- Research Article
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Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme
Fixed Point Theory and Applications volume 2009, Article number: 314581 (2009)
Abstract
A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.
1. Introduction
Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, [1–12]. A key point is that the equations under study are driven by contractive maps or at least by asymptotically nonexpansive maps. By that reason, the fixed point formalism is useful in stability theory to investigate the asymptotic convergence of the solution to stable attractors which are stable equilibrium points. The uniqueness of the fixed point is not required in the most general context although it can be sometimes suitable provided that only one such a point exists in some given problem. Therefore, the theory is useful for stability problems subject to multiple stable equilibrium points. Compared to Lyapunov's stability theory, it may be a more powerful tool in cases when searching a Lyapunov functional is a difficult task or when there exist multiple equilibrium points, [1, 12]. Furthermore, it is not easy to obtain the value of the equilibrium points from that of the Lyapunov functional in the case that the last one is very involved. A generalization of the contraction principle in metric spaces by using continuous nondecreasing functions subject to an inequality-type constraint has been performed in [2]. The concept of -times reasonable expansive mapping in a complete metric space is defined in [3] and proven to possess a fixed point. In [5], the
-stability of Picard's iteration is investigated with
being a self-mapping of
where (
) is a complete metric space. The concept of
-stability is set as follows: if a solution sequence converges to an existing fixed point of
, then the error in terms of distance of any two consecutive values of any solution generated by Picard's iteration converges asymptotically to zero. On the other hand, an important effort has been devoted to the investigation of Halpern's iteration scheme and many associate extensions during the last decades (see, e.g., [4, 6, 9, 10]). Basic Halpern's iteration is driven by an external sequence plus a contractive mapping whose two associate coefficient sequences sum unity for all samples, [9]. Recent extensions of Halpern's iteration to viscosity iterations have been proposed in [4, 6]. In the first reference, a viscosity-type term is added as extraforcing term to the basic external sequence of Halpern's scheme. In the second one, the external driving term is replaced with two ones, namely, a viscosity-type term plus an asymptotically nonexpansive mapping taking values on a left reversible semigroup of asymptotically nonexpansive Lipschitzian mappings on a compact convex subset
of the Banach space
. The final iteration process investigated in [6] consists of three forcing terms, namely, a contraction on
, an asymptotically nonexpansive Lipschitzian mapping taking values in a left reversible semigroup of mappings from a subset of that of bounded functions on its dual. It is proven that the solution converges to a unique common fixed point of all the set asymptotic nonexpansive mappings for any initial conditions on
. The objective of this paper is to investigate further generalizations for Halpern's iteration process via fixed point theory by using two more driving terms, namely, an external one taking values on
plus a nonlinear term given by a continuous nondecreasing function, subject to an inequality-type constraint as proposed in [2], whose argument is the distance between pairs of points of sequences in certain complete metric space which are not necessarily directly related to the sequence solution taking values in the subset
of the Banach space
. Another generalization point is that the sample-by-sample sum of the scalar coefficient sequences of all the driving terms is not necessarily unity but it converges asymptotically to unity.
2. Stability and Boundedness Properties of a Viscosity-Type Difference Equation
In this section a real difference equation scheme is investigated from a stability point of view by also discussing the existence of stable limiting finite points. The structure of such an iterative scheme supplies the structural basis for the general viscosity iterative scheme later discussed formally in Section 4 in the light of contractive and asymptotically nonexpansive mappings in compact convex subsets of Banach spaces. The following well-known iterative scheme is investigated for an iterative scheme which generates real sequences.
Theorem 2.1.
Consider the difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ1_HTML.gif)
such that the error sequence is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ2_HTML.gif)
for all , where
.
Assume that and
are bounded real constants and
; for all
. Then, the following properties hold.
-
(i)
The real sequences
,
, and
are uniformly bounded if
if
and
if
; for all
. If, furthermore,
if
and
, if
, with
if and only if
; for all
, then the sequences
,
, and
converge asymptotically to the zero equilibrium point as
and
is monotonically decreasing.
-
(ii)
Let the real sequence
be defined by
if
and
if
(what implies that
from (2.1) and
). Then,
is uniformly bounded if
; for all
. If, furthermore,
; for all
then
as
.
