# Interpolative Prešić Type Contractions and Related Results

Authors: Monairah Alansari; Muhammad Usman Ali
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we will extend the notion of interpolative Kannan contraction by introducing the notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions for mappings defined on product spaces. Through these notions, we will derive some results to ensure the existence of fixed points and best proximity points for such mappings.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle is the most significant and basic result of metric fixed point theory. Through this result, we can obtain a unique fixed point of a self-mapN:L⟶L, provided that N is a contraction map on a complete metric space L,dL. This result motivated Prešić to study about the existence of fixed points of the operators defined on product spaces, that is, N:Lk⟶L, for any fixed k∈ℕ. As an outcome of this motivation, Prešić [1] presented the following noteworthy extension of the Banach contraction principle.Theorem 1 (see [1]).
Suppose thatL,dL be a complete metric space and N:Lk⟶L be a map, for any fixed k∈ℕ, satisfying the following inequality:(1)dNl1,l2,…,lk,Nl2,l3,…,lk+1≤∑j=1kζjdLlj,lj+1,for everyl1,l2,…,lk,lk+1∈L, where ζ1,ζ2,…,ζk are the nonnegative real numbers with ∑j=1kζj<1. Then, there exists a unique point of L that satisfies the equation l=Nl,l,…,l︸k.This result is used to discuss the existence of equilibrium points for thekth-order nonlinear difference equation of the form(2)ln+k=Nln,ln+1,…,ln+k−1,where N:Lk⊂ℝk⟶L is a continuous map. Note that a point l∗∈L is known as an equilibrium point of (2) if l∗=Nl∗,l∗,…,l∗︸k. Such a point is also known as a fixed point of N:Lk⟶L. Some well-known generalizations of this work have been studied by several authors, for example, [2–5].Kannan and Chatterjea made a vital contribution in the development of this field through the fixed point results derived in [6, 7], respectively. Recently, Karapınar [8] modified the Kannan contraction by introducing interpolative Kannan contraction, stated as, a map N:L,dL⟶L,dL is called an interpolative Kannan contraction [8] if(3)dLNk,Nl≤ζdLk,NkϑdLl,Nl1−ϑ,for all k,l∈L with k≠Nk and l≠Nl, where ζ∈0,1 and ϑ∈0,1.After that, many existing contraction-type conditions have been generalized in the sense of interpolative Kannan contraction, for example, Karapınar et al. [9] studied interpolative Reich-Rus-Ćirić type contraction in partial metric spaces, Aydi et al. [10] studied interpolative Ćirić-Reich-Rus type contractions in Branciari metric spaces, Mohammadi et al. [11] extended the concept of F-contractions by interpolative Ćirić-Reich-Rus type F-contractions, Karapınar et al. [12] studied interpolative Hardy–Rogers type contractions, Debnath and Sen [13] studied set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-Ćirić-type contractions, Sarwar et al. [14] presented rational type interpolative contractions, Khan et al. [15] worked on interpolative ϕ,ψ-type Z-contractions, Altun and Tasdemir [16] presented interpolative proximal contractions for nonself mappings, Fulga and Yesilkaya [17] studied interpolative Suzuki-type contractions, Karapınar et al. [18] defined α,β,ψ,ϕ-interpolative contractions, and Alansari and Ali [19] studied multivalued interpolative Reich-Rus-Ćirić-type contractions.Gaba and Karapınar [20] extended the notion of interpolative Kannan contraction through exponential powers, stated as, a map N:L,dL⟶L,dL is called an ζ,ϑ1,ϑ2-interpolative Kannan contraction, if(4)dLNk,Nl≤ζdLk,Nkϑ1dLl,Nlϑ2,for all k,l∈L with k≠Nk and l≠Nl, where ϑ1,ϑ2∈0,1 with ϑ1+ϑ2<1 and ζ∈0,1. Readers can find other similar generalizations in [21].ConsiderOL and ML be nonvoid subsets of a metric space L,dL. It is well-known that a fixed point of a map N:OL⟶ML is a solution of Nl=l. If OL∩ML=∅, then fixed point of N:OL⟶ML does not exist, that is, dLNl,l>0 for all l∈OL. In this situation, we try to find l∈OL, such that dLNl,l attain the minimum value in some sense. It is obvious that the smallest value that can be obtained by dLNl,l for any l∈OL will be greater or equal to DLOL,ML, that is, distance between OL and ML. A point l∈OL is said to be a best proximity point of N:OL⟶ML, if dLNl,l=DLOL,ML. The existence of such points of nonself maps has been discussed by several researchers in different ways, for example, Caballero et al. [22] studied the existence of best proximity points for nonself maps satisfying Geraghty contraction and P-property in metric spaces, Bilgili et al. [23], Aydi et al. [24], and Pitea [25] extended the work of Caballero et al. [22] by introducing generalized Geraghty contraction, ψ-Geraghty contraction and generalized almost θ-Geraghty contraction for nonself maps, Basha and Shahzad [26] and Basha [27] defined proximal-type contractions to study the existence of best proximity points, Jleli and Samet [28] defined α-ψ-proximal contraction to ensure the existence of best proximity points, Jleli et al. [29] and Aydi et al. [30] defined generalized α-ψ-proximal contractions to extend the work of Jleli and Samet [28], Abkar and Gabeleh [31] and Kumam et al. [32] studied the existence of best proximity points for multivalued nonself maps in metric spaces, Ali et al. [33] defined implicit proximal contractions, Sahin et al. [34] defined proximal nonunique contraction, and Ali et al. [35] studied the existence of best proximity points for Prešić type nonself operators satisfying proximal type contractions.This article aims to present the notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions for mappings defined on product spaces. Through these notions, we will study the existence of fixed points and best proximity points for such mappings.


