- Research Article
- Open access
- Published:
Existence Theorems for Generalized Distance on Complete Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 397150 (2010)
Abstract
We first introduce the new concept of a distance called -distance, which generalizes
-distance, Tataru's distance, and
-distance. Then we prove a new minimization theorem and a new fixed point theorem by using a
-distance on a complete metric space. Our results extend and unify many known results due to Caristi, Ćirić, Ekeland, Kada-Suzuki-Takahashi, Kannan, Ume, and others.
1. Introduction
The Banach contraction principle [1], Ekeland's -variational principle [2], and Caristi's fixed point theorem [3] are very useful tools in nonlinear analysis, control theory, economic theory, and global analysis. These theorems are extended by several authors in different directions.
Takahashi [4] proved the following minimization theorem. Let be a complete metric space and let
be a proper lower semicontinuous function, bounded from below. Suppose that, for each
with
there exists
such that
and
Then there exists
such that
Some authors [5–7] have generalized and extended this minimization theorem in complete metric spaces.
In 1996, Kada et al. [5] introduced the concept of -distance on a metric space as follows. Let
be a metric space with metric
. Then a function
is called a
-distance on
if the followings are satisfied.
(1) for any
.
(2)For any is lower semicontinuous.
(3)For any , there exists
such that
and
imply
They gave some examples of -distance and improved Caristi's fixed point theorem [3], Ekeland's variational principle [2], and Takahashi's nonconvex minimization theorem [4]. The fixed point theorems with respect to a
-distance were proved in [8–12].
Throughout this paper we denote by the set of all positive integers, by
the set of all real numbers, and by
the set of all nonnegative real numbers.
Recently, Suzuki [6] introduced the concept of -distance on a metric space, which generalizes Tataru's distance [13] as follows. Let
be a metric space with metric
.
Then a function from
into
is called
-distance on
if there exists a function
from
into
and the followings are satisfied:
(τ1) for all
;
(τ2) and
for all
and
, and
is concave and continuous in its second variable;
(τ3) and
imply
for all
;
(τ4) and
imply
;
(τ5) and
imply
.
In this paper, we first introduce the new concept of a distance called -distance, which generalizes
-distance, Tataru's distance, and
-distance. Then we prove a new minimization theorem and a new fixed point theorem by using
-distance on a complete metric space. Our results extend and unify many known results due to Caristi [3], Ćirić [14], Ekeland [2], Takahashi [4], Kada et al. [5], Kannan [15], Suzuki [6], and Ume [7, 12] and others.
2. Preliminaries
Definition 2.1.
Let be metric space with metric
. Then a function
from
into
is called
-distance on
if there exists a function
from
into
such that
(u1) for all
;
(u2) and
for all
and
, and for any
and for every
, there exists
such that
,
,
and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ1_HTML.gif)
(u3)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ2_HTML.gif)
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ3_HTML.gif)
for all ;
(u4)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ4_HTML.gif)
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ5_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ6_HTML.gif)
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ7_HTML.gif)
(u)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ8_HTML.gif)
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ9_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ10_HTML.gif)
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ11_HTML.gif)
Remark 2.2.
Suppose that is a mapping satisfying (u2)
(u5). Then there exists a mapping
from
into
such that
is nondecreasing in its third and fourth variable, respectively, satisfying (u2)
(u5)
, where (u2)
(u5)
stand for substituting
for
in (u2)
(u5), respectively.
Proof.
Suppose that is a mapping satisfying (u2)
(u5). Define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ12_HTML.gif)
By (2.12), we have and
for all
and
. Also it follows from (2.12) that
is nondecreasing in its third and fourth variable, respectively.