-
(iii)
Let
and let
a positive real sequence (i.e., all its elements are nonnegative real constants). Define
if
and
if
. Then,
is a positive real sequence and
is uniformly bounded if
; for all
. If, furthermore,
; for all
, then
as
.
-
(iv)
If
; for all
and
, then
; for all
. If
and
; for all
, then
; for all
. If
and
; for all
for some
, with
, then
; for all
and
as
.
-
(v)
(Corollary to Venter's theorem, [7]). Assume that
for all
,
as
and
(what imply
as
and the sequence
has only a finite set of unity values). Assume also that
and
is a nonnegative real sequence with
. Then
as
.
-
(vi)
(Suzuki [8]; see also Saeidi [6]). Let
be a sequence in
with
, and let
and
be bounded sequences. Then,
.
-
(vii)
(Halpern [9]; see Hu [4]). Let
be
; for all
in (2.1) subject to
,
; for all
with
being a nonexpansive self-mapping on
. Thus,
converges weakly to a fixed point of
in the framework of Hilbert spaces endowed with the inner product
, for all
, if
for any
.
Proof.
-
(i)
Direct calculations with (2.1) lead to
(2.3)
so that if
, and equivalently, if
and
with
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ4_HTML.gif)
Thus, ,
and
. If, in addition,
and
with
then
and
is a monotonically decreasing sequence,
and
as
. Property (i) has been proven.
-
(ii)
Direct calculations with (2.2) yield for
,
(2.5)
Since is a convex parabola
for all
if real constants
exist such that
;
. The parabola zeros are
so that
if
. If
, then
with
. Thus,
if
, for all
. If
, then
as
. Property (ii) has been proven.
-
(iii)
If
is positive then
is positive from direct calculations through (2.1). The second part follows directly from Property (ii) by restricting
for uniform boundedness of
and
for its asymptotic convergence to zero in the case of nonzero
.
-
(iv)
If
; for all
and
, then from recursive evaluation of (2.1):
(2.6)
If, and
; for all
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ7_HTML.gif)
If and
, for all
for some
, then
, for all
; thus,
is monotonically strictly decreasing so that it converges asymptotically to zero.
Equation (2.1) under the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ8_HTML.gif)
with and
being a nonexpansive self-mapping on
under the weak or strong convergence conditions of Theorem 2.1(vii) is known as Halpern's iteration [4], which is a particular case of the generalized viscosity iterative scheme studied in the subsequent sections. Theorem 2.1(vi) extends stability Venter's theorem which is useful in recursive stochastic estimation theory when investigating the asymptotic expectation of the norm-squared parametrical estimation error [7]. Note that the stability result of this section has been derived by using discrete Lyapunov's stability theorem with Lyapunov's sequence
what guarantees global asymptotic stability to the zero equilibrium point if it is strictly monotonically decreasing on
and to global stability (stated essentially in terms of uniform boundedness of the sequence
) if it is monotonically decreasing on
. The links between Lyapunov's stability and fixed point theory are clear (see, e.g., [1, 2]). However, fixed point theory is a more powerful tool in the case of uncertain problems since it copes more easily with the existence of multiple stable equilibrium points and with nonlinear mappings. Note that the results of Theorem 2.1 may be further formalized in the context of fixed point theory by defining a complete metric space
, respectively,
for the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined by
.
3. Some Definitions and Background as Preparatory Tools for Section 4
The four subsequent definitions are then used in the results established and proven in Section 4.
Definition 3.1.
is a left reversible semigroup if
; for all
.
It is possible to define a partial preordering relation "" by
; for all
for any semigroup
. Thus,
, for some existing
and
, such that
if
is left reversible. The semigroup
is said to be left-amenable if it has a left-invariant mean and it is then left reversible, [6, 13].
is said to be a representation of a left reversible semigroup
as Lipschitzian mappings on
if
is a Lipschitzian mapping on
with Lipschitz constant
and, furthermore,
; for all
.
The representation may be nonexpansive, asymptotically nonexpansive, contractive and asymptotically contractive according to Definitions 3.3 and 3.4 which follow.
Definition 3.3.
