# A Generalization of Kannan's Fixed Point Theorem

- Yusuke Enjouji
^{1}, - Masato Nakanishi
^{1}and - Tomonari Suzuki
^{1}Email author

**2009**:192872

**DOI: **10.1155/2009/192872

© Yusuke Enjouji et al. 2009

**Received: **22 December 2008

**Accepted: **23 March 2009

**Published: **31 March 2009

## Abstract

In order to observe the condition of Kannan mappings, we prove a generalization of Kannan's fixed point theorem. Our theorem involves constants and we obtain the best constants to ensure a fixed point.

## 1. Introduction

*Kannan*if there exists such that

for all . Kannan [1] proved that if is complete, then every Kannan mapping has a fixed point. It is interesting that Kannan's theorem is independent of the Banach contraction principle [2]. Also, Kannan's fixed point theorem is very important because Subrahmanyam [3] proved that Kannan's theorem characterizes the metric completeness. That is, a metric space is complete if and only if every Kannan mapping on has a fixed point. Recently, Kikkawa and Suzuki proved a generalization of Kannan's fixed point theorem. See also [4–8].

Theorem 1.1 (see [9]).

for all . Then has a unique fixed point . Moreover holds for every .

Remark 1.2.

is the best constant for every .

From this theorem, we can tell that a Kannan mapping with is much stronger than a Kannan mapping with .

While and play the same role in (1.1), and do not play the same role in (1.3). So we can consider " '' instead of " .'' And it is a quite natural question of what is the best constant for each pair . In this paper, we give the complete answer to this question.

## 2. Preliminaries

Throughout this paper we denote by the set of all positive integers and by the set of all real numbers.

We use two lemmas. The first lemma is essentially proved in [5].

holds.

The second lemma is obvious. We use this lemma several times in the proof of Theorem 4.1.

Lemma 2.2.

Let , , , and be four real numbers such that and . Then holds.

## 3. Fixed Point Theorem

In this section, we prove a fixed point theorem.

Theorem 3.1.

for all . Then has a unique fixed point . Moreover holds for every .

Proof.

for all .

So is a Cauchy sequence in . Since is complete, converges to some point .

which is a contradiction. Therefore we obtain .

which is a contradiction. Therefore holds.

In the case where , we consider the following two cases.

(i)There exist at least two natural numbers satisfying

(ii) for sufficiently large .

Since , we obtain .

Since , we obtain . Therefore we have shown in all cases.

From (3.11), the fixed point is unique.

Remark 3.2.

## 4. The Best Constants

In this section, we prove the following theorem, which informs that is the best constant for every .

Theorem 4.1.

for all .

Proof.

We put and by (3.4).

for all .

This completes the proof.

## Declarations

### Acknowledgment

T. Suzuki is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.