## 1. Preliminaries

Kannan fixed-point theorem is the first significant variant of the outstanding result of Banach on the metric fixed-point theory [

1,

2]. Kannan’s theorem has been generalized in different ways. In the present note, we zoom in on one of the recent generalizations that was proposed by Karapınar [

3] as

interpolative Kannan-type contraction. It was indicated in [

3] that each interpolative Kannan-type contraction in a complete metric space admits a fixed point (see also e.g., [

4,

5,

6,

7]). More precisely, we have:

**Theorem** **1** Let $(X,d)$ be a complete metric space and $T:X\to X$ an interpolative Kannan-type contraction, i.e., T is a self-map such that there exist $\lambda \in [0,1),\phantom{\rule{4pt}{0ex}}\alpha \in (0,1)$ withfor all $x,y\in X\setminus Fix\left(T\right)$, where $Fix\left(T\right):=\{x\in X:Tx=x\}$. Then T has a fixed point in X.

Our contribution in the present manuscript aims at sharpening the inequality (

1) by increasing the degree of freedom of the powers appearing in the right-hand side in the framework of standard metric spaces. We also indicate the novelty of our results by expressing some examples.

## 2. Main Results

We start with the following definition.

**Definition** **1.** Let $(X,d)$ a metric space and $T:X\to X$ a self-map. We shall call T a $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction, if there exist $\lambda \in [0,1),\alpha ,\beta \in (0,1)$ with $\alpha +\beta <1$ such thatfor all $x,y\in X$ with $x\ne Tx,y\ne Ty.$ We are now ready to state the main result of this paper.

**Theorem** **2.** Let $(X,d)$ a complete metric space and $T:X\to X$ be a $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction with $\lambda \in [0,1),\alpha ,\beta \in (0,1)$ so that $\alpha +\beta <1$. Then T has a fixed point in X.

**Proof.** Following the steps of the proof of ([

3], Theorem 2.2), we construct the sequence

${\left({x}_{n}\right)}_{n\ge 1}$ by iterating

${x}_{n}={T}^{n}{x}_{0}$ where

${x}_{0}\in X$ is an arbitrary starting point. Then, we observe that

i.e.,

since

$\alpha <1-\beta .$As already elaborated in the proof of ([

3], Theorem 2.2), the classical procedure leads to the existence of a unique fixed point

${x}^{*}\in X.$ □

We conclude this section by presenting an example explaining why our approach is more general.

**Example** **1** (Compare [

3], Example 2.3))

**.** Take $X=\{x,y,z,w\}$ and endow it with the following metric: | x | y | z | w |

x | 0 | 5/2 | 4 | 5/2 |

y | 5/2 | 0 | 3/2 | 1 |

z | 4 | 3/2 | 0 | 3/2 |

w | 5/2 | 1 | 3/2 | 0 |

We also define the self-map T on X as We observe that the inequality:is satisfied for: In all these cases, $\alpha +\beta <1$ i.e., $\beta <1-\alpha $ and the map obviously has a unique fixed point.

In other words, the inequalitycould just be replaced by the existence of two reals $\alpha ,\beta $ such that $\alpha +\beta <1,$ Inspired by the above question, we introduce the idea of “optimal interpolative triplet $(\alpha ,\beta ,\lambda )$” for a $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction.

**Definition** **2.** Let $(X,d)$ be a metric space and $T:X\to X$ be a self-map. We shall call T a relaxed $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction, if there exist $0\le \lambda ,\alpha ,\beta $ such that **Definition** **3.** Let $(X,d)$ be a metric space and $T:X\to X$ be a relaxed $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction. The triplet $(\lambda ,\alpha ,\beta )$ will be called “optimal interpolative triplet” if for any $\epsilon >0$, the inequality (3) fails for at least one of the triplet Therefore, we formulate the following conjecture for which we currently do not have any proof.

**Theorem** **3.** Let $(X,d)$ be a complete metric space. Let $T:X\to X$ be a map such that for any $n\ge 0$, ${T}^{n}$ admits an optimal interpolative triplet $({\lambda}_{n},{\alpha}_{n},{\beta}_{n})$. If $\sum {\lambda}_{n}<\infty $ and $\sum {\alpha}_{n}+{\beta}_{n}<\infty $, then T has a unique fixed point. Moreover, this fixed point can be obtained via the Picard iteration.

Theorem 2 can easily be generalized to the case of two maps. More precisely:

**Definition** **4.** Let $(X,d)$ be a metric space and $R,T:X\to X$ be two self-maps. We shall call $(R,T)$ a $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction pair, if there exist $\lambda \in [0,1),\alpha ,\beta \in (0,1)$ with $\alpha +\beta <1$ such thatfor all $x,y\in X$ with $x\ne Rx,y\ne Ty.$ Our result then goes as follows:

**Theorem** **4.** Let $(X,d)$ be a complete metric space and $(R,T)$ be a $(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction pair. Then R and T have a common fixed point in X, i.e., there exists ${x}^{*}\in X$ such that $R{x}^{*}={x}^{*}=T{x}^{*}$.

**Proof.** We construct the sequence

${\left({x}_{n}\right)}_{n\ge 1}$ by iterating

where

${x}_{0}\in X$ is an arbitrary starting point.

The proof then follows the same steps as ([

8], Theorem 2.1). As already elaborated in the proof of ([

8], Theorem 2.1), the classical procedure leads to the existence of a unique fixed point

${x}^{*}\in X.$ □

**Example** **2.** We use the metric defined in Example 1. We also define on X the self-maps T asand R as We observe that the inequality:is satisfied for: R and T have two common fixed points x and $w.$

The above conjecture (Theorem 3) motives us in the investigation of interpolative Kannan contraction for a family of maps. Indeed Noorwali [

8] used interpolation to obtain a common fixed-point result for a Kannan-type contraction mapping. We aim at generalizing ([

8], Theorem 2.1) and Theorem 4 with the use of a

$(\lambda ,\alpha ,\beta )$-interpolative Kannan contraction for a family of maps. More precisely:

**Problem** **1.** Let $(X,d)$ be a complete metric space. Let ${T}_{n}:X\to X,n\ge 1$ be a family of self-maps such for any $x,y\in X$ What are the conditions on ${\lambda}_{i,j},{\alpha}_{i}{\beta}_{j}$ for ${T}_{n}$ to have a (unique)common fixed point.