A representation of a left reversible semigroup
as Lipschitzian mappings on
, a nonempty weakly compact convex subset of
, with Lipschitz constants
is said to be a nonexpansive (resp., asymptotically nonexpansive, [6]) semigroup on
if it holds the uniform Lipschitzian condition
(resp.,
) on the Lipschitz constants.
Definition 3.4.
A representation of a left reversible semigroup
as Lipschitzian mappings on
with Lipschitz constants
is said to be a contractive (resp., asymptotically contractive) semigroup on
if it holds the uniform Lipschitzian condition
(resp.,
) on the Lipschitz constants.
The iteration process (3.1) is subject to a forcing term generated by a set of Lipschitzian mappings where
is a sequence of means on
, with the subset
(defined in Definition 3.5 below) containing unity, where
is the Banach space of all bounded functions on
endowed with the supremum norm, such that
where
is the dual of
.
Definition 3.5.
The real sequence is a sequence of means on
if
.
Some particular characterizations of sequences of means to be invoked later on in the results of Section 4 are now given in the definitions which follow.
Definition 3.6.
The sequence of means on
is
(1)left invariant if ; for all
, for all
, for all
in
for
;
(2)strongly left regular if , for all
, where
is the adjoint operator of
defined by
; for all
, for all
.
Parallel definitions follow for right-invariant and strongly right-amenable sequences of means. is said to be left (resp., right)-amenable if it has a left (resp., right)-invariant mean. A general viscosity iteration process considered in [6] is the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ9_HTML.gif)
where
(i)the real sequences ,
, and
have elements in
of sum being identity, for all
;
(ii) is a representation of a left reversible semigroup with identity
being asymptotically nonexpansive, on a compact convex subset
of a smooth Banach space, with respect to a left-regular sequence of means defined on an appropriate invariant subspace of
;
(iii) is a contraction on
.
It has been proven that the solution of the sequence converges strongly to a unique common fixed point of the representation which is the solution of a variational inequality [6]. The viscosity iteration process (3.1) generalizes that proposed in [13] for
and
and also that proposed in [14, 15] with
,
and
; for all
. Halpern's iteration is obtained by replacing
and
in (3.1) by using the formalism of Hilbert spaces, for all
(see, e.g., [4, 9, 10]). There has been proven the weak convergence of the sequence
to a fixed point of
for any given
if
for
[9], also proven to converge strongly to one such a point if
and
as
, and
[10]. On the other hand, note that if
,
, and
with
, for all
, then the resulting particular iteration process (3.1) becomes the difference equation (2.1) discussed in Theorem 2.1 from a stability point of view provided that the boundedness of the solution is ensured on some convex compact set
; for all
.
4. Boundedness and Convergence Properties of a More General Difference Equation
The viscosity iteration process (3.1) is generalized in this section by including two more forcing terms not being directly related to the solution sequence. One of them being dependent on a nondecreasing distance-valued function related to a complete metric space while the other forcing term is governed by an external sequence . Furthermore the sum of the four terms of the scalar sequences
,
, and
and
at each sample is not necessarily unity but it is asymptotically convergent to unity.
The following generalized viscosity iterative scheme, which is a more general difference equation than (3.1), is considered in the sequel
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ10_HTML.gif)
for all for a sequence of given finite numbers
with
(if
, then the corresponding sum is dropped off) which can be rewritten as (2.1) if
; for all
(except possibly for a finite number of values of the sequence
what implies
) by defining the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ11_HTML.gif)
with , where
(i) is a strongly left-regular sequence of means on
, that is,
. See Definition 3.5;
(ii) is a left reversible semigroup represented as Lipschitzian mappings on
by
.
The iterative scheme is subject to the following assumptions.
Assumption 1.
()
,
, and
are real sequences in
,
is a real sequence in
, and
are sequences in
, for all
for some given
and
.
()
,
.
()
.
()
.
()
; for all
with
being a bounded real sequence satisfying
and
.
()
is a contraction on a nonempty compact convex subset
, of diameter
of a Banach space
, of topological dual
, which is smooth, that is, its normalized duality mapping
from
into the family of nonempty (by the Hahn-Banach theorem [6, 11]), weak-star compact convex subsets of
, defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ12_HTML.gif)
is single valued.
