Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 192872, 10 pages
doi:10.1155/2009/192872
Research Article
A Generalization of Kannan’s Fixed Point Theorem
Yusuke Enjouji, Masato Nakanishi, and Tomonari Suzuki
Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan
Correspondence should be addressed to Tomonari Suzuki, suzuki-t@mns.kyutech.ac.jp
Received 22 December 2008; Accepted 23 March 2009
Recommended by Jerzy Jezierski
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.
Copyright q 2009 Yusuke Enjouji et al. 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.
1. Introduction
A mapping T on a metric space X, d is called Kannan if there exists α ∈ 0, 1/2 such that
d T x, T y ≤ αdx, T x αd y, T y
1.1
for all x, y ∈ X. Kannan 1 proved that if X 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 X is complete if and only if every Kannan mapping on X 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 ϕ from 0, 1/2 into 1/2, 1 by
⎧
⎪
⎨1
√
if 0 ≤ α < 2 − 1,
ϕα √
⎪
⎩1 − α if 2 − 1 ≤ α < 1 .
2
1.2
2
Fixed Point Theory and Applications
Let T be a mapping on a complete metric space X, d. Assume that there exists α ∈ 0, 1/2 such that
ϕαdx, T x ≤ d x, y implies d T x, T y ≤ αdx, T x αd y, T y
1.3
for all x, y ∈ X. Then T has a unique fixed point z. Moreover limn T n x z holds for every x ∈ X.
Remark 1.2. ϕα is the best constant for every α ∈ 0, 1/2.
√
From this theorem, we can tell
√ that a Kannan mapping with α < 2−1 is much stronger
than a Kannan mapping with α ≥ 2 − 1.
While x and y play the same role in 1.1, x and y do not play the same role in 1.3.
So we can consider “αdx, T x βdy, T y” instead of “αdx, T x αdy, T y.” And it is a
quite natural question of what is the best constant for each pair α, β. In this paper, we give
the complete answer to this question.
2. Preliminaries
Throughout this paper we denote by N the set of all positive integers and by R the set of all
real numbers.
We use two lemmas. The first lemma is essentially proved in 5.
Lemma 2.1 see 5, 9. Let X, d be a metric space and let T be a mapping on X. Let x ∈ X satisfy
dT x, T 2 x ≤ rdx, T x for some r ∈ 0, 1. Then for y ∈ X, either
1 r−1 dx, T x ≤ d x, y
or
1 r−1 d T x, T 2 x ≤ d T x, y
2.1
holds.
The second lemma is obvious. We use this lemma several times in the proof of
Theorem 4.1.
Lemma 2.2. Let a, A, b, and B be four real numbers such that a ≤ A and b ≤ B. Then aB Ab ≤
ab AB holds.
3. Fixed Point Theorem
In this section, we prove a fixed point theorem.
We first put Δ and Δj j 1, . . . , 4 by
Δ
Δ1 α, β : α ≥ 0, β ≥ 0, α β < 1 ,
α, β ∈ Δ: α ≤ β, α β α2 < 1 ,
Fixed Point Theory and Applications
3
β1
4
1
3
β0
2
α0
α1
Figure 1: Δj j 1, . . . , 4
Δ2 Δ3 Δ4 α, β ∈ Δ: α ≥ β, α β β2 < 1 ,
α, β ∈ Δ: α ≥ β, α β β2 ≥ 1 ,
α, β ∈ Δ: α ≤ β, α β α2 ≥ 1 .
3.1
See Figure 1.
Theorem 3.1. Define a nonincreasing function ψ from Δ into 1/2, 1 by
ψ α, β ⎧
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪1
⎨
⎪1 − β
⎪
⎪
⎪
⎪
⎪
1−β
⎪
⎪
⎩
1−βα
if α, β ∈ Δ1 ,
if α, β ∈ Δ2 ,
if α, β ∈ Δ3 ,
3.2
if α, β ∈ Δ4 .
Let T be a mapping on a complete metric space X, d. Assume that there exists α, β ∈ Δ such that
ψ α, β dx, T x ≤ d x, y implies d T x, T y ≤ αdx, T x βd y, T y
3.3
for all x, y ∈ X. Then T has a unique fixed point z. Moreover limn T n x z holds for every x ∈ X.
Proof. We put
q :
β
∈ 0, 1,
1−α
r :
α
∈ 0, 1.
1−β
3.4
4
Fixed Point Theory and Applications
Since ψα, β ≤ 1, ψα, βdx, T x ≤ dx, T x holds. From the assumption, we have
d T x, T 2 x ≤ αdx, T x βd T x, T 2 x
3.5
d T x, T 2 x ≤ rdx, T x
3.6
ψ α, β d T x, T 2 x ≤ d T x, T 2 x ≤ rdx, T x ≤ dT x, x,
3.7
d T 2 x, T x ≤ αd T x, T 2 x βdx, T x
3.8
d T x, T 2 x ≤ qdx, T x
3.9
and hence
for all x ∈ X. Since
we have
and hence
for all x ∈ X.
