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A Generalization of Kannan's Fixed Point Theorem
Fixed Point Theory and Applications volume 2009, Article number: 192872 (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
A mapping on a metric space
is called Kannan if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ1_HTML.gif)
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]).
Define a nonincreasing function from
into
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ2_HTML.gif)
Let be a mapping on a complete metric space
. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ3_HTML.gif)
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].
Let be a metric space and let
be a mapping on
. Let
satisfy
for some
. Then for
, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ4_HTML.gif)
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.
We first put and
(
) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ5_HTML.gif)
See Figure 1.
Theorem 3.1.
Define a nonincreasing function from
into
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ6_HTML.gif)
Let be a mapping on a complete metric space
. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ7_HTML.gif)
for all . Then
has a unique fixed point
. Moreover
holds for every
.
Proof.
We put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ8_HTML.gif)
Since ,
holds. From the assumption, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ9_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ10_HTML.gif)
for all . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ11_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ12_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ13_HTML.gif)
for all .
Fix and put
for
. From (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ14_HTML.gif)
So is a Cauchy sequence in
. Since
is complete,
converges to some point
.
We next show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ15_HTML.gif)
Since converges, for sufficiently large
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ16_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ17_HTML.gif)
Therefore we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ18_HTML.gif)
for all . By (3.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ19_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ20_HTML.gif)
Let us prove that is a fixed point of
. In the case where
, arguing by contradiction, we assume
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ21_HTML.gif)
So for sufficiently large ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ22_HTML.gif)
holds and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ23_HTML.gif)
Thus we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ24_HTML.gif)
which is a contradiction. Therefore we obtain .
In the case where , if we assume
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ25_HTML.gif)
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
.
In the first case, if we assume , then
cannot be Cauchy. Therefore
. In the second case, we have by (3.16),
for sufficiently large
. From the assumption,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ26_HTML.gif)
Since , we obtain
.
In the case where , we note that
. By Lemma 2.1, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ27_HTML.gif)
holds for every . Thus there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ28_HTML.gif)
for . From the assumption, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ29_HTML.gif)
Since , we obtain
. Therefore we have shown
in all cases.
From (3.11), the fixed point is unique.
Remark 3.2.
We have shown , dividing four cases. It is interesting that the four methods are all different. We can rewrite
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ30_HTML.gif)
4. The Best Constants
In this section, we prove the following theorem, which informs that is the best constant for every
.
Theorem 4.1.
Define a function as in Theorem 3.1. For every
, there exist a complete metric space
and a mapping
on
such that
has no fixed points and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ31_HTML.gif)
for all .
Proof.
We put and
by (3.4).
In the case where , define a complete subset
of the Euclidean space
by
. We also define a mapping
on
by
for
. Then
does not have any fixed points and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ32_HTML.gif)
for all .
In the case where , we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ33_HTML.gif)
We note that . Define a complete subset
of the Euclidean space
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ34_HTML.gif)
where for
. Define a mapping
on
by
,
and
for
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ35_HTML.gif)
for . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ36_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ37_HTML.gif)
for . For
with
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ38_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ39_HTML.gif)
In the case where , we note that
. We also note that
. Let
be the Banach space consisting of all functions
from
into
(i.e.,
is a real sequence) such that
. Let
be the canonical basis of
. Define a complete subset
of
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ40_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ41_HTML.gif)
for . We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ42_HTML.gif)
for with
. Define a mapping
on
by
,
, and
for
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ43_HTML.gif)
for . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ44_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ45_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ46_HTML.gif)
for . We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ47_HTML.gif)
for . For
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192872/MediaObjects/13663_2008_Article_1121_Equ48_HTML.gif)
This completes the proof.
References
Kannan R: Some results on fixed points. II. American Mathematical Monthly 1969, 76: 405–408. 10.2307/2316437
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 1922, 3: 133–181.
Subrahmanyam PV: Completeness and fixed-points. Monatshefte für Mathematik 1975,80(4):325–330. 10.1007/BF01472580
Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):2942–2949. 10.1016/j.na.2007.08.064
Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society 2008,136(5):1861–1869.
Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications 2008,340(2):1088–1095. 10.1016/j.jmaa.2007.09.023
Suzuki T, Kikkawa M: Some remarks on a recent generalization of the Banach contraction principle. In Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications (ICFPTA '08), July 2008, Chiang Mai, Thailand. Edited by: Dhompongsa S, Goebel K, Kirk WA, Plubtieng S, Sims B, Suantai S. Yokohama; 151–161.
Suzuki T, Vetro C: Three existence theorems for weak contractions of Matkowski type. International Journal of Mathematics and Statistics 2010,6(supplement 10):110–120.
Kikkawa M, Suzuki T: Some similarity between contractions and Kannan mappings. Fixed Point Theory and Applications 2008, Article ID 649749, 2008:-8.
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.
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Enjouji, Y., Nakanishi, M. & Suzuki, T. A Generalization of Kannan's Fixed Point Theorem. Fixed Point Theory Appl 2009, 192872 (2009). https://doi.org/10.1155/2009/192872
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DOI: https://doi.org/10.1155/2009/192872