# Interpolative Kannan Contractions inT0-Quasi-Metric Spaces

Authors: Yaé Ulrich Gaba; Maggie Aphane; Hassen Aydi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we update the well-known fixed point theorem of Kannan using the interpolation notion in the realm of quasi-metric spaces. We consider some asymmetric versions. We also present some illustrative examples in support of the obtained results.


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## Body

## 1. Introduction and Preliminaries

In 2018, Karapinar [1] published a new type of contractions obtained from the definition of the Kannan contraction by interpolation as follows.Theorem 1 (see [11, Theorem 2.2]).
LetX,d be a complete metric space and T:X⟶X be an interpolative Kannan type contraction, i.e., a self-map such that there exist λ∈0,1,α∈0,1 so that(1)dTx,Ty≤λdx,Txαdy,Ty1−α,for all x,y∈X with x≠Tx. Then, T has a unique fixed point in X.This theorem has been generalized in 2019 by Gaba et al. [2] where they introduced the concept of λ,α,β-interpolative Kannan contractions as follows.Definition 1.
LetX,d be a metric space and T:X⟶X be a self-map. We shall call T a relaxed λ,α,β-interpolative Kannan contraction, if there exist 0≤λ<1,0<α,β≤1 with α+β<1 such that(2)dTx,Ty≤λdx,Txαdy,Tyβ.The interpolative method has been successfully applied to a diverse range of contractions (see [3–8]).In 1982, Reilly et al.[9] obtained a quasi-metric version of the celebrated Banach contraction principle. Since then, and especially in the last decade, several authors have contributed to the development of the fixed point theory in the framework of quasi-metric spaces (see [10, 11]). It is in the same spirit that we provide here a quasi-metric version of the interpolative Kannan fixed point theorem.Our basic reference for quasi-metric spaces is [12].Definition 2.
(see [13]). Let q:X×X⟶0,∞ be a function where X is a nonempty set. This function is called a quasi-pseudometric (respectively, T0-quasi-metric) on X if q1 and q2 (respectively, q1∗ and q2) hold, whereq1qξ,ξ=0,for allξ∈Xq1∗qξ,η=0=qη,ξimpliesξ=yq2qξ,ζ≤qξ,η+qη,ζfor allξ,η,ζ∈XThe conditionq1∗ is known as the T0-condition. Furthermore, given a quasi-pseudometric q on X, there is a natural function q−1:X×X⟶0,∞, defined by q−1ξ,η=qη,ξ for all ξ,η∈X. It is named as the conjugate of q. For a T0-quasi-metric q on X, the distance function qs:X×X⟶0,∞ defined by qsξ,η=maxqξ,η,qη,ξ for all ξ,η∈X×X is a metric on X.The classical example of aT0-quasi-metric is the truncated difference.Example 1.
(truncated difference). Setℝ0+:=0,∞. Given δ:ℝ0+×ℝ0+⟶ℝ0+ as(3)δξ,η=max0,ξ−ηξ,η∈X.
Under these conditions,δ forms a T0-quasi-metric. Further, the pair ℝ0+,δ becomes a T0-quasi-metric space.Each quasi-metricd on X induces a T0 topology τd on X which has as a base the family of open balls Bdξ,η:x∈X,ε>0 where Bdξ,η=ξ∈X:dξ,η<ε.Ifτd satisfies the separation axiom T1 (resp. T2 ) on X, we say that X,d is a T1 (resp. a Hausdorff) quasi-metric space. Note that a T0-quasi-metric space X,d is T1 if and only if for each x,y∈X, the condition dx,y=0 implies x=y.A quasi-metric spaceX,d is called bicomplete if the metric space X,ds is complete.Definition 3.
(convergence in quasi-pseudometric spaces, see [12]).
LetX,q be a quasi-pseudometric space. We say that the sequence ξnq-converges to ξ, or left-converges to ξ, if(4)qξn,ξ⟶0.
We denote this byξn⟶dξ. More precisely, ξn converges to ξ with respect to τq.
In a similar manner, a sequenceξnq−1-converges to ξ or right-converges to ξ with respect to τq−1, if(5)qξ,ξn⟶0.
We denote this byξn⟶q−1ξ. A sequence ξn in the setting of a quasi-pseudometric space X,qqs-converges to ξ in the case that the sequence converges to ξ from left and right, that is,(6)ξn⟶qξ,ξn⟶q−1ξ.
Moreover, it is denoted asξn⟶qsξ (or, ξn⟶ξ, if there is no confusion).Remark 1.
From the definition ofqs-convergence, we have(7)qs−convergence impliesq−convergence.As demonstrated in [7, Example 1.7.], the reverse implication does not hold in general.Definition 4.
(compare [13]).(1)
A sequenceξn in a quasi-pseudometric X,q is called leftK-Cauchy if for every ε>0, there exists n0∈ℕ such that(8)for alln,k:n0≤k≤n,qξk,ξn<ε.(2)
Similarly, we definerightK-Cauchyness. A sequence ξn in a quasi-metric space X,q is called right K-Cauchy if it is a left K-Cauchy sequence in the quasi-metric space X,q−1.(3)
The quasi-metric spaceX,d is called left (right) K-sequentially complete if every left (right) K-Cauchy sequence converges with respect to the topology τd.(4)
X,q is called q-sequentially complete if every Cauchy sequence in the metric space X,qs converges with respect to the topology τd.Remark 2.
One can easily be convinced that both bicompleteness and left (right)K-sequential completeness imply q-sequential completeness. However, the rest of implications does not hold in general. For more details, the reader can consult [12].


