrdv0,v1.
Now, applying the triangle inequality and using (11), for all j≥1, we get(12)dvr,vr+j≤dvr,vr+1+dvr+1,vr+2+⋯+dvr+j−1,vr+j≤ψrdv0,v1+ψr+1dv0,v1+⋯+ψr+j−1dv0,v1=∑m=rr+j−1ψmdv0,v1=Pr+j−1−Pr−1,where Pk=∑m=0kψmdv0,v1. But, ψ∈Ψ, the series ∑m=0∞ψmdv0,v1 is convergent, so there exists a positive real number P such that limk⟶∞Pk=P. Consequently, letting r,j⟶∞ in the above inequality, we get(13)dvr,vr+j⟶0.
Therefore,vr is a Cauchy sequence, and taking into account the completeness of the space K,d, it follows that there exists ω∈K such that(14)limr⟶∞vr=ω,and we claim that this ω is a fixed point of S.
In case that the assumption (a) holds, we havewvr,ω≥1, and we claim that(15)12dvr,Svr≤dvr,ωor(16)12dSvr,SSvr≤dSvr,ω,for every r∈ℕ. Supposing(17)12dvr,Svr>dvr,ω,12dSvr,SSvr>dSvr,ω,on the account of the triangle inequality, we have(18)dvr,vr+1=dvr,Svr≤dvr,ω+dω,Svr<12dvr,Svr+12dSvr,SSvr=12dvr,vr+1+12dvr+1,vr+2≤12dvr,vr+1+12dvr,vr+1=dvr,vr+1,which is a contradiction. Thereupon, for every r∈ℕ, either(19)12dvr,Svr≤dvr,ω,or(20)12dSvr,SSvr≤dSvr,ω,holds. In the case that (19) holds, we obtain(21)dvr+1,Sω≤wvr,ωdSvr,Sω≤ψddvr,Svrβ·dω,Sω1−β=ψdvr,vr+1β·dω,Sω1−β<dvr,vr+1β·dω,Sω1−β.
If the second condition, (20), holds, we have(22)dvr+2,Sω≤wvr+1,ωdS2vr,Sω≤ψddSvr,S2vrβ·dω,Sω1−β=ψdvr+1,vr+2β·dω,Sω1−β<dvr+1,vr+2β·dω,Sω1−β.
Therefore, lettingr⟶∞ in (21) and (22), we get that dω,Sω=0, that is, ω=Sω.
In the case that the assumption (b) is true, that is, the mappingS is continuous,(23)Sω=limr⟶∞Svr=limr⟶∞vr+1=ω.
If the last assumption, (c), holds, as above, we haveS2ω=limr⟶∞S2vr=limr⟶∞vr+2=ω and we want to show that also Sω=ω. Supposing on the contrary, that ω≠Sω, since(24)12dSω,S2ω=12dSω,ω≤dSω,ω,by (6), we get(25)dω,Sω≤wSω,ωdS2ω,Sω≤ψdSω,S2ωβ·dω,Sω1−β<dSω,ωβ·dω,Sω1−β=dSω,ω,which is a contradiction. Consequently, ω=Sω, that is, ω is a fixed point of the mapping S.☐Example 7.
LetK=0,3andd:K×K⟶0,+∞be the usual distance onℝ. Consider the mappingS:K⟶Kbe defined as(26)Sv=45,ifv∈0,1,13,ifv∈1,2,v3+6v+ln4−v35,ifv∈2,3.
Let alsow:K×K⟶0,∞, where(27)wv,t=2,ifv,t∈0,1,v2+t2,ifv,t∈1,2,1,ifv=0,t=3,0,otherwise.We remark that the space is not regular since, for example, considering the sequencevr, with vr=r+8/2r+4 we have vr⟶1/2 as r⟶∞, wvr,vr+1=vr2+vr+12≤1, but wvr,1/2=0. On the other hand, the mapping S is not continuous, but since S2=4/5, we have that S2 is a continuous mapping. Let the function ψ∈Ψ defined as ψz=z/3 and we choose β=1/9. Thus, we have to check that (6) holds. We have to consider the following cases:(1)
Forv,t∈0,1, respectively, v,t∈1,2, we have dSv,St=0, so (6) holds(2)
Forv=0 and t=3(28)12d0,S0=25<3=d0,3⇒w0,3dS0,S3=0,088621339≤0,678785019=130.81/9·2,2886213398/9=ψd0,S01/9·d3,S38/9(3)
All other cases are not interesting becausewv,t=0Consequently, the assumptions of Theorem6 being satisfied, it follows that the mapping S has a fixed point, which is v=4/5.Corollary 8.
