- Research Article
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Stable Iteration Procedures in Metric Spaces which Generalize a Picard-Type Iteration
Fixed Point Theory and Applications volume 2010, Article number: 953091 (2010)
Abstract
This paper investigates the stability of iteration procedures defined by continuous functions acting on self-maps in continuous metric spaces. Some of the obtained results extend the contraction principle to the use of altering-distance functions and extended altering-distance functions, the last ones being piecewise continuous. The conditions for the maps to be contractive for the achievement of stability of the iteration process can be relaxed to the fulfilment of being large contractions or to be subject to altering-distance functions or extended altering functions.
1. Introduction
Banach contraction principle is a very basic and useful result of Mathematical Analysis [1–7]. Basic applications of this principle are related to stability of both continuous-time and discrete-time dynamic systems [4, 8], including the case of high-complexity models for dynamic systems consisting of functional differential equations by the presence of delays [4, 9]. Several generalizations of the contraction principle are investigated in [2] by proving that the result still holds if altering-distance functions [1] are replaced with a difference of two continuous monotone nondecreasing real functions which take zero values only at the origin. The so-called -times reasonable expansive mappings and the associated existence of unique fixed points are investigated in [7]. The so-called Halpern's iteration [10] and several of its extensions in the context of fixed-point theory have been investigated in [11–13]. Further extended viscosity iteration schemes with nonexpansive mappings based on the above one have been investigated in [9, 10, 12–18], while proving the common existence of unique fixed points for the related schemes and the strong convergence of the iterations to those points for any arbitrary initial conditions. The stability of Picard iteration has been investigated exhaustively (see, e.g., [5, 19–22]). The Picard and approximate Picard methods have been also used in classical papers for proving the existence and uniqueness of solutions in many differential equations including those of Sobolev type (see, e.g., [23]).
This paper presents some generalizations of results concerning the stability of iterations in the sense that the iteration scheme subject to error sequences converges asymptotically to its nominal fixed point provided that the iteration error converges asymptotically to zero. Several generalizations are discussed in the framework of stability of iteration schemes in complete metric spaces including:
(a)the use of altering-distance functions (Definition 1.1) [1, 2], and the so-called then defined extended altering functions (Definition 2.1 in Section 2) where the continuous altering functions are allowed to be piecewise continuous;
(b)the use of iteration schemes which are based on continuous functions which modify the Picard iteration scheme [5, 6];
(c)the removal of the common hypothesis in the context of -stability that the set of fixed points of the iteration scheme is nonempty by guaranteeing that this is in fact true under contractive mappings, large contractions, or altering- and extended altering-distance functions, [1–4, 6].
Definition 1.1 (see [1] (altering-distance function)).
A monotone nondecreasing function , with
, if and only if
, is said to be an altering-distance function.
If is a complete metric space,
is a self-mapping on
, and
, for all
and some real constant
, then
has a unique fixed point [1, 2]. This result is extendable to the use of monotone nondecreasing functions
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ1_HTML.gif)
for some monotone nondecreasing function satisfying
. Those results are directly extended to monotone nondecreasing piecewise continuous functions being continuous at "0" after a preliminary "ad hoc" definition in the subsequent section.
2. Fixed Point Properties Related to Altering- and Extended Altering-Distance Functions
Since but continuous at
it can possess bounded isolated discontinuities on
and it is necessary to reflect this fact in the notation as follows. The left (resp., right) limit of
at
is simply denoted by
, instead of using the more cumbersome classical notation
(resp., by
instead of using the more cumbersome
). Since
is an extended altering-distance function, then continuous at
,
. If
is continuous at a given
, then
. If
is has a discontinuity point (of second class), then
, with
.
Definition 2.1 (extended altering-distance function).
A monotone nondecreasing function being continuous at "0", with
, if and only if
, is said to be an extended altering-distance function.
Theorem 2.2.
Let be a complete metric space and
be a self-mapping on
. Then, the following properties hold.
(i)Assume that is an altering-distance function such that
, for all
for some real constant
. Then
has a unique fixed point [1].
(ii)Assume that is an extended altering-distance function such that
and
, for all
for some real function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ2_HTML.gif)
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ3_HTML.gif)
for all for some monotone nondecreasing function
satisfying
, for all
and
, if and only if
Then
is monotone nondecreasing and
has a unique fixed point. In particular, if
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ4_HTML.gif)
for all for some monotone nondecreasing function
satisfying
for all
and
, if and only if
, then
has a unique fixed point.
