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Fixed Point Theorems for Contractive Mappings in Complete
-Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 917175 (2009)
Abstract
We prove some fixed point results for mappings satisfying various contractive conditions on Complete -metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are
-continuous on such fixed points.
1. Introduction
Metric spaces are playing an increasing role in mathematics and the applied sciences.
Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory.
Different generalizations of the notion of a metric space have been proposed by Gahler [1, 2] and by Dhage [3, 4]. However, HA et al. [5] have pointed out that the results obtained by Gahler for his metrics are independent, rather than generalizations, of the corresponding results in metric spaces, while in [6]the current authors have pointed out that Dhage's notion of a
-metric space is fundamentally flawed and most of the results claimed by Dhage and others are invalid.
In 2003 we introduced a more appropriate and robust notion of a generalized metric space as follows.
Definition 1.1 ([7]).
Let X be a nonempty set, and let be a function satisfying the following axioms:
() if
,
(),
(),
() (symmetry in all three variables),
(), for all
, (rectangle inequality).
Then the function is called a generalized metric, or, more specifically a
-metric on
, and the pair
is called a
-metric space.
Example 1.2 ([7]).
Let be a usual metric space, then
and
are
-metric space, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ1_HTML.gif)
We now recall some of the basic concepts and results for -metric spaces that were introduced in ([7]).
Definition 1.3.
Let be a
-metric space, let
be a sequence of points of
, we say that
is
-convergent to
if
; that is, for any
there exists
such that
, for all
(throughout this paper we mean by
the set of all natural numbers). We refer to
as the limit of the sequence
and write
.
Proposition 1.4.
Let be a
-metric space then the following are equivalent.
(1) is
-convergent to
.
(2), as
.
(3), as
.
Definition.
Let be a
-metric space, a sequence
is called
-Cauchy if given
, there is
such that
for all
that is if
as
.
Proposition 1.6.
In a -metric space
, the following are equivalent.
(1)The sequence is
-Cauchy.
(2)For everythere exists
such that
for all
.
Definition 1.7.
Let and
be
-metric spaces and let
be a function, then
is said to be
-continuous at a point
if given
, there exists
such that
;
implies
. A function
is
-continuous on
if and only if it is
-continuous at all
.
Proposition 1.8.
Let ,
be
-metric spaces, then a function
is
-continuous at a point
if and only if it is
-sequentially continuous at
; that is, whenever
is
-convergent to
,
is
-convergent to
.
Proposition 1.9.
Let be a
-metric space, then the function
is jointly continuous in all three of its variables.
Definition 1.10.
A -metric space
is said to be
-complete (or a complete
-metric space) if every
-Cauchy sequence in
is
-convergent in
.
2. The Main Results
We begin with the following theorem.
Theorem 2.1.
Let be a complete
-metric space and let
be a mapping which satisfies the following condition, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ2_HTML.gif)
where . Then
has a unique fixed point (say
) and
is
-continuous at
.
Proof.
Suppose that satisfies condition (2.1), let
be an arbitrary point, and define the sequence
by
, then by (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ3_HTML.gif)
so,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ4_HTML.gif)
But, by (G5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ5_HTML.gif)
So, (2.3)becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ6_HTML.gif)
So, it must be the case that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ7_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ8_HTML.gif)
Let , then
and by repeated application of (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ9_HTML.gif)
Then, for all we have by repeated use of the rectangle inequality and (2.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ10_HTML.gif)
Then, , as
, since
, as
. For
(G5) implies that
, taking limit as
, we get
. So
is
-Cauchy a sequence. By completeness of
, there exists
such that
is
-converges to
. Suppose that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ11_HTML.gif)
taking the limit as , and using the fact that the function
is continuous on its variables, we have
, which is a contradiction since
. So,
. To prove uniqueness, suppose that
is such that
, then (2.1) implies that
, thus
again by the same argument we will find
, thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ12_HTML.gif)
which implies that , since
. To see that
is
-continuous at
, let
be a sequence such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ13_HTML.gif)
and we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ14_HTML.gif)
but (G5) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ15_HTML.gif)
and (2.13) leads to the following cases,
(1),
(2)
(3)
In each case take the limit as to see that
and so, by Proposition 1.4, we have that the sequence
is
-convergent to
, therefor Proposition 1.8 implies that
is
-continuous at
.
Remark 2.2.
If the -metric space is bounded (that is, for some
we have
for all
) then an argument similar to that used above establishes the result for
.
Corollary 2.3.