We shall prove the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ13_HTML.gif)
Suppose that (2.13) does not hold. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ14_HTML.gif)
By virtue of (2.12) and (2.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ15_HTML.gif)
Combining (u2) and (2.14), we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ16_HTML.gif)
Due to (2.16), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ17_HTML.gif)
From (2.16) and (2.17), we obtain the following.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ20_HTML.gif)
In terms of (2.19) and (2.20), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ21_HTML.gif)
In view of (2.21), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ22_HTML.gif)
On account of (2.20), we know the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ23_HTML.gif)
Using (2.16), (2.18), (2.19), and (2.23), we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ24_HTML.gif)
By (2.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ25_HTML.gif)
By virtue of (2.15), (2.19), (2.20), (2.22), and (2.25), we have which is a contradiction. Hence (u2)
holds. From (2.12) and (u2)~(u5), it follows that (u3)
~(u5)
are satisfied.
Remark 2.3.
From Remark 2.2, we may assume that is nondecreasing in its third and fourth variables, respectively, for a function
satisfying (u2)
(u5).
We give some examples of -distance.
Example 2.4.
Let be the set of real numbers with the usual metric and let
be defined by
. Then
is a
-distance on
but not a
-distance on
.
Proof.
Define by
for all
and
. Then
and
satisfy (u1)~(u5). But for an arbitrary function
and for all sequences
,
and
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ26_HTML.gif)
since the limit of the sequence and the limit of the sequence
do not depend on
and
, the limit of the sequence
may not be
. This does not satisfy (
5). Hence
is not a
-distance on
. Therefore
is a
-distance on
but not a
-distance on
.
Example 2.5.
Let be a
-distance on a metric space
. Then
is also a
-distance on
.
Proof.
Since is a
-distance, there exists a function
satisfying (
1)~(
5). Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ27_HTML.gif)
Then it is easy to see that and
satisfy (u2)~(u5). Thus
is a
-distance on
.
Example 2.6.
Let be a normed space with norm
. Then a function
defined by
for every
is a
-distance on
but not a
-distance.
Proof.
Let be as in the proof of Example 2.4. Then it is clear that
satisfies
and
satisfies (u2)~(u5) on
but
does not satisfy
. Thus
is a
-distance on
but not a
-distance.
Example 2.7.
Let be a normed space with norm
. Then a function
defined by
for every
is a
-distance on
.
Proof.
Define by
for all
and
. Then
satisfies
and
satisfies (u2)~(u5). Thus
is a
-distance on
.
Example 2.8.
Let be a
-distance on a metric space
and let
be a positive real number. Then a function
from
into
defined by
for every
is also a
-distance on
.
Proof.
Since is a
-distance on
, there exists a function
satisfying (u2)
~(u5)
and
satisfies (u1). Define
by
for all
and
. Then it is clear that
satisfies
and
satisfies (u2)~(u5). Thus
is a
-distance on
.
The following examples can be easily obtained from Remark 2.3.
Example 2.9.
Let be a metric space with metric
and let
be a
-distance on
such that
is a lower semicontinuous in its first variable. Then a function
defined by
for all
is a
-distance on
.
Example 2.10.
Let be a metric space with metric
. Let
be a
-distance on
and let
be a function from
into
. Then a function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ28_HTML.gif)
is a -distance on
.
Remark 2.11.
It follows from Example 2.4 to Example 2.10 that -distance is a proper extension of
-distance.
Definition 2.12.
Let be a metric space with a metric
and let
be a
-distance on
. Then a sequence
of
is called
-Cauchy if there exists a function
satisfying (u2)~(u5) and a sequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ29_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ30_HTML.gif)
The following lemmas play an important role in proving our theorems.
Lemma 2.13.
Let be a metric space with a metric
and let
be a
-distance on
. If
is a
-Cauchy sequence, then
is a Cauchy sequence.
Proof.
By assumption, there exists a function from
into
satisfying (u2)~(u5) and a sequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ31_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ32_HTML.gif)
Then from (u5), we have . This means that
is a Cauchy sequence.
Lemma 2.14.
Let be a metric space with a metric
and let
be a
-distance on
.
(1)If sequences and
of
satisfy
and
for some
, then
.