Proof.
Note that from l'Hopital rule and the fact that both functions
and
are continuous at "0" with
. Note that after taking left and right limits at each nonnegative real argument
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ5_HTML.gif)
for all , such that
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ6_HTML.gif)
Then, is monotone nondecreasing from simple inspection of the above properties. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ7_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ8_HTML.gif)
Now, it is proven by contradiction that there is no such that
, for any given
Take two arbitrary
Assume that
, for all
, and some given
, so that if
also for all
, then for some
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ9_HTML.gif)
for all But, it always exist a finite
such that
, for all
since
;
, what leads to a contradiction. Thus, there is no
such that
, for all
for any given
. As a result, the subsequent relations are true:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ10_HTML.gif)
for all , with the above limits since
is continuous at
and
, if and only if
. Furthermore, any sequence
with
, for all
, is a Cauchy sequence since for any arbitrarily small prefixed constant
, there exist sequences
,
, and
of nonnegative integers satisfying
;
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ11_HTML.gif)
Thus, there is a unique which is in
, the set of fixed points of
, that is,
, for all
. Since
is a complete metric space and the sequence
with
is a Cauchy sequence, for all
then
. It holds trivially that all the above proof also holds for special case
and some monotone nondecreasing function
satisfying
(i.e.,
is a weak contraction) as may be proven [3] (see also [24]). Property (ii) has been fully proven.
Theorem 2.2 might be linked to the concept of large contraction which is less restrictive than that of contraction. The related discussion follows.
Definition 2.3 (see [4] (large contraction)).
Let be a complete metric space. Then, the self-mapping
on
is said to be a large contraction, if
, for all
, and if for any given
, such that
, then there exist
, such that
.
It turns out that a contraction is also a large contraction with being independent of
in Definition 2.3. The following result proves that the self-mapping
on
satisfying Theorem 2.2(ii) is a large contraction.
Proposition 2.4.
Let be a complete metric space and
be a self-mapping on
. If
is a modified altering-distance function which satisfies the conditions of Theorem 2.2(ii), then
is a large contraction.
Proof.
Given , for all
, since
if
. Since
is an extended altering-distance function it is monotone nondecreasing of nonnegative values and taking the zero value only at "0". Thus,
. Furthermore, it is proven by contradiction that for any given
, such that
,
, such that
. Take
, such that
, and assume that
Since
is monotone nondecreasing, then
, for all
, and one also gets that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ12_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ13_HTML.gif)
which are two contradictions. Thus, , for some
and
is a large contraction.
It is now proven that the sequence is uniformly bounded if Theorem 2.2(ii) holds.
Proposition 2.5.
Let be a complete metric space and
be a self-mapping on
. If
is a modified altering-distance function which satisfies the conditions of Theorem 2.2(ii), then
, for all
and
.
Proof.
Proceed by contradiction by assuming that , for all
and
, is false so that
,
is unbounded. Thus, there is a subsequence
of self-mappings on
, with
, as
, such that the real subsequence
is strictly monotone increasing so that it diverges to
, so that
and
, as
. Since
is monotone nondecreasing, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ14_HTML.gif)
since is monotone nondecreasing. Since the inequalities are nonstrict, the above subsequences might either converge to nonnegative real limits
and
, or diverge to
. The event that
and
are one finite and the other infinity is not possible since
so that any existing discontinuity is a finite-jump type discontinuity. Thus, both limits are either finite, although eventually distinct or both are
so that
and
(and simultaneously finite or infinity) as
, where
for the given
. Such a
always exists in
for each given
since
is a self-mapping on
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ15_HTML.gif)
which is a contradiction unless , as
, as
, since
is continuous at
and
, if and only if
But, if the subsequence
has a zero limit as
, then it is a bounded sequence. Thus,
as
is false and then
,
being unbounded fails so that the contradiction follows. The right-limit convergence
leads to the same conclusion. As a result, there is no
such that
,
is unbounded and the result is fully proven.
An alternative proof to that of Theorem 2.2(ii) related to the existence of a unique fixed point in , follows directly by using Theorem
in [4] since
is a large contraction and the sequence
is uniformly bounded (Propositions 2.4 and 2.5).