() The representation
of the left reversible semigroup
with identity is asymptotically nonexpansive on
(see Definition 3.3) with respect to
, with
which is strongly left regular so that it fulfils
.
()
.
()
is a complete metric space and
is a self-mapping satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ13_HTML.gif)
where , for all
are continuous monotone nondecreasing functions satisfying
if and only if
; for all
.
()
is a sequence in
generated as
,
with
and
is a finite given number.
Note that Assumption 1() is stronger than the conditions imposed on the sequence
in Theorem 2.1 for (2.1). However, the whole viscosity iteration is much more general than the iterative equation (2.1). Three generalizations compared to existing schemes of this class are that an extracoefficient sequence
is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients
being unity for all
is replaced by a limit-type constraint
as
while during the transient such a constraint can exceed unity or be below unity at each sample (see Assumption 1(
). Another generalization is the inclusion of a nonnegative term with generalized contractive mapping
involving another iterative scheme evolving on another, and in general distinct, complete metric space
(see Assumptions 1(
) and 1(
). Some boundedness and convergence properties of the iterative process (4.1) are formulated and proven in the subsequent result.
Theorem 4.1.
The difference iterative scheme (4.1) and equivalently the difference equation (2.1) subject to (4.2) possess the following properties under Assumption 1.
(i). Also,
and
for any norm defined on the smooth Banach space
and there exists a nonempty bounded compact convex set
such that the solution of (4.2) is permanent in
, for all
and some sufficiently large finite
with
.
(ii) and
as
.
-
(iii)
(4.5)
(iv)Assume that such that each sequence element
(the first closed orthant of
); for all
, for some
so that (4.1) is a positive viscosity iteration scheme. Then,
(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all
, for all
), denoted as
; for all
.
(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point
.
(iv.3)Property (iii) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ15_HTML.gif)
what implies that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ16_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ17_HTML.gif)
Proof.
From (4.2) and substituting the real sequence from the constraint Assumption 1(
), we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ18_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ19_HTML.gif)
where is an arbitrary finite sufficiently large integer, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ20_HTML.gif)
since the functions are continuous on
with
and
as
, [2] with
being prefixed and arbitrarily small. The constants
and
are finite for sufficiently large
since
(Assumption 1(
),
is a contraction on
(Assumption 1(
), and
is a self-mapping on
satisfying Assumption 1(
). Since
,
and
as
from Assumptions 1(
) and 1(
) and
is finite,
as
and
; for all
being arbitrarily small since
is arbitrarily large. Since from Assumption 1(
),
is an asymptotically nonexpansive semigroup on
, and
,
, and
as
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ21_HTML.gif)
with as
. One gets from (4.12) into (4.10),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ22_HTML.gif)
what implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ23_HTML.gif)
(see [8]) since as
since
is in
and
is a strongly left-regular sequence of means on
such that
; furthermore,
,
,
,
,
as
and
as
. Thus, from (4.14) and using the above technical result in [8] for difference equations of the class (2.1) (see also [2]), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ24_HTML.gif)
since from Assumption 1(
) since
,
, and
as
. From (4.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ25_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ26_HTML.gif)
Using Assumption 1 and using (4.15) into (4.17) yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ27_HTML.gif)
since ,
,
,
as
. Also, it follows that
as
from (4.15) and (4.18). Note that it has not been proven yet that the sequences
and
converge to a finite limit as
since it has not been proven that they are bounded. Thus, the four sequences
and
converge asymptotically to the same finite or infinite real limit. Proceed recursively with the solution of (4.1). Thus, for a given sufficiently large finite
and for all
, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ29_HTML.gif)
for all , for some positive real sequences
,
, and
satisfying
and
,
and
as
with
and
being arbitrarily small for sufficiently large
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ30_HTML.gif)
for sufficiently large and a sufficiently small
which exists from Assumptions 1(
) and 1(
). Note that the sequences
and
may be chosen to satisfy
and
; for all
. Now, proceed by complete induction by assuming that
for given sufficiently large
and finite
. Then, one gets from (4.20) that
for any prescribed
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ31_HTML.gif)
with and
which always holds for sufficiently large finite
since
as
. It has been proven by complete induction that the first part of Property (i) holds with the set
being built such that
for the given initial condition
. For a set of initial conditions
with any set
convex and bounded, a common set
might be defined for any initial condition of (4.1) in
with a redefinition of the constant
as
. The second part of Property (i) follows for any norm on
from the property of equivalence of norms. Furthermore, the real sequences
and
converge strongly to a finite limit in
since they are uniformly bounded so that Property (ii) has also been proven. Property (iii) follows directly from (4.1) and Property (ii). Property (iv.