(2)If and
, then
.
(3)Suppose that sequences and
of
satisfy
and
for some
, then
.
(4)If and
, then
.
Proof.
-
(1)
Let
be a function from
into
satisfying (u2)~(u5). From Remark 2.3 and hypotheses,
(2.33)
By (u5), .
-
(2)
In (1), putting
and
for all
, (2) holds.
By method similar to (1) and (2), results of (3) and (4) follow.
Lemma 2.15.
Let be a metric space with a metric
and let
be a
-distance on
. Suppose that a sequence
of
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ34_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ35_HTML.gif)
Then is a
-Cauchy sequence and
is a Cauchy sequence.
Proof.
Since is a
-distance on
, there exists a function
satisfying (u2)
(u5). Suppose
. Let
. Then we have
. Let
be an arbitrary subsequence of
. By assumption and (u2), there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ36_HTML.gif)
From (u4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ37_HTML.gif)
Since is an arbitrary sequence of
,
is also an arbitrary sequence of
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ38_HTML.gif)
Therefore we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ39_HTML.gif)
This implies that is a
-Cauchy sequence. By Lemma 2.13,
is a Cauchy sequence. Similarly, if
we can prove that
is also a Cauchy sequence.
3. Minimization Theorems and Fixed Point Theorems
The following theorem is a generalization of Takahashi's minimization theorem [4].
Theorem 3.1.
Let be a metric space with metric
, let
be a proper function which is bounded from below, and let
be a function such that, one has the following.
(i) for all
.
(ii)For any sequence in
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ40_HTML.gif)
there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ41_HTML.gif)
(iii) imply
.
(iv)For every with
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ42_HTML.gif)
where a function is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ43_HTML.gif)
for all . Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ44_HTML.gif)
Proof.
Suppose for all
. For each
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ45_HTML.gif)
Then, by condition (iv) and (3.6), is nonempty for each
. From condition (i) and (3.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ46_HTML.gif)
For each , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ47_HTML.gif)
Choose with
. Then, from (3.7) and (3.8), there exists a sequence
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ48_HTML.gif)
for all .
From (3.6), (3.8) and (3.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ50_HTML.gif)
By (3.10), is a nonincreasing sequence of real numbers and so it converges. Therefore, from (3.11) there is some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ51_HTML.gif)
From condition (i) and (3.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ52_HTML.gif)
for all . From (3.12) and (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ53_HTML.gif)
Thus, by condition (ii), (3.12), and (3.13), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ55_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ56_HTML.gif)
From (3.13), (3.16), and (3.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ57_HTML.gif)
From (3.6), (3.8), and (3.18), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ58_HTML.gif)
Taking the limit in inequality (3.19) when tends to infinity, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ59_HTML.gif)
From (3.12), (3.16), and (3.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ60_HTML.gif)
On the other hand, by condition (iv) and (3.6), we have the following property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ61_HTML.gif)
From (3.7), (3.8), (3.19), and (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ62_HTML.gif)
From (3.6), (3.12), (3.21), (3.22), (3.23), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ63_HTML.gif)
From (3.21), (3.22), and (3.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ64_HTML.gif)
By method similar to (3.22)(3.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ65_HTML.gif)
From (3.25), (3.26), and condition (i), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ66_HTML.gif)
From (3.25), (3.27), and condition (iii), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ67_HTML.gif)
This is a contradiction from (3.26).
Corollary 3.2.
Let be a complete metric space with metric
, and let
be a proper lower semicontinuous function which is bounded from below. Assume that there exists a
-distance
on
such that for each
with
, there exists
with
and
. Then there exists
such that
.
Proof.
Let be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ68_HTML.gif)
for all . It follows easily from Definition 2.12, Lemmas 2.13, 2.14, and 2.15, and (u3) that conditions of Corollary 3.2 satisfy all conditions of Theorem 3.1. Thus, we obtain result of Corollary 3.2.
Remark 3.3.