## 2. Main Results

We begin this section with the following definition.Definition 1.
A mapN:L×L⟶L is called an interpolative Prešić type-I contraction, if for each s,w,t,v∈L∖FixN, we get(5)dLNs,w,Nt,vminγs,w,γt,v≤ζdLw,Ns,wϑ1dLv,Nt,vϑ2,where γ:L×L⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l.
The following theorem is used to study the existence of fixed points for the above map.Theorem 2.
Consider an interpolative Prešić type-I contraction mapN:L×L⟶L on a complete metric space L,dL. Also, consider that(i)
Ifminγs,w,γt,v=1, then γNs,w,Nt,v=1.(ii)
There exist two elementss,w∈L with minγs,w,γw,Ns,w=1.(iii)
For every sequencelm in L with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l=1.
Then, there exists at least one point ofL that satisfies the equation l=Nl,l.Proof.
Hypothesis (ii) assures that there are two points, say,l0 and l1 in L with(6)minγl0,l1,γl1,Nl0,l1=1.
By using these two points, we can define a sequencelm with lm+1=Nlm−1,lm∀m∈ℕ. From hypothesis (i), it can be concluded that γlm,lm+1=1,∀m>1. Hence,(7)minγlm−1,lm,γlm,lm+1=1∀m∈ℕ.
By (5), we get(8)dLNlm−1,lm,Nlm,lm+1=dLNlm−1,lm,Nlm,lm+1minγlm−1,lm,γlm,lm+1,≤ζdLlm,Nlm−1,lmϑ1dLlm+1,Nlm,lm+1ϑ2∀m∈ℕ,that is,(9)dLlm+1,lm+2≤ζdLlm,lm+1ϑ1dLlm+1,lm+2ϑ2,∀m∈ℕ.
By (9), we obtain(10)dLlm+1,lm+21−ϑ2≤ζdLlm,lm+1ϑ1,∀m∈ℕ.
Since1−ϑ2=ϑ1, thus, by (10), we get(11)dLlm+1,lm+2≤ζ1/1−ϑ2dLlm,lm+1,≤ζdLlm,lm+1∀m∈ℕ.
Hence, by (11), we conclude that(12)dLlm+1,lm+2≤ζmdLl1,l2,∀m∈ℕ.
By triangle inequality and (12), for each k,n∈ℕ with k>n, we obtain(13)dLln,lk≤∑j=nk−1dLlj,lj+1≤∑j=nk−1ζj−1dLl1,l2.
In view of the above inequality and the convergence of∑j=1∞ζj, we say that the sequence lm is a Cauchy in L. By the completeness of L,dL, we get a point l∗∈L, such that lm⟶l∗. Also, by (iii), we get γl∗,l∗=1, since γlm,lm+1=1,∀m∈ℕ and lm⟶l∗.
Here, the claim isl∗=Nl∗,l∗. If the claim is wrong, then by (5), for each m∈ℕ, we get(14)dLNlm−1,lm,Nl∗,l∗=dLNlm−1,lm,Nl∗,l∗minγlm−1,lm,γl∗,l∗,≤ζdLlm,Nlm−1,lmϑ1dLl∗,Nl∗,l∗ϑ2,≤ζdLlm+1,lm+2ϑ1dLl∗,Nl∗,l∗ϑ2.
By triangle inequality and (14), we obtain(15)dLl∗,Nl∗,l∗≤dLl∗,Nlm,lm+1+dLNlm,lm+1,Nl∗,l∗,≤dLl∗,lm+2+ζdLlm+1,lm+2ϑ1dLl∗,Nl∗,l∗ϑ2.
Lettingm⟶∞ in (15), we get dLl∗,Nl∗,l∗=0. Hence, the claim is true, that is, l∗=Nl∗,l∗.Example 1.
ConsiderL=ℝ equipped with a metric dLk,l=k−l for each k,l∈L. Define N:L×L⟶L and γ:L×L⟶ℝ∖0 by(16)Nk,l=k+l2,ifk,l≥0,0,otherwise,and(17)γk,l=1,ifk,l≥0.1/2,otherwise.
Then, one can easily verify that the axioms of Theorem2 are satisfied. Hence, there is at least one element l∗∈L, such that l∗=Nl∗,l∗.
In the following, we present interpolative Prešić type-II contraction map and related fixed point result.Definition 2.