## 2. Revisiting the Interpolative Kannan Mappings

Let us recall that an interpolative Kannan contraction on a metric spaceX,m is a self-mapping T:X⟶X such that there exist λ∈0,1,α∈0,1 for which(9)mTx,Ty≤λmx,Txαmy,Ty1−α,for all x,y∈X with x≠Tx.Before proceeding to the main results of this paper, we would like to correct an inaccuracy that appears in the proof of Theorem 2.2 in the paper by Karapinar [1].Theorem 2.
(see [11, Theorem 2.2]).
LetX,d be a complete metric space and T:X⟶X be an interpolative Kannan type contraction, i.e., a self-map such that there exist λ∈0,1,α∈0,1 so that(10)dTx,Ty≤λdx,Txαdy,Ty1−α,for all x,y∈X with x≠Tx. Then, T has a unique fixed point in X.Proof.
Following the proof presented by Karapinar, letx0∈X, and construct the sequence xn by xn+1=Tnx0 for all positive integers n. Taking x=xn and y=xn−1 in (10), we derive that(11)dxn+1,xn=dTxn,Txn−1≤λdxn,Txnαdxn−1,Txn−11−α=λdxn,xn+1αdxn−1,xn1−α,which yields(12)dxn+1,xn1−α≤λdxn−1,xn1−α.
It is not difficult to see that inequality (10) is asymmetric, and to fill in this gap, we also consider the case of x=xn−1 and y=xn in (10). So taking x=xn−1 and y=xn in (10), we have(13)dxn,xn+1=dTxn−1,Txn≤λdxn−1,Txn−1αdxn,Txn1−α=λdxn−1,xnαdxn,xn+11−α,which yields(14)dxn,xn+1α≤λdxn−1,xnα.
Since0<α<1, it is obvious that 0<1−α<1; hence, it is routine to check that xn is a Cauchy sequence which converges to the unique fixed point x∗ of T.This gap was revealed in Example 2.3 of [1]. Indeed, according to [11, Example 2.3], let X=x,y,z,w be a set endowed with a metric m such that mx,x=my,y=mz,z=mw,w=0,mx,y=my,x=3,mx,z=mz,x=4, my,z=mz,y=3/2,mw,x=mx,w=5/2,mw,y=my,w=2,mw,z=mz,w=3/2. Define the self-mapping T on X by Tx=x,Ty=w,Tz=x and Tw=y. The author claimed that for α=1/8 and λ=9/10, the self-mapping T forms an interpolative Kannan type contraction. However, considering the pair z,y, note that Tz≠z. We have(15)2.5=mTz,Ty=mx,w=52>910mz,Tz1/8my,Ty7/8=41/827/8≃1.962913.That is, inequality (10) fails for mTz,Ty.However, considering the pairy,z, note that Ty≠y. We have(16)2.5=mTy,Tz=mw,x=52≤910my,Ty1/8mz,Tz7/8=91021/847/8≃3.301214.This suggests a modification of the so-calledinterpolative Kannan contraction in the following way.Definition 5.
LetX,m be a metric space. A self-mapping T:X⟶X is called a generalized interpolative Kannan contraction if there exist λ∈0,1,α∈0,1 for which(17)mTx,Ty≤λmaxmx,Txαmy,Ty1−α,mx,Tx1−αmy,Tyα,for all x,y∈X with x≠Tx and y≠Ty.Remark 3.
Observe then that forα=1/2 in (17), a generalized interpolative Kannan contraction is actually just an interpolative Kannan contraction in the sense and spirit of Karapinar.Using this definition, we get the following.Theorem 3 (compare [11, Theorem 2.2]).
LetX,d be a complete metric space and T:X⟶X be a generalized interpolative Kannan contraction. Then, T has a unique fixed point in X.Proof.
The proof is merely a copy of that of [11, Theorem 2.2.] and needs not be repeated. One just has to observe that the right hand side of inequality (17) is symmetric in x,y∈X. This has actually been captured in the proof of Theorem 2. Moreover, [11, Example 2.3.] satisfies the hypothesis of Theorem 3.