Corollary 3.2 is a generalization of Kadaet al. [5, Theorem ] and Suzuki [6, Theorem
].
From Lemmas 2.13, 2.14, and 2.15, we have the following fixed point theorem.
Theorem 3.4.
Let be a complete metric space with metric
, let
be a
-distance on
and let
be a selfmapping of
. Suppose that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ69_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ70_HTML.gif)
for every with
. Then there exists
such that
and
. Moreover, if
, then
,
.
Proof.
By method similar to [12, Lemma ], for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ71_HTML.gif)
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ72_HTML.gif)
for every . By Example 2.10,
is a
-distance on
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ73_HTML.gif)
for all . Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ74_HTML.gif)
for all . Now we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ75_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ76_HTML.gif)
By Lemma 2.15, is a
-Cauchy and hence
is a Cauchy from Lemma 2.13. Since
is complete and
is a
-Cauchy, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ77_HTML.gif)
Suppose . Then, by hypothesis, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ78_HTML.gif)
This is a contradiction. Therefore we have . If
, we have
and hence
. To prove unique fixed point of
, let
and
. Then, by hypothesis, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ79_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ80_HTML.gif)
By Lemma 2.14, we have .
From Theorem 3.4, we have the following corollary which generalizes the results of Ćirić [14], Kannan [15], and Ume [12].
Corollary 3.5.
Let be a complete metric space with metric
, let
be a
-distance on
and let
be a selfmapping of
. Suppose that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ81_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ82_HTML.gif)
for every with
. Then there exists
such that
and
. Moreover, if
, then
and
.
Proof.
Since a -distance is a
-distance, Corollary 3.5 follows from Theorem 3.4.
The following corollary is a generalization of Suzuki's fixed point theorem [6].
Corollary 3.6.
Let and
be as in Corollary 3.5. Suppose that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ83_HTML.gif)
for all . Assume that if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ84_HTML.gif)
then . Then there exists
such that
and
. Moreover, if
, then
and
.
Proof.
Let and
be as in Theorem 3.4. Then from Theorem 3.4 and hypotheses of Corollary 3.6, we have the following properties.
(1) is a Cauchy sequence.
(2)There exists such that
.
(3)One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ85_HTML.gif)
(4)There exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ86_HTML.gif)
(5)One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ87_HTML.gif)
By (1)~(5) and hypotheses, we have . The remainders are same as Theorem 3.4.
The following theorem is a generalization of Caristi's fixed point theorem [3].
Theorem 3.7.
Let be a metric space with metric
, let
be a proper function which is bounded from below, and let
be a function satisfying (i), (ii), and (iii) of Theorem 3.1. Let
be a selfmapping of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ88_HTML.gif)
where a function is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ89_HTML.gif)
for all . Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ90_HTML.gif)
Proof.
Suppose for all
. Then, by Theorem 3.1, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ91_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ92_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ93_HTML.gif)
By hypothesis, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ94_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ95_HTML.gif)
By conditions (i) and (iii) of Theorem 3.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ96_HTML.gif)
This is a contradiction.
Corollary 3.8.
Let be a complete metric space with metric
and let
be a proper lower semicontinuous function which is bounded from below. Let
be a
-distance on
. Suppose that
is a selfmapping of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ97_HTML.gif)
for all . Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ98_HTML.gif)
Proof.
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ99_HTML.gif)
for all . Then, by Definition 2.12 and Lemmas 2.13, 2.14, and 2.15, we can easily show that conditions of Corollary 3.8 satisfy all conditions of Theorem 3.7. Thus, Corollary 3.8 follows from Theorem 3.7.
Remark 3.9.
Since a -distance and a
-distance are a
-distance, Corollary 3.8 is a generalization of Kada-Suzuki-Takahashi [5, Theorem
] and Suzuki [6, Theorem
].
The following theorem is a generalization of Ekeland's -variational principle [2].
Theorem 3.10.
Let be a complete metric space with metric
, let
be a proper lower semicontinuous function which is bounded from below, and let
be a function satisfying (i), (ii), and (iii) of Theorem 3.1. Then the following (1) and (2) hold.