A mapN:L×L⟶L is called an interpolative Prešić type-II contraction, if for each s,w,t,v∈L∖FixN with minγs,w,γt,v≥1, we get(18)dLNs,w,Nt,v≤ζdLw,Ns,wϑ1dLv,Nt,vϑ2,where γ:L×L⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l.Theorem 3.
Consider an interpolative Prešić type-II contraction mapN:L×L⟶L on a complete metric space L,dL. Also, consider that(i)
Ifminγs,w,γt,v≥1, then γNs,w,Nt,v≥1.(ii)
There exist two elementss,w∈L with minγs,w,γw,Ns,w≥1.(iii)
For every sequencelm in L with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l≥1.Then, there exists at least one point ofL that satisfies the equation l=Nl,l.Proof.
In view of the hypothesis (ii), we get(19)minγl0,l1,γl1,Nl0,l1≥1.For some l0 and l1 in L. Through these two points, we can construct a sequence lm with lm+1=Nlm−1,lm∀m∈ℕ. Also, hypothesis (i) implies that γlm,lm+1≥1,∀m>1. Hence,(20)minγlm−1,lm,γlm,lm+1≥1∀m∈ℕ.
By (18), we get(21)dLNlm−1,lm,Nlm,lm+1≤ζdLlm,Nlm−1,lmϑ1dLlm+1,Nlm,lm+1ϑ2∀m∈ℕ.
By performing some calculations, we get(22)dLlm+1,lm+2≤ζdLlm,lm+1ϑ1≤ζmdLl1,l2,∀m∈ℕ.
Hence, it can be seen thatlm is a Cauchy sequence in L with lm⟶l∗∈L. Also, by (iii), we get γl∗,l∗≥1. Suppose that l∗≠Nl∗,l∗. Then, by (18), for each m∈ℕ, we get(23)dLNlm,lm+1,Nl∗,l∗≤ζdLlm+1,Nlm,lm+1ϑ1dLl∗,Nl∗,l∗ϑ2,≤ζdLlm+1,lm+2ϑ1dLl∗,Nl∗,l∗ϑ2.
By triangle inequality and (23), we obtain(24)dLl∗,Nl∗,l∗≤dLl∗,Nlm,lm+1+dLNlm,lm+1,Nl∗,l∗,≤dLl∗,lm+2+ζdLlm+1,lm+2ϑ1dLl∗,Nl∗,l∗ϑ2.
Lettingm⟶∞ in (24), we get dLl∗,Nl∗,l∗=0. Hence, our supposition is wrong and l∗=Nl∗,l∗.Example 2.
ConsiderL=ℤ equipped with a metric dLk,l=k−l for each k,l∈L. Define N:L×L⟶L and γ:L×L⟶ℝ by(25)Nk,l=l,ifk,l≥0,k+lotherwise,and(26)γk,l=1,ifk,l≥0.0,otherwise.
Then, one can easily verify that the axioms of Theorem3 are satisfied. Hence, there is at least one element l∗∈L, such that l∗=Nl∗,l∗.
Some consequences of the above results can be obtained in the form of the following listed corollaries. The following corollary is obtained from Theorem2 by considering t=w.Corollary 1.
Consider a mapN:L×L⟶L on a complete metric space L,dL, such that for each s,w,v∈L∖FixN, we get(27)dLNs,w,Nw,vminγs,w,γw,v≤ζdLw,Ns,wϑ1dLv,Nw,vϑ2,where γ:L×L⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, and ζ∈0,1. Also, consider that(i)
Ifminγs,w,γw,v=1, then γNs,w,Nw,v=1(ii)
There exist two elementss,w∈L with minγs,w,γw,Ns,w=1(iii)
For every sequencelm in L with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l=1Then, there exists at least one point ofL that satisfies the equation l=Nl,l.The following corollary is a special case of Theorem3 which can be obtained by considering t=w.Corollary 2.
Consider a mapN:L×L⟶L on a complete metric space L,dL, such that for each s,w,v∈L∖FixN with minγs,w,γw,v≥1, we get(28)dLNs,w,Nw,v≤ζdLw,Ns,wϑ1dLv,Nw,vϑ2,where γ:L×L⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1 and ζ∈0,1. Also, consider that(i)
Ifminγs,w,γw,v≥1, then γNs,w,Nw,v≥1(ii)
There exist two elementss,w∈L with minγs,w,γw,Ns,w≥1(iii)
For every sequencelm in L with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l≥1Then, there exists at least one point ofL that satisfies the equation l=Nl,l.By definingγs,w=1 for each s,w∈L in Theorem 2 or Theorem 3, we get the following.Corollary 3.
Consider a mapN:L×L⟶L on a complete metric space L,dL that satisfies(29)dLNs,w,Nt,v≤ζdLw,Ns,wϑ1dLv,Nt,vϑ2,for each s,w,v∈L∖FixN, where ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1 and ζ∈0,1. Then, there exists at least one point of L that satisfies the equation l=Nl,l.From the above corollary, we can also obtain the following result.Corollary 4.