Proposition 2.6.
Let be a complete metric space and
be a large contraction. If
is a modified altering-distance function which satisfies the conditions of Theorem 2.2(ii), then
has a unique fixed point in
.
Proof.
is a large contraction from Proposition 2.4, since it fulfils Theorem 2.2(ii). Also,
, for all
and
from Proposition 2.5. Thus, from [4, Theorem
],
has a unique fixed point in
.
The following result is a direct consequence of Theorem 2.2, Propositions 2.4 and 2.5.
Proposition 2.7.
Let be a complete metric space and
be a weak contraction on
. Then,
, for all
and
and
has a unique fixed point on
.
3.
-Stability Related to a Class of Nonlinear Iterations Related to Distance and Altering-Distance Functions
Assume that is a complete metric space,
is a self-mapping on
. The iteration process
has a fixed point if
. A necessary condition for
to have a fixed point is that it to be injective. The
-stability of the Picard iteration has been investigated in a set of papers (see, e.g., [5, 19, 20]). The Picard iteration is said to be
-stable if
, for all
. The subsequent result is an extension of a previous one in [5] for the so-called
-stability of the iteration
, for all
if the pair
satisfies the so-called
property defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ16_HTML.gif)
for all ;
, with
and
, provided that the set of fixed points
is nonempty. If
is identity, then the above property is stated as
satisfying the
property.
Theorem 3.1.
Assume that
(1) is a complete metric space,
is a continuous mapping, and
is a self-mapping on
such that the set of fixed points
of the iteration procedure
, for all
, is nonempty;
(2)the pair satisfies the
property; that is,
; for all
with
and
;
(3), for all
.
Then, the iteration procedure , for all
, is
-stable and it possesses a unique fixed point.
Proof.
For any given , which exists since
, and for all
such that
with
being the computation error sequence, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ17_HTML.gif)
Since ,
, and
, as
, then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ18_HTML.gif)
from (3.2), [5, 6]. Also, , as
, as
. Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ19_HTML.gif)
as , since
, as
. From the above inequalities either
, as
, or
, with
as
, but in this second case,
, as
, is false so that
. Then,
, as
. Thus,
, as
from (3.4). Then,
and
, as
. Since
is injective,
, as
so that f is
-stable. It is proven by contradiction that the fixed point of the iteration procedure
, for all
is unique. Assume that there exists
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ20_HTML.gif)
what is impossible if . Then,
.
Theorem 3.1 is now extended by extending the property of the pair
to that of the triple
, where
is an appropriate continuous function.
Theorem 3.2.
Assume that
(1) is a complete metric space,
is a continuous mapping, and
is a self-mapping on
such that the set of fixed points
of the iteration procedure
, for all
, is nonempty;
(2) is continuous, satisfies
, possesses the subadditive property, and, furthermore, the triple
satisfies the
property defined by
+
for all
with
and
;
(3), for all
.
Then, the iteration procedure , for all
is
-stable and it possesses a unique fixed point.
Proof.
For any given , which exists since
, and for all
such that
, with
being the computation error sequence and, since
possesses the sub-additive property, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ21_HTML.gif)
according to Hypothesis () since
and
from Hypothesis (
). Since
is everywhere continuous and satisfies
, then
as
. Also,
as
as
. The remaining of the proof follows with the same arguments as in that of Theorem 3.1.
Theorem 3.3.
Assume that
(1) is a complete metric space,
is a continuous mapping, and
is a self-mapping on
such that the set of fixed points
of the iteration procedure
, for all
, is nonempty;
(2) and
are both continuous and monotone nondecreasing while satisfying
, and, furthermore, the quadruple
satisfies the
property:
, for all
; and for all
with
and
;
(3), for all
.
Then, the iteration procedure , for all
is
-stable and it possesses a unique fixed point.
Proof.
Since and
are both continuous and monotone nondecreasing then Hypothesis (
) implies that
, for all
and
; for all
with
and
which is Hypothesis (
) of Theorem 3.1. Furthermore,
from the continuity of
everywhere within its definition domain
and its property
. Thus, the proof follows as in Theorem 3.1 since Hypothesis (
) to (
) of this theorem hold.
The following direct particular result of Theorems 3.1 to 3.3 follows.
Corollary 3.4.