) follows since
is a nonnegative
-vector sequence provided that
if
what follows from simple inspection of (4.1). Properties (iv.
)-(iv.
) follow directly from separating nonnegative positive and nonpositive terms in the right-hand side of the expression in Property (iii).
The convergence properties of Theorem 4.1(ii) are now related to the limits being fixed points of the asymptotically nonexpansive semigroup which is the representation as Lipschitzian mappings on
of a left reversible semigroup
with identity.
Theorem 4.2.
The following properties hold.
-
(i)
Let
be the set of fixed points of the asymptotically nonexpansive semigroup
on
. Then, the common strong limit
of the sequences
and
in Theorem 4.1(ii) is a fixed point of
located in
and, thus, a stable equilibrium point of the iterative scheme (4.1) provided that
, and then
, is sufficiently large.
(ii).
Proof.
-
(i)
Proceed by contradiction by assuming that
so that there exists
such that
(4.23)
since , where the above two limits exist and are zero from Theorem 4.1(ii). Then,
, with
being nonempty since, at least one such finite fixed point exists in
.
Property (ii) follows directly from Theorem 4.1(iii)-(iv).
Remark 4.3.
Note that the boundedness property of Theorem 4.1(i) does not require explicitly the condition of Assumption 1() that
is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1(
).
Definition 4.4 (see [8]).
Let the sequence of means be in
, and let
be a representation of a left reversible semigroup
. Then
is
-stable if the functions
and
on
are also in
; for all
, for all
.
Let and
be convex subsets of the Banach space
, with
under proper inclusion, and let
be a retraction of
onto
. Then
is said to be sunny if
; for all
, for all
provided that
.
Definition 4.6.
is said t be a sunny nonexpansive retract of
if there exists a sunny nonexpansive retraction
of
onto
.
It is known that if is weakly compact,
is a mean on
(see Definition 3.5), and
is in
for each
, then there is a unique
such that
for each
. Also, if
is smooth, that is, the duality mapping
of
is single valued then a retraction
of
onto
is sunny and nonexpansive if and only if
, for all
[6, 11].
Remark 4.7.
Note that Theorem 4.2 proves the convergence to a fixed point in , with
being constructively proven to be nonempty by first building a sufficiently large convex compact
so that the solution of the iterative scheme (4.1) is always bounded on
. Note also that Theorems 4.1 and 4.2 need not the assumption of
being a left-invariant
-stable subspace of containing "
" and to be a left-invariant mean on
, although it is assumed to be strongly left regular so that it fulfils
; for all
(Assumption 1(
), see Definition 3.6. However, the convergence to a unique fixed point in the set
is not proven under those less stringent assumptions. Note also that Assumption 1(
) required by Theorem 4.1 and also by Theorem 4.2 as a result is one of the two properties associated with the
-stability of
.
The results of Theorems 4.1 and 4.2 with further considerations by using Definitions 4.4 and 4.5 allow to obtain the convergence to a unique fixed point under more stringent conditions for the semigroup of self-mappings,
as follows.
Theorem 4.8.
If Assumption 1 hold and, furthermore, is a left-invariant
-stable subspace of
then the sequence
, generated by (4.1), converges strongly to a unique
; for all
, for all
, for all
which is the unique solution of the variational inequality
. Equivalently,
where
is the unique sunny nonexpansive retraction of
onto
.
The proof follows under similar tools as those used in [6] since is a nonempy sunny nonexpansive retract of
which is unique since
is nonexpansive for all
.
Proof.