Consider a mapN:L×L⟶L on a complete metric space L,dL that satisfies(30)dLNs,w,Nw,v≤ζdLw,Ns,wϑ1dLv,Nw,vϑ2,for each s,w,v∈L∖FixN, where ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1 and ζ∈0,1. Then, there exists at least one point of L that satisfies the equation l=Nl,l.In the following, we will study about the interpolative Prešić type proximal contractions and related results.LetL,dL be a metric space and OL,ML be nonvoid subsets of L. We will use the following notations.(31)DLOL,ML=infdLo,m:o∈OL,m∈ML,dLo,ML=infdLo,m:m∈ML,OL0=o∈OL:dLo,m=DLOL,MLfor somem∈ML,ML0=m∈ML:dLo,m=DLOL,MLfor someo∈OL.Note that a pointo∗∈OL is known as a best proximity point of N:OL×OL⟶ML if dLo∗,No∗,o∗=DLOL,ML. The collection of all such points for N:OL×OL⟶ML is denoted by BesN.Definition 3.
A mapN:OL×OL⟶ML is called an interpolative Prešić type-I proximal contraction, if for each s,w,t,v,p,q∈OL∖BesN with dLp,Ns,w=DLOL,ML=dLq,Nt,v, we get(32)dLp,qminγs,w,γt,v≤ζdLw,pϑ1dLv,qϑ2,where γ:OL×OL⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1 and ζ∈0,1.
The following theorem is used to ensure the existence of best proximity points for the above defined maps.Theorem 4.
Consider an interpolative Prešić type-I proximal contraction mapN:OL×OL⟶ML on a metric space L,dL. Also, consider that(i)
Ifminγs,w,γt,v=1 and dLp,Ns,w=DLOL,ML=dLq,Nt,v, then γp,q=1.(ii)
There exist elementss,w,p∈OL with dLp,Ns,w=DLOL,ML and minγs,w,γw,p≥1.(iii)
NOL×OL0⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l=1.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l=DLOL,ML.Proof.
From hypothesis (ii), we havel0, l1, and l2 in OL with minγl0,l1γl1,l2=1 and dLl2,Nl0,l1=DLOL,ML. Hypothesis (iii) implies that Nl1,l2∈ML0, and there is l3∈OL0 satisfying dLl3,Nl1,l2=DLOL,ML. Thus, from hypothesis (i), we get γl2,l3=1. Hence, by using hypotheses (i) and (ii) repeatedly, we conclude that γlm−1,lm=1 and dLlm+1,Nlm−1,lm=DLOL,ML for all m∈ℕ.
SincedLlm+1,Nlm−1,lm=DLOL,ML=dLlm+2,Nlm,lm+1 for each m∈ℕ and minγlm−1,lm,γlm,lm+1=1 for each m∈ℕ, then, by (32), we get(33)dLlm+1,lm+2≤ζdLlm+1,lm+2minγlm−1,lm,γlm,lm+1,≤ζdLlm,lm+1ϑ1dLlm+1,lm+2ϑ2,∀m∈ℕ.
Now, by following the proof of Theorem2, we say that lmm≥2 is a Cauchy sequence in OL0. Since OL0 is complete, we have a point l∗∈OL0, such that lm⟶l∗. Also, by (v), we get γl∗,l∗=1, since γlm,lm+1=1 and lm⟶l∗. Clearly, Nl∗,l∗∈ML0, and there is w∗∈OL0 with dLw∗,Nl∗,l∗=DLOL,ML. Here, the claim is w∗=l∗. Suppose it is wrong, then by (33), for each m∈ℕ, we get(34)dLlm+1,w∗=dLlm+1,w∗minγlm−1,lm,γl∗,l∗,≤ζdLlm,lm+1ϑ1dLl∗,w∗ϑ2.
Lettingm⟶∞ in (34), we obtain dLl∗,w∗=0, and it contradicts our assumption. Hence, our claim is true, that is, l∗=w∗. Therefore, dLl∗,Nl∗,l∗=DLOL,ML.
In the following, we present the notion of interpolative Prešić type-II proximal contraction.Definition 4.
A mapN:OL×OL⟶ML is called an interpolative Prešić type-II proximal contraction, if for each s,w,t,v,p,q∈OL∖BesN with dLp,Ns,w=DLOL,ML=dLq,Nt,v and minγs,w,γt,v≥1, we get(35)dLp,q≤ζdLw,pϑ1dLv,qϑ2,where γ:OL×OL⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1 and ζ∈0,1.
The existence of best proximity points for above map can be ensured by the result given.Theorem 5.
Consider an interpolative Prešić type-II proximal contraction mapN:OL×OL⟶ML on a metric space L,dL. Also, consider that(i)
Ifminγs,w,γt,v≥1 and dLp,Ns,w=DLOL,ML=dLq,Nt,v, then γp,q≥1.(ii)
There exist elementss,w,p∈OL with dLp,Ns,w=DLOL,ML and minγs,w,γw,p≥1.(iii)
NOL×OL0⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γl,l≥1.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l=DLOL,ML.Proof.