LetK,dbe a complete metric space andSbe a self-mapping onK, such that,(29)12dv,Sv≤dv,timpliesdSv,St≤ψdv,Svβ·dt,St1−β,for each v,t∈K\FixS, where ψ∈Ψ and β∈0,1. Then, S possesses a fixed point in K.Proof.
Theorem6 is sufficient to get wv,t=1 for the proof.☐Moreover, takingψz=zκ, with κ∈0,1 in Corollary (8), we obtain the following consequence.Corollary 9.
LetK,dbe a complete metric space andSbe a self-mapping onK, such that(30)12dv,Sv≤dv,timpliesdSv,St≤κdv,Svβ·dt,St1−β,for each v,t∈K\FixS, where β∈0,1. Then, the mapping S possesses a fixed point in K.Definition 10.
LetK,dbe a metric space. The mappingS:K⟶Kis called anw-ψ-interpolative Ćirić-Reich-Rus contraction of Suzuki type if there existψ∈Ψ, w:K×K⟶0,∞, and positive realsβ,α>0, withβ+α<1, such that(31)12dv,Sv≤dv,t⇒wv,tdSv,St≤ψdv,tβ·dv,Svα·dt,St1−α−βfor each v,t∈K\FixS.Theorem 11.
LetK,dbe a complete metric space and the mappingS:K⟶Kbe anw-ψ-interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Suppose thatSisw-orbital admissible andwv0,Sv0≥1for somev0∈K. IfK,disw-regular or either(1)
S is continuous or(2)
S2 is continuous and wSω,ω≤1 for any v∈FixS2, then the mapping S has a fixed point in KProof.
Letv0∈K satisfy wv0,Sv0≥1 and vr be the sequence defined by Srv0=vr for each positive integer r. If vr0=vr0+1 for some r0∈ℕ, we get vr0=Svr0, that means vr0 is a fixed point of S. Then, we can assume that vr≠vr+1 for each positive integer r. Moreover, due to the assumption that S is w-orbital admissible, as in the previous proof, we have(32)wvr,vr+1≥1.
By lettingv=vr−1 and t=Svr−1=vr in (31), we obtain(33)12dvr−1,Svr−1=12dvr−1,vr≤dvr−1,vr⇒dvr,vr+1≤wvr−1,vrdSvr−1,Svr≤ψdvr−1,vrβ·dvr−1,Svr−1α·dvr,Svr1−α−β=ψdvr−1,vrβ·dvr−1,vrα·dvr,vr+11−α−β,
then, usingψz<z for every z>0.(34)dvr,vr+1≤dvr−1,vrβ+α·dvr,vr+11−α−β,or equivalent(35)dvr,vr+1α+β<dvr−1,vrα+β.
So,(36)dvr,vr+1<dvr−1,vr,for every r∈ℕ. Therefore, the positive sequence dvr−1,vr is decreasing. Eventually, by (33), we have(37)dvr,vr+1≤ψdvr−1,vr,and by repeating this process, we find that(38)dvr,vr+1≤ψrdv0,v1.
We assert thatvr is a fundamental sequence in K,d. Thus, using the triangle inequality with (38), we can write(39)dvr,vr+l≤dvr,vr+1+dvr+1,vr+2+⋯+dvr+l−1,vr+l≤ψrdv0,v1+ψr+1dv0,v1+⋯+ψr+l−1dv0,v1≤∑k=r∞ψkdv0,v1.
Takingr⟶∞ in (39), we deduce that vr is a fundamental sequence in K,d, and using the completeness K,d, there exists ω∈K such that(40)limr⟶∞dvr,ω=0.
We claim that the pointω is a fixed point of S. In the case of the space K,d being w-regular and vr verifies (32), that is, wvr,vr+1≥1 for every r∈ℕ, we get wvr,ω≥1. On the other hand, we know (see the proof of Theorem 6) that either(41)12dvr,Svr≤dvr,ω,or(42)12dSvr,SSvr≤dSvr,ω,
holds, for everyr∈ℕ. If (41) is holds, we obtain(43)dvr+1,Sω≤wvr,ωdSvr,Sω≤ψdvr,ωβ·dvr,Svrα·dω,Sω1−α−β,=ψdvr,ωβ·dvr,vr+1α·dω,Sω1−α−β<dvr,ωβ·dvr,vr+1α·dω,Sω1−α−β.
