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On
-Stability of Picard Iteration in Cone Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 751090 (2009)
Abstract
The aim of this work is to investigate the -stability of Picard's iteration procedures in cone metric spaces and give an application.
1. Introduction and Preliminary
Let be a real Banach space. A nonempty convex closed subset
is called a cone in
if it satisfies the following:
(i) is closed, nonempty, and
,
(ii), and
imply that
(iii) and
imply that
The space can be partially ordered by the cone
; by defining,
if and only if
. Also, we write
if
int
, where int
denotes the interior of
.
A cone is called normal if there exists a constant
such that
implies
.
In the following we always suppose that is a real Banach space,
is a cone in
, and
is the partial ordering with respect to
.
Definition 1.1 (see [1]).
Let be a nonempty set. Assume that the mapping
satisfies the following:
(i) for all
and
if and only if
,
(ii) for all
,
(iii) for all
.
Then is called a cone metric on
, and
is called a cone metric space.
Definition 1.2.
Let be a map for which there exist real numbers
satisfying
. Then
is called a Zamfirescu operator if, for each pair
,
satisfies at least one of the following conditions:
(1),
(2),
(3).
Every Zamfirescu operator satisfies the inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ1_HTML.gif)
for all , where
max
, with
. For normed spaces see [2].
Lemma 1.3 (see [3]).
Let and
be nonnegative real sequences satisfying the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ2_HTML.gif)
where for all
and
as
Then
Remark 1.4.
Let and
be nonnegative real sequences satisfying the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ3_HTML.gif)
where for all
and for some positive integer number
. If
as
Then
Lemma 1.5.
Let be a normal cone with constant
, and let
and
be sequences in
satisfying the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ4_HTML.gif)
where and
as
Then
Proof.
Let be a positive integer such that
By recursion we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ5_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ6_HTML.gif)
and then by Remark 1.4?? Therefore
2.
-Stability in Cone Metric Spaces
Let be a cone metric space, and
a self-map of
. Let
be a point of
, and assume that
is an iteration procedure, involving
, which yields a sequence
of points from
.
Definition 2.1 (see [4]).
The iteration procedure is said to be
-stable with respect to
if
converges to a fixed point
of
and whenever
is a sequence in
with
we have
In practice, such a sequence could arise in the following way. Let
be a point in
. Set
. Let
. Now
. Because of rounding or discretization in the function
, a new value
approximately equal to
might be obtained instead of the true value of
. Then to approximate
, the value
is computed to yield
, an approximation of
. This computation is continued to obtain
an approximate sequence of
.
One of the most popular iteration procedures for approximating a fixed point of is Picard's iteration defined by
. If the conditions of Definition 2.1 hold for
then we will say that Picard's iteration is
-stable.
Recently Qing and Rhoades [5] established a result for the -stability of Picard's iteration in metric spaces. Here we are going to generalize their result to cone metric spaces and present an application.
Theorem 2.2.
Let be cone metric space,
a normal cone, and
with
If there exist numbers
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ7_HTML.gif)
for each and in addition, whenever
is a sequence with
as
, then Picard's iteration is
-stable.
Proof.
Suppose and
We shall show that
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ8_HTML.gif)
if we put and
in Lemma 1.5, then we have
Note that the fixed point of
is unique. Because if
is another fixed point of
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ9_HTML.gif)
which implies
Corollary 2.3.
Let be a cone metric space,
a normal cone, and
with
If there exists a number
such that
for each
then Picard's iteration is
-stable.
Corollary 2.4.
Let be a cone metric space,
a normal cone, and
is a Zamfirescu operator with
and whenever
is a sequence with
as
, then Picard's iteration is
-stable.
Definition 2.5 (see [6]).
Let be a cone metric space. A map
is called a quasicontraction if for some constant
and for every
there exists
such that
Lemma 2.6.
If is a quasicontraction with
, then
is a Zamfirescu operator and so satisfies (2.1).
Proof.
Let for every
we have
for some
In the case that
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ10_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ11_HTML.gif)
Put so
The other cases are similarly proved. Therefore
is a Zamfirescu operator.
Theorem 2.7.
Let be a nonempty complete cone metric space,
be a normal cone, and
a quasicontraction and self map of
with some
Then Picard's iteration is
-stable.
Proof.
By [6, Theorem??2.1], has a unique fixed point
Also
satisfies (2.1). So by Theorem 2.2 it is enough to show that
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ12_HTML.gif)
Put and
Therefore
as
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ13_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F751090/MediaObjects/13663_2009_Article_1173_Equ14_HTML.gif)
Hence we have or
where
or
and
or
Therefore by (2.7),
by
Now by Lemma 1.5 we have
3. An Application
Theorem 3.1.
Let with
for
and let
be a self map of
defined by
where
(a) is a continuous function,
(b)the partial derivative of
with respect to
exists and
for some
(c)for every real number one has
for every
Let be a normal cone and
the complete cone metric space defined by
where
Then,
(i)Picard's iteration is -stable if
,
(ii)Picard's iteration fails to be -stable if
and
Proof.
-
(i)
We have
being a continuous quasicontraction map with
so by Theorem 2.7, Picard's iteration is
-stable.
-
(ii)
Put
so
and
where
Also
since
(3.1)
as But
and
is not a fixed point for
Therefore Picard's iteration is not
-stable.
Example 3.2.
Let and
Therefore
and
satisfy the hypothesis of Theorem 3.1 where
has property (i) and
has property (ii). So the self maps
of
defined by
and
have unique fixed points but Picard's iteration is
-stable for
but not
-stable for
References
Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
Zhiqun X: Remarks of equivalence among Picard, Mann, and Ishikawa iterations in normed spaces. Fixed Point Theory and Applications 2007, 2007:-5.
Vasilev FP: Numerical Methodes for Solving Extremal Problems. 2nd edition. Nauka, Moscow, Russian; 1988:550.
Harder AM, Hicks TL: Stability results for fixed point iteration procedures. Mathematica Japonica 1988,33(5):693–706.
Qing Y, Rhoades BE: -stability of Picard iteration in metric spaces. Fixed Point Theory and Applications 2008, 2008:-4.
Ilic D, Rakocevic V: Quasi-contraction on a cone metric space. Applied Mathematics Letters 2009,22(5):728–731. 10.1016/j.aml.2008.08.011
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Asadi, M., Soleimani, H., Vaezpour, S.M. et al. On -Stability of Picard Iteration in Cone Metric Spaces.
Fixed Point Theory Appl 2009, 751090 (2009). https://doi.org/10.1155/2009/751090
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DOI: https://doi.org/10.1155/2009/751090