The proof can be derived on the same steps as the proof of Theorem4 is done.


### 2.1. Results for Extended Interpolative Prešić Type Maps

This subsection presents the extensions of the above listed results. Theorems6 and 7 can be considered as an extended version of Theorem 2 and Theorem 3, respectively.Theorem 6.
Consider an extended interpolative Prešić type-I contraction mapN:Lk⟶L, for any fixed k∈ℕ, on a complete metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk∈L∖FixN, we get(36)dLNs1,s2,…,sk,Nw1,w2,…,wkminγs1,w1,γs2,w2,…,γsk,wk≤ζdLsk,Ns1,s2,…,skϑ1dLwk,Nw1,w2,…,wkϑ2,where γ:L×L⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l,…,l. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk=1, then γNs1,s2,…,sk,Nw1,w2,…,wk=1.(ii)
There exist elementss1,s2,…,sk∈L, such that(37)minγs1,s2,γs2,s3,…,γsk,Ns1,s2,…,sk=1.(iii)
For every sequencelm in L with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l=1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation l=Nl,l,…,l︸k−times.Proof.
Hypothesis (ii) implies the existence of elementsl1,l2,…,lk in L with(38)minγl1,l2,γl2,l3,…,γlk,Nl1,l2,…,lk=1.
Through these points, we can define a sequencelm with lm+k=Nlm,lm+1,…,lm+k−1 for all m∈ℕ. Hence, by considering hypothesis (i), it can be concluded that γlm,lm+1=1∀m≥m0. Then, we say that(39)minγlm,lm+1,γlm,lm+1,…,γlm+k−1,lm+k=1∀m∈ℕ.
By (36), we get(40)dLNlm,lm+1,…,lm+k−1,Nlm+1,lm+2,…,lm+k=dLNlm,lm+1,…,lm+k−1,Nlm+1,lm+2,…,lm+kminγlm,lm+1,γlm+1,lm+2,…,γlm+k−1,lm+k≤ζdLlm+k−1,Nlm,lm+1,…,lm+k−1ϑ1dLlm+k,Nlm+1,lm+2,…,lm+kϑ2∀m∈ℕ,that is,(41)dLlm+k,lm+k+1≤ζdLlm+k−1,lm+kϑ1dLlm+k,lm+k+1ϑ2,∀m∈ℕ.
By (41), we obtain(42)dLlm+k,lm+k+11−ϑ2≤ζdLlm+k−1,lm+kϑ1,∀m∈ℕ.
Since1−ϑ2=ϑ1, thus, by (42), we get(43)dLlm+k,lm+k+1≤ζ1/1−ϑ2dLlm+k−1,lm+kϑ1,≤ζdLlm+k−1,lm+k∀m∈ℕ.
Hence, by (43), we get(44)dLlm+k,lm+k+1≤ζmdLlk,lk+1,∀m∈ℕ,that is,(45)dLlm,lm+1≤ζm−kdLlk,lk+1,∀m≥k+1.
From triangle inequality and (45), for each q,n∈ℕ with q>n≥k+1, we obtain(46)dLln,lq≤∑j=nq−1dLlj,lj+1≤∑j=nq−1ζj−kdLlk,lk+1.
Above inequality yields thatlm is a Cauchy sequence in a complete space L. Hence, we get a point l∗∈L with lm⟶l∗. Also, by (iii), we get γlm,l∗=1,∀m∈ℕ, since γlm,lm+1=1,∀m∈ℕ and lm⟶l∗.
Here, the claim isl∗=Nl∗,l∗,…,l∗. If the claim is wrong, then by (36), for each m∈ℕ, we get(47)dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗=dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗minγlm,l∗,γlm+1,l∗,…,γlm+k−1,l∗≤ζdLlm+k−1,Nlm,lm+1,…,lm+k−1ϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2,≤dLlm+k−1,lm+kϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2.
By triangle inequality and (47), for each m, we obtain(48)dLl∗,Nl∗,l∗,…,l∗≤dLl∗,Nlm,lm+1,…,lm+k−1+dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗≤dLl∗,lm+k+ζdLlm+k−1,lm+kϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2.
Lettingm⟶∞ in(48), we get dLl∗,Nl∗,l∗,…,l∗=0. Hence, the claim is true, that is, l∗=Nl∗,l∗,…,l∗.Theorem 7.
Consider an extended interpolative Prešić type-II contraction mapN:Lk⟶L, for any fixed k∈ℕ, on a complete metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,∈L∖FixN with minγs1,w1,γs2,w2,…,γsk,wk≥1, we get(49)dLNs1,s2,…,sk,Nw1,w2,…,wk≤ζdLsk,Ns1,s2,…,skϑ1dLwk,Nw1,w2,…,wkϑ2,where γ:L×L⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l,…,l. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk≥1, then γNs1,s2,…,sk,Nw1,w2,…,wk≥1.(ii)
There exist elementss1,s2,…,sk∈L, such that(50)minγs1,s2,γs2,s3,…,γsk,Ns1,s2,…,sk≥1.(iii)
For every sequencelm in L with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l≥1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation l=Nl,l,…,l︸k−times.Proof.