Fix u ∈ X and put un T n u for n ∈ N. From 3.6, we have
∞
∞
dun , un1 ≤
r n du, T u < ∞.
n1
3.10
n1
So {un } is a Cauchy sequence in X. Since X is complete, {un } converges to some point z ∈ X.
We next show
dz, T x ≤ βdx, T x ∀x ∈ X \ {z}.
3.11
Since {un } converges, for sufficiently large n ∈ N, we have
ψ α, β dun , T un ≤ dun , un1 ≤ dun , x
3.12
dT un , T x ≤ αdun , T un βdx, T x.
3.13
and hence
Fixed Point Theory and Applications
5
Therefore we obtain
dz, T x lim dun1 , T x lim dT un , T x
n→∞
n→∞
≤ lim αdun , T un βdx, T x βdx, T x
3.14
n→∞
for all x ∈ X \ {z}. By 3.11, we have
dx, T x ≤ dx, z dz, T x ≤ dx, z βdx, T x
3.15
and hence
1 − β dx, T x ≤ dx, z
∀x ∈ X \ {z}.
3.16
Let us prove that z is a fixed point of T . In the case where α, β ∈ Δ1 , arguing by
contradiction, we assume T z /
z. Then we have
d T z, T 2 z ≤ rdz, T z < dz, T z lim dT z, un .
n→∞
3.17
So for sufficiently large n ∈ N,
ψ α, β d T z, T 2 z d T z, T 2 z ≤ dT z, un 3.18
holds and hence
d T 2 z, z lim d T 2 z, T un
n→∞
≤ lim αd T z, T 2 z βdun , T un αd T z, T 2 z .
3.19
n→∞
Thus we obtain
dz, T z ≤ d z, T 2 z d T z, T 2 z ≤ 1 αd T z, T 2 z
≤ 1 αrdz, T z α α2
dz, T z
1−β
< dz, T z,
which is a contradiction. Therefore we obtain T z z.
3.20
6
Fixed Point Theory and Applications
z, then we have
In the case where α, β ∈ Δ2 , if we assume T z /
dz, T z ≤ d z, T 2 z d T z, T 2 z ≤ 1 β d T z, T 2 z
β β2
dz, T z
≤ 1 β qdz, T z 1−α
3.21
< dz, T z,
which is a contradiction. Therefore T z z holds.
In the case where α, β ∈ Δ3 , we consider the following two cases.
i There exist at least two natural numbers n satisfying un z.
ii un / z for sufficiently large n ∈ N.
In the first case, if we assume T z / z, then {un } cannot be Cauchy. Therefore T z z. In the
second case, we have by 3.16, ψα, βdun , T un ≤ dun , z for sufficiently large n ∈ N. From
the assumption,
dz, T z lim dT un , T z ≤ lim αdun , T un βdz, T z βdz, T z.
n→∞
n→∞
3.22
Since β < 1, we obtain T z z.
In the case where α, β ∈ Δ4 , we note that ψα, β 1 r−1 . By Lemma 2.1, either
ψ α, β dun , T un ≤ dun , z
or
ψ α, β d T un , T 2 un ≤ dT un , z
3.23
holds for every n ∈ N. Thus there exists a subsequence {nj } of {n} such that
ψ α, β d unj , T unj ≤ d unj , z
3.24
for j ∈ N. From the assumption, we have
dz, T z lim d T unj , T z ≤ lim αd unj , T unj βdz, T z βdz, T z.
j →∞
j →∞
3.25
Since β < 1, we obtain T z z. Therefore we have shown T z z in all cases.
From 3.11, the fixed point z is unique.
Remark 3.2. We have shown T z z, dividing four cases. It is interesting that the four methods
are all different. We can rewrite ψ by
ψ α, β ⎧
⎪
⎪
⎨1
⎪
⎪
⎩
1−β
1 − β min α, β
if α β min {α, β}2 < 1,
if α β min {α, β}2 ≥ 1.
3.26
Fixed Point Theory and Applications
7
4. The Best Constants
In this section, we prove the following theorem, which informs that ψα, β is the best constant
for every α, β ∈ Δ.
Theorem 4.1. Define a function ψ as in Theorem 3.1. For every α, β ∈ Δ, there exist a complete
metric space X, d and a mapping T on X such that T has no fixed points and
ψ α, β dx, T x < d x, y implies d T x, T y ≤ αdx, T x βd y, T y
4.1
for all x, y ∈ X.
Proof. We put q and r by 3.4.
In the case where α, β ∈ Δ1 ∪ Δ2 , define a complete subset X of the Euclidean space
R by X {−1, 1}. We also define a mapping T on X by T x −x for x ∈ X. Then T does not
have any fixed points and
ψ α, β dx, T x 2 ≥ d x, y
4.2
for all x, y ∈ X.
In the case where α, β ∈ Δ3 , we put
p :
β
∈ 0, 1.
