A Generalization of Kannan's Fixed Point Theorem
© Yusuke Enjouji et al. 2009
Received: 22 December 2008
Accepted: 23 March 2009
Published: 31 March 2009
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.
for all . Kannan  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 . Also, Kannan's fixed point theorem is very important because Subrahmanyam  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 ).
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.
We use two lemmas. The first lemma is essentially proved in .
The second lemma is obvious. We use this lemma several times in the proof of Theorem 4.1.
3. Fixed Point Theorem
In this section, we prove a fixed point theorem.
4. The Best Constants
This completes the proof.
T. Suzuki is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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