The proof can be obtained on the same steps as the proofs of Theorems6 and 2 are done.
The following theorems can be considered as an extended form of Theorems3 and 5, respectively.Theorem 8.
Consider an extended interpolative Prešić type-I proximal contraction mapN:OLk⟶ML, for any fixed k∈ℕ, on a metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,p,q∈OL∖BesN with(51)dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk,we get(52)dLp,qminγs1,w1,γs2,w2,…,γsk,wk≤ζdLsk,pϑ1dLwk,qϑ2,where γ:OL×OL⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, BesN=o∈OL:dLo,No,o,…,o=DLOL,ML, and OL,ML are the nonvoid subsets of L. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk=1 and dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk, then γp,q=1.(ii)
There exist elementss1,s2,…,sk,p∈OL with dLp,Ns1,s2,…,sk=DLOL,ML and(53)minγs1,s2,γs2,s3,…,γsk,p=1.(iii)
NOL×⋯×OL×OL0︸⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l=1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l,…,l︸k−times=DLOL,ML.Proof.
By hypothesis (ii), we getl1,l2,…,lk, lk+1 in OL with dLlk+1,Nl1,l2,…,lk=DLOL,ML and(54)minγl1,l2,γl2,l3,…,γlk,lk+1=1.
Hypothesis (iii) implies thatNl2,l3,…,lk+1∈ML0, and there is lk+2∈OL0 satisfying(55)dLlk+2,Nl2,l3,…,lk+1=DLOL,ML.
Then, from hypothesis (i), we getγlk+1,lk+2=1. Repeated use of hypotheses (i), (ii), and (iii) yields γlm,lm+1=1 and dLlm+k,Nlm,lm+1,…,lm+k−1=DLOL,ML for all m∈ℕ. As(56)dLlm+k,Nlm,lm+1,…,lm+k−1=DLOL,ML,and(57)dLlm+k+1,Nlm+2,lm+2,…,lm+k=DLOL,ML,∀m∈ℕ,and(58)minγlm,lm+1,γlm+1,lm+2,…,γlm+k−1,lm+k=1,∀m∈ℕ.
Then, by (52), we get(59)dLlm+k,lm+k+1=dLlm+k,lm+k+1minγlm,lm+1,…,γlm+k−1,lm+k,≤ζdLlm+k−1,lm+kϑ1dLlm+k,lm+k+1ϑ2,∀m∈ℕ.
Following the proof of Theorem6, we say that lmm≥k+1 is Cauchy in OL0, and by the completeness of OL0, we get a point l∗∈OL0 with lm⟶l∗. Also, by (v), we get γlm,l∗=1∀m∈ℕ, since γlm,lm+1=1 and lm⟶l∗. Clearly, Nl∗,l∗,…,l∗∈ML0, and there is w∗∈OL0 with dLw∗,Nl∗,l∗,…,l∗=DLOL,ML. Here, the claim is w∗=l∗. Suppose it is wrong, then by (52), for each m∈ℕ, we get(60)dLlm+k,w∗=dLlm+1,w∗minγlm,l∗,γlm+1,l∗,…,γlm+k−1,l∗≤ζdLlm+k−1,lm+kϑ1dLl∗,w∗ϑ2.
Lettingm⟶∞ in (60), we obtain dLl∗,w∗=0, and it contradicts our assumption. Hence, our claim is true, that is, l∗=w∗. Therefore, dLl∗,Nl∗,l∗,…,l∗=DLOL,ML.Theorem 9.
Consider an extended interpolative Prešić type-II proximal contraction mapN:OLk⟶ML, for any fixed k∈ℕ, on a metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,p,q∈OL∖BesN with(61)dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk,and(62)minγs1,w1,γs2,w2,…,γsk,wk≥1,we get(63)dLp,q≤ζdLsk,pϑ1dLwk,qϑ2,where γ:OL×OL⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, and ζ∈0,1. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk≥1 and dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk, then γp,q≥1.(ii)
There exist elementss1,s2,…,sk,p∈OL with dLp,Ns1,s2,…,sk=DLOL,ML and(64)minγs1,s2,γs2,s3,…,γsk,p≥1.(iii)
NOL×⋯×OL×OL0︸k−1times⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l≥1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l,…,l︸k−times=DLOL,ML.The proof of the above theorem can be derived by viewing the proof of Theorem8.


## 2.1. Results for Extended Interpolative Prešić Type Maps

This subsection presents the extensions of the above listed results. Theorems6 and 7 can be considered as an extended version of Theorem 2 and Theorem 3, respectively.Theorem 6.