Let be a complete
-metric spaces and let
be a mapping which satisfies the following condition for some
and for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ16_HTML.gif)
where , then
has a unique fixed point (say
), and
is
-continuous at
.
Proof.
From the previous theorem, we have that has a unique fixed point (say u), that is,
. But
, so
is another fixed point for
and by uniqueness
.
Theorem 2.4.
Let be a complete
-metric space, and let
be a mapping which satisfies the following condition for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ17_HTML.gif)
where , then
has a unique fixed point (say
), and
is
-continuous at
.
Proof.
Suppose that satisfies the condition (2.16), let
be an arbitrary point, and define the sequence
by
, then by (2.16) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ18_HTML.gif)
since , then it must be the case that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ19_HTML.gif)
but from (G5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ20_HTML.gif)
so (2.18) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ21_HTML.gif)
let , then
and by repeated application of (2.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ22_HTML.gif)
Then, for all we have, by repeated use of the rectangle inequality,
So,
, as
and
is
-Cauchy sequence. By the completeness of
, there exists
such tha
is
-convergent to
.Suppose that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ23_HTML.gif)
Taking the limit as , and using the fact that the function
is continuous in its variables, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ24_HTML.gif)
since , this contradiction implies that
.To prove uniqueness, suppose that
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ25_HTML.gif)
so we deduce that . This implies that
and by repeated use of the same argument we will find
. Therefor we get
, since
, this contradiction implies that
. To show that
is
-continuous at
let
be a sequence such that
in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ26_HTML.gif)
Thus, (2.25) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ27_HTML.gif)
but by (G5) we have , therefor (2.26) implies that
and we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ28_HTML.gif)
Taking the limit of (2.27) as , we see that
and so, by Proposition 1.8, we have
which implies that
is
-continuous at
.
Corollary 2.5.
Let be a complete
-metric space, and let
be a mapping which satisfies the following condition for some
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ29_HTML.gif)
where , then
has a unique fixed point (say
), and
is
-continuous at
.
Proof.
The proof follows from the previous theorem and the same argument used in Corollary 2.3.
Theorem 2.6.
Let be a complete
-metric space, and let
be a mapping which satisfies the following condition, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ30_HTML.gif)
where , then
has a unique fixed point, say
, and
is
-continuous at
.
Proof.
Suppose that satisfies the condition (2.29). Let
be an arbitrary point, and define the sequence
by
, then by (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ31_HTML.gif)
thus and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ32_HTML.gif)
But by (G5) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ33_HTML.gif)
Let , then
since
and from (2.31) we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ34_HTML.gif)
Continuing this procedure we get Then, for all
we have by repeated use of the rectangle inequality that
Thus,
, as
, so,
is
-Cauchy a sequence. By completeness of
, there exists
such that
is
-convergent to
. Suppose that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ35_HTML.gif)
taking the limit as , and using the fact that the function
is continuous in its variables, we obtain
. Since
this is a contradiction so,
. To prove uniqueness, suppose that
is such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ36_HTML.gif)
thus and we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ37_HTML.gif)
By the same argument we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ38_HTML.gif)
hence, which implies that
( since
). To show that
is
-continuous at
, let
be a sequence such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ39_HTML.gif)
therefore, (2.38) implies two cases.
Case 1.
.
Case 2.
.
But, by (G5) we have , so case 2 implies that
In each case taking the limit as
, we see that
and so, by Proposition 1.8, we have
which implies that
is
-continuous at
.
Corollary 2.7.
Let be a complete
-metric spaces, and let
be a mapping which satisfies the following condition for some
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ40_HTML.gif)
where , then
has a unique fixed point, say
, and
is
-continuous at
.
Proof.
The proof follows from the previous theorem and the same argument used in Corollary 2.3. The following theorem has been stated in [8] without proof, but this can be proved by using Theorem (2.6) as follows.
Theorem 2.8 ([8]).
Let be a complete
-metric space and let
be a mapping which satisfies the following condition, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917175/MediaObjects/13663_2008_Article_1191_Equ41_HTML.gif)
where , then
has a unique fixed point, say
, and
is
-continuous at
.
Proof.
Setting in condition (2.40), then it reduced to condition (2.29), and the proof follows from Theorem (2.6).
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Mustafa, Z., Sims, B. Fixed Point Theorems for Contractive Mappings in Complete -Metric Spaces.
Fixed Point Theory Appl 2009, 917175 (2009). https://doi.org/10.1155/2009/917175
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DOI: https://doi.org/10.1155/2009/917175