(1)For each with
, there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ100_HTML.gif)
for all with
where a function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ101_HTML.gif)
for all .
(2)For each and
with
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ102_HTML.gif)
there exists such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ103_HTML.gif)
for all with
Proof.
() Let
be such that
, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ104_HTML.gif)
Then, by hypotheses, is nonempty and closed. Thus
is a complete metric space. Hence we may prove that there exists an element
such that
for all
with
Suppose not. Then, for every
, there exists
such that
and
By Theorem 3.1, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ105_HTML.gif)
Again for , there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ106_HTML.gif)
Hence we have and
Similarly, there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ107_HTML.gif)
Thus we have and
From conditions (i) and (iii) of Theorem 3.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ108_HTML.gif)
This is a contradiction. The proof of (1) is complete.
-
(2)
Let
(3.70)
Then is nonempty and closed. Hence
is complete. As in the proof of (1), we have that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ110_HTML.gif)
for every with
On the other hand, since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ111_HTML.gif)
This completes the proof of (2).
Corollary 3.11.
Let and
be as in Corollary 3.8. Then the following (1) and (2) hold.
(1)For each with
, there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ112_HTML.gif)
for all with
(2)For each and
with
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ113_HTML.gif)
there exists such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F397150/MediaObjects/13663_2009_Article_1275_Equ114_HTML.gif)
for all with
Proof.
By method similar to Corollary 3.8, Corollary 3.11 follows from Theorem 3.10.
Remark 3.12.
Corollary 3.11 is a generalization of Suzuki [6, Theorem ].
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 1992, 3: 133–181.
Ekeland I: Nonconvex minimization problems. Bulletin of the American Mathematical Society 1979,1(3):443–474. 10.1090/S0273-0979-1979-14595-6
Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.
Takahashi W: Minimization theorems and fixed point theorems. In Nonlinear Analysis and Mathematical Economics. Volume 829. Edited by: Maruyama T. RIMS Kokyuroku; 1993:175–191.
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.
Suzuki T: Generalized distance and existence theorems in complete metric spaces. Journal of Mathematical Analysis and Applications 2001,253(2):440–458. 10.1006/jmaa.2000.7151
Ume J-S: Some existence theorems generalizing fixed point theorems on complete metric spaces. Mathematica Japonica 1994,40(1):109–114.
Shioji N, Suzuki T, Takahashi W: Contractive mappings, Kannan mappings and metric completeness. Proceedings of the American Mathematical Society 1998,126(10):3117–3124. 10.1090/S0002-9939-98-04605-X
Suzuki T: Fixed point theorems in complete metric spaces. In Nonlinear Analysis and Convex Analysis. Volume 939. Edited by: Takahashi W. RIMS Kokyuroku; 1996:173–182.
Suzuki T: Several fixed point theorems in complete metric spaces. Yokohama Mathematical Journal 1997,44(1):61–72.
Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topological Methods in Nonlinear Analysis 1996,8(2):371–382.
Ume J-S: Fixed point theorems related to Ćirić's contraction principle. Journal of Mathematical Analysis and Applications 1998,225(2):630–640. 10.1006/jmaa.1998.6030
Tataru D: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. Journal of Mathematical Analysis and Applications 1992,163(2):345–392. 10.1016/0022-247X(92)90256-D
Ćirić LjB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.
Kannan R: Some results on fixed points. II. The American Mathematical Monthly 1969, 76: 405–408. 10.2307/2316437
Acknowledgments
The author would like to thank the referees for useful comments and suggestions. This work was supported by the Korea Research Foundation (KRF) Grant funded by the Korea government (MEST) (2009-0073655).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ume, J. Existence Theorems for Generalized Distance on Complete Metric Spaces. Fixed Point Theory Appl 2010, 397150 (2010). https://doi.org/10.1155/2010/397150
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/397150