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.
-times reasonable expansive mapping in a complete metric space is defined in [
-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., [
of the Banach space 
. The final iteration process investigated in [
, 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 [
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.
is generated by
, where 
.
and 
are bounded real constants and 
; for all 
. Then, the following properties hold.
, 
, 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.
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 
.
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 
.
; 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 
.
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 
.
be a sequence in 
with 
, and let 
and 
be bounded sequences. Then, 
.
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 
.
if 
, and equivalently, if 
and 
with 
, and
, 
and 
. If, in addition, 
and 
with 
then 
and 
is a monotonically decreasing sequence, 
and 
as 
. Property (i) has been proven.
, 
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.
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 
.
; for all 
and 
, then from recursive evaluation of (2.1): 
and 
; for all 
, then
and 
, for all 
for some 
, then 
, for all 
; thus, 
is monotonically strictly decreasing so that it converges asymptotically to zero.
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 [
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., [
, respectively, 
for the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined by 
.
is a left reversible semigroup if 
; for all 
.
" 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, [
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 
.
may be nonexpansive, asymptotically nonexpansive, contractive and asymptotically contractive according to Definitions 3.3 and 3.4 which follow.
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, [
if it holds the uniform Lipschitzian condition 
(resp., 
) on the Lipschitz constants.
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.
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 
.
is a sequence of means on 
if 
.
on 
is
; for all 
, for all 
, for all 
in 
for 
;
, for all 
, where 
is the adjoint operator of 
defined by 
; for all 
, for all 
.
is said to be left (resp., right)-amenable if it has a left (resp., right)-invariant mean. A general viscosity iteration process considered in [
, 
, and 
have elements in 
of sum being identity, for all 
;
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 
;
is a contraction on 
.
which is the solution of a variational inequality [
and 
and also that proposed in [
, 
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., [
to a fixed point of 
for any given 
if 
for 
[
and 
as 
, and 
[
, 
, and 
with 
, for all 
, then the resulting particular iteration process (3.1) becomes the difference equation (
; for all 
.
. 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.
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
, where
is a strongly left-regular sequence of means on 
, that is, 
. See Definition 3.5;
is a left reversible semigroup represented as Lipschitzian mappings on 
by 
.
) 
, 
, 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 [
, defined by
) 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
, 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.
) 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 (
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.
. 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 
.
and 
as 
.
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,
is a nonnegative sequence (i.e., all its components are nonnegative for all 
, for all 
), denoted as 
; for all 
.
and Property (ii) also holds for a limiting point 
.
from the constraint Assumption 1(
), we have the following:
is an arbitrary finite sufficiently large integer, and
are continuous on 
with 
and 
as 
, [
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 
:
as 
. One gets from (4.12) into (4.10),
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 [
from Assumption 1(
) since 
, 
, and 
as 
. From (4.1),
, 
, 
, 
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
, for some positive real sequences
, 
, and 
satisfying 
and 
, 
and 
as 
with 
and 
being arbitrarily small for sufficiently large 
, and
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
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).
which is the representation as Lipschitzian mappings on 
of a left reversible semigroup 
with identity.
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.
.
so that there exists 
such that 
, 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 
.
) that 
is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1(
).
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 
.
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 
.
is said t be a sunny nonexpansive retract of 
if there exists a sunny nonexpansive retraction 
of 
onto 
.
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 
[
, 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 
.
, 
as follows.
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 
.
is a nonempy sunny nonexpansive retract of 
which is unique since 
is nonexpansive for all 
.
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
it follows that a unique fixed point exists satisfying
is unique since 
is also unique from Theorem 4.8. Assume that 
with 
. Then,
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:
, where
is a strongly left-regular sequence of means on 
, that is, 
(the dual of 
). See Definitions 3.5 and 3.6;
is a left reversible semigroup represented as Lipschitzian mappings on 
by 
.
)–1(
), 1(
) for the nonidentically zero parameterizing sequences 
, 
and 
. Assumptions 1(
) and 1(
) are modified with the replacements 
, 
, and 
.
, 
:
; 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 
.
and 
as 
.
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,
is a nonnegative sequence (i.e., all its components are nonnegative for all 
, for all 
), denoted as 
; for all 
.
and Property (ii) also holds for a limiting point 
.
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
and 
and the 
-functions are identically zero in the iterative scheme (4.1), then 
.
) 
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 
.
which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is, 
and that 
.
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).
is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.
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.