## 3. Interpolative Kannan Mappings and Fixed Points in Quasi-Metric Spaces

The discussion we made in the previous section suggests, in a natural way, the following notions.Definition 6.
LetX,d be a quasi-metric space.
Ad-interpolative Kannan contraction on X,d is a mapping T:X⟶X such that there exist λ∈0,1,α∈0,1 for which(18)dTx,Ty≤λdTx,xαdy,Ty1−α,for all x,y∈X with x≠Tx and y≠Ty.
Ad−1-interpolative Kannan contraction on X,d is a mapping T:X⟶X such that there exist λ∈0,1,α∈0,1 for which(19)dTx,Ty≤λdx,TxαdTy,y1−α,for all x,y∈X with x≠Tx,y≠Ty.Then, we have the following easy, but useful, consequence of the interpolative Kannan contraction for metric spaces.Proposition 1.
LetX,d be a T0-quasi-metric space. If T is a d-interpolative Kannan contraction or a d−1-interpolative Kannan contraction on X,d, then the following conditions hold:(1)
T is a generalized interpolative Kannan contraction on the metric space X,ds.(2)
For anyx0∈X, Tnx0n∈ℕ is a Cauchy sequence in the metric space X,ds.Proof.
For (1), suppose thatT is a d-interpolative Kannan contraction on X,d. So there exist λ∈0,1 and α∈0,1 for which(20)dTx,Ty≤λdTx,xαdy,Ty1−α≤λdsy,TyαdsTx,x1−α,for all x,y∈X with x≠Tx and y≠Ty. Similarly, given x,y∈X, we also have(21)dTy,Tx≤λdTy,yαdx,Tx1−α≤λdsx,TxαdsTy,y1−α,for all x,y∈X with x≠Tx and y≠Ty.
Thus, givenx,y∈X,(22)dsTx,Ty≤λmaxdsy,TyαdsTx,x1−α,dsy,TyαdsTx,x1−α.
Then,T is a generalized interpolative Kannan contraction on the metric space X,ds.
(2) Since, by (1),T is a generalized interpolative Kannan contraction on the metric space X,ds, it follows from the modified proof of classical interpolative Kannan contraction principle that Tnx0n∈ℕ is a Cauchy sequence in the metric space X,ds.By using the preceding proposition, three quasi-metric versions of the generalized interpolative Kannan principle are easily deduced.Theorem 4.
Everyd-(resp. every d−1-) interpolative Kannan contraction on a bicomplete T0-quasi-metric space on X,d has a unique fixed point.Proof.
LetT be a d or a d−1 interpolative Kannan contraction. Since X,ds is a complete metric space and, by Proposition 1 (1), T is a generalized interpolative Kannan on X,ds, we deduce a modified proof of classical interpolative Kannan contraction principle that T has a unique fixed point.Theorem 5.
Everyd-interpolative Kannan contraction on a Hausdorff d-sequentially complete T0-quasi-metric space X,d has a unique fixed point.Proof.
LetT be a d-interpolative Kannan contraction on the Hausdorff d-sequentially complete T0-quasi- metric space X,d. Fix an x0∈X. By Proposition 1 (2), the sequence Tnx0n∈ℕ is a Cauchy sequence in the metric space X,ds. Hence, there is y∈X such that Tnx0n∈ℕ converges to y with respect to τd, i.e., dy,Tnx0⟶0 as n⟶∞. Since T is a d-interpolative Kannan contraction, there exist λ∈0,1,α∈0,1 for which(23)dTn+1x0,Ty≤λdTn+1x0,Tnx0αdy,Ty1−α.
Consequently,dTn+1x0,Ty⟶0 as n⟶∞. From Hausdorffness of X,d, we deduce that y=Ty. Finally, suppose that z∈X is a fixed point of T. From (10), we have(24)dy,z=dTy,Tz≤λdTy,yαdz,Tz1−α⇒dy,z=0,dz,y=dTz,Ty≤λdTz,zαdy,Ty1−α⇒dz,y=0,and thus y=z since the space X,d is T0. This concludes the proof.Corollary 1.