Consider an extended interpolative Prešić type-I contraction mapN:Lk⟶L, for any fixed k∈ℕ, on a complete metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk∈L∖FixN, we get(36)dLNs1,s2,…,sk,Nw1,w2,…,wkminγs1,w1,γs2,w2,…,γsk,wk≤ζdLsk,Ns1,s2,…,skϑ1dLwk,Nw1,w2,…,wkϑ2,where γ:L×L⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l,…,l. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk=1, then γNs1,s2,…,sk,Nw1,w2,…,wk=1.(ii)
There exist elementss1,s2,…,sk∈L, such that(37)minγs1,s2,γs2,s3,…,γsk,Ns1,s2,…,sk=1.(iii)
For every sequencelm in L with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l=1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation l=Nl,l,…,l︸k−times.Proof.
Hypothesis (ii) implies the existence of elementsl1,l2,…,lk in L with(38)minγl1,l2,γl2,l3,…,γlk,Nl1,l2,…,lk=1.
Through these points, we can define a sequencelm with lm+k=Nlm,lm+1,…,lm+k−1 for all m∈ℕ. Hence, by considering hypothesis (i), it can be concluded that γlm,lm+1=1∀m≥m0. Then, we say that(39)minγlm,lm+1,γlm,lm+1,…,γlm+k−1,lm+k=1∀m∈ℕ.
By (36), we get(40)dLNlm,lm+1,…,lm+k−1,Nlm+1,lm+2,…,lm+k=dLNlm,lm+1,…,lm+k−1,Nlm+1,lm+2,…,lm+kminγlm,lm+1,γlm+1,lm+2,…,γlm+k−1,lm+k≤ζdLlm+k−1,Nlm,lm+1,…,lm+k−1ϑ1dLlm+k,Nlm+1,lm+2,…,lm+kϑ2∀m∈ℕ,that is,(41)dLlm+k,lm+k+1≤ζdLlm+k−1,lm+kϑ1dLlm+k,lm+k+1ϑ2,∀m∈ℕ.
By (41), we obtain(42)dLlm+k,lm+k+11−ϑ2≤ζdLlm+k−1,lm+kϑ1,∀m∈ℕ.
Since1−ϑ2=ϑ1, thus, by (42), we get(43)dLlm+k,lm+k+1≤ζ1/1−ϑ2dLlm+k−1,lm+kϑ1,≤ζdLlm+k−1,lm+k∀m∈ℕ.
Hence, by (43), we get(44)dLlm+k,lm+k+1≤ζmdLlk,lk+1,∀m∈ℕ,that is,(45)dLlm,lm+1≤ζm−kdLlk,lk+1,∀m≥k+1.
From triangle inequality and (45), for each q,n∈ℕ with q>n≥k+1, we obtain(46)dLln,lq≤∑j=nq−1dLlj,lj+1≤∑j=nq−1ζj−kdLlk,lk+1.
Above inequality yields thatlm is a Cauchy sequence in a complete space L. Hence, we get a point l∗∈L with lm⟶l∗. Also, by (iii), we get γlm,l∗=1,∀m∈ℕ, since γlm,lm+1=1,∀m∈ℕ and lm⟶l∗.
Here, the claim isl∗=Nl∗,l∗,…,l∗. If the claim is wrong, then by (36), for each m∈ℕ, we get(47)dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗=dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗minγlm,l∗,γlm+1,l∗,…,γlm+k−1,l∗≤ζdLlm+k−1,Nlm,lm+1,…,lm+k−1ϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2,≤dLlm+k−1,lm+kϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2.
By triangle inequality and (47), for each m, we obtain(48)dLl∗,Nl∗,l∗,…,l∗≤dLl∗,Nlm,lm+1,…,lm+k−1+dLNlm,lm+1,…,lm+k−1,Nl∗,l∗,…,l∗≤dLl∗,lm+k+ζdLlm+k−1,lm+kϑ1dLl∗,Nl∗,l∗,…,l∗ϑ2.
Lettingm⟶∞ in(48), we get dLl∗,Nl∗,l∗,…,l∗=0. Hence, the claim is true, that is, l∗=Nl∗,l∗,…,l∗.Theorem 7.
Consider an extended interpolative Prešić type-II contraction mapN:Lk⟶L, for any fixed k∈ℕ, on a complete metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,∈L∖FixN with minγs1,w1,γs2,w2,…,γsk,wk≥1, we get(49)dLNs1,s2,…,sk,Nw1,w2,…,wk≤ζdLsk,Ns1,s2,…,skϑ1dLwk,Nw1,w2,…,wkϑ2,where γ:L×L⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, and FixN=l∈L:l=Nl,l,…,l. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk≥1, then γNs1,s2,…,sk,Nw1,w2,…,wk≥1.(ii)
There exist elementss1,s2,…,sk∈L, such that(50)minγs1,s2,γs2,s3,…,γsk,Ns1,s2,…,sk≥1.(iii)
For every sequencelm in L with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l≥1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation l=Nl,l,…,l︸k−times.Proof.
The proof can be obtained on the same steps as the proofs of Theorems6 and 2 are done.
The following theorems can be considered as an extended form of Theorems3 and 5, respectively.Theorem 8.