Theorems 3.1, 3.2, and 3.3 hold "mutatis-mutandis" stated for the function being the identity mapping on
and
.
Corollary 3.4 referred to Theorem 3.1 was first proven in [5]. It is now of interest the removal of the condition of the set of fixed points to be nonempty by guaranteeing that is in fact nonempty consisting of a unique element under extra contractive properties of the pair . The following result holds.
Theorem 3.5.
The following two properties hold.
(i)Consider the Picard -iteration process
, for all
and
. If
satisfies the
property while it is a
-contraction (i.e., a contractive mapping with constant
, then
for some
, and, furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ23_HTML.gif)
(ii)Consider the iteration process , for all
and
. If
satisfies the
property while the pair
is a
-contraction (i.e., a contractive mapping with constant
, then
for some
, and, furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ25_HTML.gif)
Proof.
-
(i)
Equation (3.7) follows from the
property leading to
(3.11)
since is
-contractive, so that it possesses a unique fixed point
, and it satisfies the
property with
. Equation (3.8) follows directly from the following three inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ27_HTML.gif)
For , for all
, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ28_HTML.gif)
after using the property, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ29_HTML.gif)
again with the use of the property.
-
(ii)
Equation (3.9) follows from the
property leading to
(3.15)
since the pair is
-contractive and it satisfies the
property with
. Equation (3.10) follows directly from the following three inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ31_HTML.gif)
since the pair (f, ) is
-contractive. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ32_HTML.gif)
for all and some
independent of
.
. Therefore, for any
, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ33_HTML.gif)
after using the property, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ34_HTML.gif)
again with the use of the property.
Theorem 3.5(ii) allows directly extending Theorem 3.1 as follows by removing the requirement of the set of fixed points to be nonempty with a unique element since this is guaranteed by the Banach contractive mapping principle.
Theorem 3.6.
Assume that: is a complete metric space,
is a continuous mapping on
, and
is a self-mapping on
such that the pair
is
-contractive and satisfies the
property (provided that the set of fixed points is nonempty), that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ35_HTML.gif)
with being any fixed point; that is,
and some real constants
,
, and
, and, furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ36_HTML.gif)
Then, the iteration procedure , for all
is
-stable with
being its unique fixed point, that is,
.
A direct particular case of Theorem 3.5 applies directly to the case of f being the identity map on via Theorem 3.5(i).
Corollary 3.7.
Assume that is a complete metric space, and
is a
-contractive self-mapping on
satisfying the
property, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ37_HTML.gif)
with being any fixed point; that is,
and some real constants
,
, and
, and, furthermore,
, for all
. Then, the Picard iteration procedure
, for all
, is
-stable with
being its unique fixed point, that is,
.
Theorems 3.5 and 3.6 and Corollary 3.7 are directly extendable to the case that the pair is a large contraction. Also, Theorem 3.6 and Corollary 3.7 can be extended directly for the use of distance functions or extended altering-distance functions as follows.
Theorem 3.8.
Assume that
(1) is a complete metric space,
is a continuous mapping on
, and
is a self-mapping on
;
(2) is an extended altering-distance function (Definition 2.1) satisfying the conditions of Theorem 2.2(ii) with some monotone nondecreasing function
satisfying
, for all
, and
, if and only if
;
(3)the quadruple satisfies the following
property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ38_HTML.gif)
for all , and
with real constants
and
;
(4), for all
.
Then, the iteration procedure , for all
, is
-stable with
being its unique fixed point, that is,
.
Corollary 3.9.
Assume that Assumptions () and (
) of Theorem 3.8 hold, and, furthermore,
(1) is an altering-distance function satisfying the conditions of Theorem 2.2(i), for all
, satisfying
, if and only if
;
(2)the triple satisfies the following
property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F953091/MediaObjects/13663_2010_Article_1371_Equ39_HTML.gif)
for all , and
with real constants
and
;
(3), for all
.
Then, the iteration procedure , for all
is
-stable with
being its unique fixed point, that is,
.
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Acknowledgments
The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government for its support through Grants IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK SPE07UN04.
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De la Sen, M. Stable Iteration Procedures in Metric Spaces which Generalize a Picard-Type Iteration. Fixed Point Theory Appl 2010, 953091 (2010). https://doi.org/10.1155/2010/953091
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DOI: https://doi.org/10.1155/2010/953091