1−β
4.3
We note that ψα, β1 p 1. Define a complete subset X of the Euclidean space R by
X {0, 1} ∪ {xn : n ∈ N ∪ {0}},
4.4
where xn 1 − q−pn for n ∈ N ∪ {0}. Define a mapping T on X by T 0 1, T 1 x0 , and
T xn xn1 for n ∈ N ∪ {0}. Then we have
dT 1, T 0 q αd1, T 1 βd0, T 0 ≤ αd0, T 0 βd1, T 1,
ψ α, β d0, T 0 > ψ α, β dxn , T xn 1 − q pn d0, xn 4.5
for n ∈ N ∪ {0}. Since
dT xn , T 1 − αdxn , T xn βd1, T 1
β2
α n1
n1
1 − q 1 − −p − p −
β
1−α−β
β2
α
1 − q pn1 1 −
≤ 0,
≤ 1−q 1−
1−α−β
β
4.6
8
Fixed Point Theory and Applications
we have
dT xn , T 1 ≤ αdxn , T xn βd1, T 1 ≤ αd1, T 1 βdxn , T xn 4.7
for n ∈ N ∪ {0}. For m, n ∈ N ∪ {0} with m < n, since
dT xn , T xm − αdxn , T xn βdxm , T xm α
1 − q −pn1 − −pm1 − pn1 − pm1
β
n1
α n1
m1
m1
− p −p
≤ 1−q p p
≤ 0,
β
4.8
dT xn , T xm ≤ αdxn , T xn βdxm , T xm ≤ αdxm , T xm βdxn , T xn .
4.9
we have
In the case where α, β ∈ Δ4 , we note that ψα, β1 r 1. We also note that r ≥
2−1/2 > 1/2. Let ∞ be the Banach space consisting of all functions f from N into R i.e., f is a
real sequence such that f
: supn |fn| < ∞. Let {en } be the canonical basis of ∞ . Define
a complete subset X of ∞ by
X {0, e1 } ∪ {xn : n ∈ N ∪ {0}},
4.10
xn 1 − rr n en1 − 1 − rr n en2
4.11
where
for n ∈ N ∪ {0}. We note that
dxm , xn ⎧
⎨ 1 − r2 rm
if m 1 n,
⎩1 − rr m
if m 1 < n,
4.12
for m, n ∈ N with m < n. Define a mapping T on X by T 0 e1 , T e1 x0 , and T xn xn1 for
n ∈ N ∪ {0}. Then we have
dT 0, T e1 r αd0, T 0 βde1 , T e1 ≤ αde1 , T e1 βd0, T 0,
ψ α, β d0, T 0 > ψ α, β dxn , T xn 1 − rr n d0, xn 4.13
for n ∈ N ∪ {0}. Since
dT e1 , T x0 − αde1 , T e1 βdx0 , T x0 1 − β 1 − 2r 2 ≤ 0,
4.14
Fixed Point Theory and Applications
9
we have
dT e1 , T x0 ≤ αde1 , T e1 βdx0 , T x0 ≤ αdx0 , T x0 βde1 , T e1 .
4.15
Since α β α2 ≥ 1, we have
dT e1 , T xn 1 − r ≤ αr αde1 , T e1 < αde1 , T e1 βdxn , T xn ≤ αdxn , T xn βde1 , T e1 4.16
for n ∈ N. We have
dT xn , T xn1 1 − r 2 r n1 αdxn , T xn βdxn1 , T xn1 4.17
≤ αdxn1 , T xn1 βdxn , T xn for n ∈ N ∪ {0}. For m, n ∈ N ∪ {0} with m 1 < n, we have
ψ α, β dxm , T xm 1 − rr m dxm , xn ,
dT xn , T xm − αdxn , T xn βdxm , T xm < dT xn , T xm − βdxm , T xm r m1 1 − r − βr m 1 − r 2
4.18
r m 1 − r α − β ≤ 0.
This completes the proof.
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.
References
1 R. Kannan, “Some results on fixed points. II,” American Mathematical Monthly, vol. 76, pp. 405–408,
1969.
2 S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations
intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
3 P. V. Subrahmanyam, “Completeness and fixed-points,” Monatshefte für Mathematik, vol. 80, no. 4, pp.
325–330, 1975.
4 M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants
in complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–
2949, 2008.
5 T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,”
Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
6 T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive
mappings,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1088–1095, 2008.
10
Fixed Point Theory and Applications
7 T. Suzuki and M. Kikkawa, “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), S. Dhompongsa, K. Goebel, W. A. Kirk, S. Plubtieng, B. Sims, and S. Suantai, Eds., pp.
151–161, Yokohama, Chiang Mai, Thailand, July 2008.
8 T. Suzuki and C. Vetro, “Three existence theorems for weak contractions of Matkowski type,”
International Journal of Mathematics and Statistics, vol. 6, supplement 10, pp. 110–120, 2010.
9 M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,” Fixed Point
Theory and Applications, vol. 2008, Article ID 649749, 8 pages, 2008.