Let be the sequence solution generated by the particular iterative scheme resulting from (4.1) for any initial conditions
when all the functions
and
are zeroed. It is obvious by the calculation of the recursive solution of (4.1) from (4.19) that the error from both solutions satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ33_HTML.gif)
Since the convergence of the solution to fixed points of Theorems 4.1, 4.2, and 4.8 follows also for the sequence it follows that a unique fixed point exists satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ34_HTML.gif)
where is unique since
is also unique from Theorem 4.8. Assume that
with
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ35_HTML.gif)
If and
; for all
and the
-functions are zero then both fixed points are related by the constraint
. Thus, consider a representation
of a left reversible semigroup
as Lipschitzian mappings on
(see Definitions 3.2 and 3.3), a nonempty compact subset of the smooth Banach space
with Lipschitz constants
which is asymptotically nonexpansive. Consider the iteration scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ37_HTML.gif)
with , where
(i) is a strongly left-regular sequence of means on
, that is,
(the dual of
). See Definitions 3.5 and 3.6;
(ii) is a left reversible semigroup represented as Lipschitzian mappings on
by
.
Assumption 2.
The iterative scheme (4.27) keeps the applicable parts of Assumptions 1()–1(
), 1(
) for the nonidentically zero parameterizing sequences
,
and
. Assumptions 1(
) and 1(
) are modified with the replacements
,
, and
.
Theorems 4.1 and 4.8 result in the following result for the iterative scheme (4.27) for ,
:
Theorem 4.9.
The following properties hold under Assumption 2.
(i); for all
. Also,
and
for any norm defined on the smooth Banach space
and there exists a nonempty bounded compact convex set
such that the solution of (4.2) is permanent in
, for all
and some sufficiently large finite
with
.
(ii) and
as
.
(iii)
(iv)Assume that the nonempty convex subset of the smooth Banach space
, which contains the sequence
of means on
, is such that each element
; for all
, for some
so that (4.1) is a positive viscosity iteration scheme (4.27). Then,
(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all
, for all
), denoted as
; for all
.
(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point
.
(iv.3)Property (iii) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ38_HTML.gif)
what implies that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ39_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ40_HTML.gif)
(v)If, furthermore, is a left-invariant
-stable subspace of
, then the sequence
, generated by (4.27), converges strongly to a unique
; for all
, for all
which is the unique solution of the variational inequality
. Equivalently,
where
is the unique sunny nonexpansive retraction of
onto
. Furthermore, the unique fixed points of the iterative schemes (4.1) and (4.27) are related by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F314581/MediaObjects/13663_2009_Article_1130_Equ41_HTML.gif)
If, in addition, and
and the
-functions are identically zero in the iterative scheme (4.1), then
.
Remark 4.10.
Note that the results of Section 4 generalize those of Section 2 since the iterative process (4.1) possesses simultaneously a nonlinear contraction and a nonexpansive mapping plus terms associated to driving terms combining both external driving forces plus the contribution of a nonlinear function evaluating distances over, in general, distinct metric spaces than that generating the solution of the iteration process. Therefore, the results about fixed points in Theorem 2.1(vi)-(vii) are directly included in Theorem 4.1.
Venter's theorem can be used for the convergence to the equilibrium points of the solutions of the generalized iterative schemes (4.1) and (4.27), provided they are positive, as follows.
Corollary 4.11.
Assume that
()
are both contractive mappings with
being compact and convex,
, such that
is a left-invariant
-stable subspace of
with
being a left reversible semigroup;
()
, with
being compact and convex,
,
,
and
; for all
for some real constants
, and
if
;
()
and
.
Then, the sets of fixed points of the positive iteration schemes (4.1) and (4.27) contain a common stable equilibrium point which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is,
and that
.
Outline of Proof
The fact that the mappings are both contractive,
and
imply that the generated sequences
,
are both nonnegative and bounded for any
and they have unique zero limits from Theorem 2.1(v).
The following result is obvious since if the representation is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.
Corollary 4.12.
If the representation is nonexpansive, contractive or asymptotically contractive, then Theorems 4.1, 4.2, and 4.8 still hold under Assumption 1, and Theorem 4.9 still holds under Assumption 2.
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Acknowledgments
The author is very grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2006-00714. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07 and SAIOTEK S-PE08UN15. The author is also grateful to the reviewers for their interesting comments which helped him to improve the final version of the manuscript.
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De la Sen, M. Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme. Fixed Point Theory Appl 2009, 314581 (2009). https://doi.org/10.1155/2009/314581
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DOI: https://doi.org/10.1155/2009/314581