Consider an extended interpolative Prešić type-I proximal contraction mapN:OLk⟶ML, for any fixed k∈ℕ, on a metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,p,q∈OL∖BesN with(51)dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk,we get(52)dLp,qminγs1,w1,γs2,w2,…,γsk,wk≤ζdLsk,pϑ1dLwk,qϑ2,where γ:OL×OL⟶ℝ∖0 is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, ζ∈0,1, BesN=o∈OL:dLo,No,o,…,o=DLOL,ML, and OL,ML are the nonvoid subsets of L. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk=1 and dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk, then γp,q=1.(ii)
There exist elementss1,s2,…,sk,p∈OL with dLp,Ns1,s2,…,sk=DLOL,ML and(53)minγs1,s2,γs2,s3,…,γsk,p=1.(iii)
NOL×⋯×OL×OL0︸⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1=1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l=1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l,…,l︸k−times=DLOL,ML.Proof.
By hypothesis (ii), we getl1,l2,…,lk, lk+1 in OL with dLlk+1,Nl1,l2,…,lk=DLOL,ML and(54)minγl1,l2,γl2,l3,…,γlk,lk+1=1.
Hypothesis (iii) implies thatNl2,l3,…,lk+1∈ML0, and there is lk+2∈OL0 satisfying(55)dLlk+2,Nl2,l3,…,lk+1=DLOL,ML.
Then, from hypothesis (i), we getγlk+1,lk+2=1. Repeated use of hypotheses (i), (ii), and (iii) yields γlm,lm+1=1 and dLlm+k,Nlm,lm+1,…,lm+k−1=DLOL,ML for all m∈ℕ. As(56)dLlm+k,Nlm,lm+1,…,lm+k−1=DLOL,ML,and(57)dLlm+k+1,Nlm+2,lm+2,…,lm+k=DLOL,ML,∀m∈ℕ,and(58)minγlm,lm+1,γlm+1,lm+2,…,γlm+k−1,lm+k=1,∀m∈ℕ.
Then, by (52), we get(59)dLlm+k,lm+k+1=dLlm+k,lm+k+1minγlm,lm+1,…,γlm+k−1,lm+k,≤ζdLlm+k−1,lm+kϑ1dLlm+k,lm+k+1ϑ2,∀m∈ℕ.
Following the proof of Theorem6, we say that lmm≥k+1 is Cauchy in OL0, and by the completeness of OL0, we get a point l∗∈OL0 with lm⟶l∗. Also, by (v), we get γlm,l∗=1∀m∈ℕ, since γlm,lm+1=1 and lm⟶l∗. Clearly, Nl∗,l∗,…,l∗∈ML0, and there is w∗∈OL0 with dLw∗,Nl∗,l∗,…,l∗=DLOL,ML. Here, the claim is w∗=l∗. Suppose it is wrong, then by (52), for each m∈ℕ, we get(60)dLlm+k,w∗=dLlm+1,w∗minγlm,l∗,γlm+1,l∗,…,γlm+k−1,l∗≤ζdLlm+k−1,lm+kϑ1dLl∗,w∗ϑ2.
Lettingm⟶∞ in (60), we obtain dLl∗,w∗=0, and it contradicts our assumption. Hence, our claim is true, that is, l∗=w∗. Therefore, dLl∗,Nl∗,l∗,…,l∗=DLOL,ML.Theorem 9.
Consider an extended interpolative Prešić type-II proximal contraction mapN:OLk⟶ML, for any fixed k∈ℕ, on a metric space L,dL, that is, for each s1,s2,…,sk, w1,w2,…,wk,p,q∈OL∖BesN with(61)dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk,and(62)minγs1,w1,γs2,w2,…,γsk,wk≥1,we get(63)dLp,q≤ζdLsk,pϑ1dLwk,qϑ2,where γ:OL×OL⟶ℝ is a map, ϑ1,ϑ2∈0,1 with ϑ1+ϑ2=1, and ζ∈0,1. Also, consider that(i)
Ifminγs1,w1,γs2,w2,…,γsk,wk≥1 and dLp,Ns1,s2,…,sk=DLOL,ML=dLq,Nw1,w2,…,wk, then γp,q≥1.(ii)
There exist elementss1,s2,…,sk,p∈OL with dLp,Ns1,s2,…,sk=DLOL,ML and(64)minγs1,s2,γs2,s3,…,γsk,p≥1.(iii)
NOL×⋯×OL×OL0︸k−1times⊆ML0.(iv)
OL0 is nonempty and complete with respect to dL.(v)
For every sequencelm in OL0 with γlm,lm+1≥1∀m≥m0 for some natural number m0 and lm⟶l, we have γlm,l≥1∀m≥m0.Then, there exists at least one point ofL that satisfies the equation dLl,Nl,l,…,l︸k−times=DLOL,ML.The proof of the above theorem can be derived by viewing the proof of Theorem8.


## 3. Conclusion

This article provides a few results dealing with fixed points and best proximity points of the mappings defined on product spaces. The notions of interpolative Prešić type contractions and interpolative Prešić type proximal contractions are introduced in the context of metric spaces to discuss the existence of fixed points and best proximity points of such maps, respectively. These notions are derived by considering the concept of interpolative Kannan contraction.



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