Lettingr⟶∞ in the above inequality, we get that dω,Sω=0, that is, ω=Sω. If the second condition (42) is true, we get that ω is a fixed point S by a similar argument.
Furthermore, if thew-regular of K,d is removed and, instead, S is continuous, we get that S has a fixed point in K, because(44)ω=limr⟶∞vr+1=limr⟶∞Svr=Slimr⟶∞vr=Sω.
Finally, if the mappingS is such that S2 is continuous, we easily obtain S2ω=ω. Supposing that Sω≠ω, since wSω,ω≤1 for any v∈FixS2 and 1/2dSω,S2ω=1/2dSω,ω≤dSω,ω, we have(45)dω,Sω=dS2ω,Sω≤wSω,ωdS2ω,Sω≤ψdSω,ωα·dSω,S2ωβ·dω,Sω1−α−β<dSω,ωα·dSω,ωβ·dω,Sω1−α−β=dω,Sω.
That is a contradiction. Thereupon,Sω=ω.☐Example 12.
LetK=0,1/8,1/4,1/2,1, d:K×K⟶0,+∞, dv,t=v−t, andS:K⟶K, whereS0=S1/2=1/8, S1/8=1/2, andS1=S1/4=1/4. Consider the functionw:K×K⟶0,∞,(46)wv,t=1,ifv,t∈0,1,0,12,3,ifv,t∈K∪14,t+1,ifv,t∈14∪K,0,otherwise,let ψ∈Ψ, ψz=2z/5, and the real constants α=β=1/3. Taking into account the definition of the function w, the only interesting situations are for v=0,y=1/2, respectively, v=0,y=1. For the first case, we have(47)12d0,S0=12d0,18=116<12=d0,12⇒w0,12dS0,S12=0≤ψd0,121/3·d0,S01/3·d12,S121/3.
For the second case,(48)12d0,S0=12d0,18=116<1=d0,1⇒w0,1dS0,S1=18≤0,158740105=25·181/3·341/3=ψd0,11/3·d0,S01/3·d1,S11/3.Definition 13.
LetK,dbe a metric space. The mappingS:K⟶Kis called anψ-interpolative Ćirić-Reich-Rus contraction of Suzuki type if there existψ∈Ψand the constantsβ,α>0, withβ+α<1, such that(49)12dv,Sv≤dv,t⇒dSv,St≤ψdv,tβ·dv,Svα·dt,St1−α−β,for each v,t∈K\FixS.Theorem 14.
LetK,dbe a complete metric space and the mappingS:K⟶Kbe anψ-interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Then, the mappingShas a fixed point inK.Proof.
Putwv,t=1 in Theorem 11.☐Definition 15.
LetK,dbe a metric space. A mappingS:K⟶Kis called an interpolative Ćirić-Reich-Rus contraction of the Suzuki type if there existκ∈0,1and positive realsβ,α>0, withβ+α<1, such that(50)12dv,Sv≤dv,t⇒dSv,St≤κdv,tβ·dv,Svα·dt,St1−α−β,for each v,t∈K\FixS.Theorem 16.
LetK,dbe a complete metric space andS:K⟶Kbe an interpolative Ćirić-Reich-Rus contraction of the Suzuki type. Therefore,Shas a fixed point inK.Proof.
Putψz=κz, for all z>0, in Theorem 14.☐


## 3. Conclusions

In this manuscript, we introduce new concepts on completeness ofw-ψ-interpolative Kannan contraction of Suzuki type and w-ψ-interpolative Ćirić-Reich-Rus contraction of Suzuki type mappings in metric space. We prove the existence of some fixed point theorems for mappings these concepts. Further, we obtain some fixed point results and give examples to show that the new results are applicable. Interpolation contraction, which is generalized from the Kannan type contraction, is a new and interesting contraction in fixed point theory, and different interpolation contractions of Suzuki type studies can be obtained by combining it with a Suzuki type contraction in the future. Additionally, these proposed contractions can be generalized in other well-known spaces and can give new fixed point results.



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