Everyd-interpolative Kannan contraction on a T0-quasi-metric space X,d such that X,d−1 is Hausdorff and d−1-sequentially complete has a unique fixed point.Proof.
LetT be a d-interpolative Kannan contraction on X,d. Put q=d−1; then, T is a q-interpolative Kannan contraction on a Hausdorff q-sequentially complete T0-quasi-metric space X,q. From Theorem 5, we deduce that T has a unique fixed point.Theorem 6.
Everyd−1-interpolative Kannan contraction on a T1d-sequentially complete T0-quasi-metric space X,d has a unique fixed point.Proof.
LetT be a d−1-interpolative Kannan contraction on the T1d-sequentially complete T0-quasi-metric space X,d. Fix x0∈X. As in the proof of Theorem 5 (see Proposition 1), Tnx0n∈ℕ is a Cauchy sequence in the metric space X,ds. Hence, there is y∈X such that Tnx0n∈ℕ converges to y with respect to τd, i.e., dy,Tnx0⟶0 as n⟶∞. Since T is a d−1-interpolative Kannan contraction, there exist λ∈0,1,α∈0,1 for which(25)dTn+1x0,Ty≤λdTnx0,Tn+1x0αdTy,y1−α,for all n∈ℕ. Consequently, dTn+1x0,Ty⟶0 as n⟶∞. From the triangle inequality,(26)dy,Ty≤dy,Tnx0+dTnx0,Ty.
We deducedy,Ty=0. Therefore, y=Ty because X,d is a T0-quasi-metric space, which is T1. Finally, suppose that z∈X is a fixed point of T. Then,(27)dy,z=dTy,Tz≤λdy,TyαdTz,z1−α,and thus y=z. This concludes the proof.Corollary 2.
Everyd-contraction on a T0-quasi-metric space X,d which is T1 such that X,d−1 is d−1-sequentially complete has a unique fixed point.Proof.
LetT be a d-interpolative Kannan contraction on X,d. Put q=d−1. Then, T is a q−1-interpolative Kannan contraction on the T1q-sequentially complete T0-quasi-metric space X,q. From Theorem 6, we deduce that T has a unique fixed point.Remark 4.
Note that Theorems5 and 6 remain valid if “d-sequentially complete” is replaced with “left K-sequentially complete” or “right K-sequentially complete.”We conclude the paper with an example which shows that Theorem5 cannot be generalized to T1d-sequentially complete T0-quasi-metric spaces.Example 2.
LetX=ℕ and let d be the T1T0-quasi-metric on X given by dn,n=0 for all n∈X, and dn,m=1/m otherwise. Clearly, X,d is both left and right K-sequentially complete, so it is d-sequentially complete. Now, define T:X⟶X as Tn=2n for all n∈X. Of course, T has no fixed point. However, it is a d-interpolative Kannan contraction with λ=3/4 and α=1/2 since for each n,m∈X with n≠m, one has(28)dTn,Tm=d2n,2m=12m≤34n2m=34dTn,n1/2dm,Tm1/2.In light of the above example, a question is in order.Open Question 1. Can Theorem6 be generalized to d-sequentially complete T0-quasi-metric spaces? It is our belief that the answer is “No”; however, so far, we have failed to provide a counter-example.


## 4. Conclusion

As mentioned in the introduction, the interesting concept ofλ,α,β-interpolative Kannan contractions was introduced in [2]. So, the authors plan, in a different manuscript, to discuss asymmetric versions of such a concept by first outlining the asymmetric nature of this notion, even for a classical interpolative Kannan contraction in a metric space.



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