# String-Averaging Methods for Best Approximation to Common Fixed Point Sets
of Operators: The Finite and Infinite Cases

Yair Censor and Ariel Nisenbaum
Department of Mathematics
The University of Haifa
Haifa 3498838, Israel Corresponding author: Yair Censor, Email:
<EMAIL_ADDRESS>
(December 22, 2020. Revised: March 5, 2021.)

###### Abstract

String-averaging is an algorithmic structure used when handling a family of
operators in situations where the algorithm at hand requires to employ the
operators in a specific order. Sequential orderings are well-known and a
simultaneous order means that all operators are used simultaneously (in
parallel). String-averaging allows to use strings of indices, constructed by
subsets of the index set of all operators, to apply the operators along these
strings and then to combine their end-points in some agreed manner to yield
the next iterate of the algorithm. String-averaging methods were discussed and
used for solving the common fixed point problem or its important special case
of the convex feasibility problem. In this paper we propose and investigate
string-averaging methods for the problem of best approximation to the common
fixed point set of a family of operators. This problem involves finding a
point in the common fixed point set of a family of operators that is closest
to a given point, called an anchor point, in contrast with the common fixed
point problem that seeks any point in the common fixed point set.

We construct string-averaging methods for solving the best approximation
problem to the common fixed points set of either finite or infinite families
of firmly nonexpansive operators in a real Hilbert space. We show that the
simultaneous Halpern-Lions-Wittman-Bauschke algorithm, the Halpern-Wittman
algorithm and the Combettes algorithm, which were not labeled as string-
averaging methods, are actually special cases of these methods. Some of our
string-averaging methods are labeled as “static” because they use a fixed pre-
determined set of strings. Others are labeled as “quasi-dynamic” because they
allow the choices of strings to vary, between iterations, in a specific manner
and belong to a finite fixed pre-determined set of applicable strings. For the
problem of best approximation to the common fixed point set of a family of
operators, the full dynamic case that would allow strings to unconditionally
vary, between iterations, remains unsolved although it exists and is validated
in the literature for the convex feasibility problem where it is called
“dynamic string-averaging”.

Keywords: String-averaging; common fixed points; best approximation problem;
firmly nonexpansive operators; HLWB algorithm; Halpern theorem; projection
methods.

##  1 Introduction

String-averaging algorithmic structures are used for handling a family of
operators in situations where the algorithm needs to employ the operators in a
specific order. String-averaging allows to use strings of indices, taken from
the index set of all operators, to apply the operators along these strings and
to combine their end-points in some agreed manner to yield the next iterate of
the algorithm.

If a point $p$ of a set $C$ is nearest to a given point $x$ in space, known as
the anchor, then $p$ is a best approximation to $x$ from $C$. In the case when
$C$ is the common fixed points set of a family of self-mapping operators, the
problem of finding such a $p$ is known as the best approximation problem
(BAP). In case the fixed point sets are all convex, this problem is a special
case of two well-known problems: The convex feasibility problem (CFP), which
is to find a (any) point in the intersection of closed convex sets, and the
common fixed point problem (CFPP), where the closed convex sets in CFP are the
fixed point sets of operators of a given family. The CFP, the CFPP and BAP are
widely studied, and are useful in mathematics and various physical science
(see, e.g., Bauschke and Borwein [6], Reich and Zalas [33] and Cegielski [13],
to name but a few), including in the context of string-averaging for the CFP
and the CFPP (see, e.g., Censor and Zaslavski [19, 20], Bargetz, Reich and
Zalas [3] and [33]).

Nevertheless, string-averaging algorithmic approaches for solving the BAP
were, to the best of our knowledge, not proposed neither investigated.
Motivated by this, we devote our research presented here, to developing
string-averaging algorithmic schemes for finding the best approximation to the
common fixed points set of a, either finite or infinite, family of firmly
nonexpansive operators. Besides pure mathematical interest, see, e.g., Dye,
Khamsi and Reich [27] and references therein, motivation for the infinite case
can also come from practical real-world situations. This point is succinctly
made in the introduction of the recent paper of Kong, Pajoohesh and Herman
[30]. Although they refer to the infinite CFP their arguments can serve also
to justify the infinite case for the BAP. The case of infinitely many sets is
useful in applications where there is a potential infinity of samples or
measurements, each one of which gives rise to a convex set that contains the
point we wish to recover, see, for example Blat and Hero [10].

###  1.1 Contribution and structure of the paper

Our string-averaging methods are applied to families of firmly nonexpansive
operators and they follow the principles of the original string-averaging
process suggested by Censor, Elfving and Herman in [16]. Consider a set
$\mathcal{M}$ containing all pairs of the form $(\varOmega,w)$, where
$\varOmega$ is a set of finite strings of indices, by which the string
operators are formed, and $w$ is a function that attaches to every string
$t\in\varOmega$ a positive real weight $w(t)$ such that the sum of weights
equals $1$.

We construct what we call a “static” string-averaging methods in which a fixed
pre-determined single pair $(\varOmega,w)\in\mathcal{M}$ is used throughout
the iterative process. This approach solves the BAP for a common fixed points
set of either a finite or an infinite family of firmly nonexpansive operators
and we show that the simultaneous Halpern-Lions-Wittman-Bauschke algorithm
(see, e.g., in Censor [14, Algorithm 5]), the Halpern-Wittman algorithm (see,
e.g., [6, Algorithm 4.1]) and the Combettes algorithm [21], which were labeled
as sequential and simultaneous algorithms, respectively, are special cases of
our static string-averaging methods.

We extend the “static” notion in the finite case to the situation where a
finite number of pairs $(\varOmega,w)\in\mathcal{M}$ can be used, and call it
a “quasi” dynamic string-averaging method due to its resemblance to the
dynamic string-averaging scheme (see [19]). This is done by ordering a finite
pre-defined set of pairs $(\varOmega,w)\in\mathcal{M}$, and forming a finite
family of operators of the form $\sum_{t\in\Omega}w(t)T[t](x)$, for every such
$(\varOmega,w)$, where $T[t]$ are string operators. These are, in turn, used
in a cyclic manner.

With the aid of a construction similar to the quasi dynamic string-averaging
algorithm, we propose a simultaneous string-averaging method.

We focus here on Halpern’s algorithm and the Halpern-Lions-Wittman-Bauschke
algorithm. There are, however, several other iterative processes for solving
the BAP, which are not treated here, but can possibly be also extended via the
string-averaging algorithmic concept, e.g., Dykstra’s algorithm, see [7,
Subsection 30.2] and Haugazeau’s method, see [7, Subsection 30.3], where
further references can be found.

The results presented here for the infinite case complement our earlier work
in Aleyner and Censor [1] where a sequential algorithm for solving the BAP to
the common fixed points set of a semigroup of nonexpansive operators in
Hilbert space was studied. Only a sequential iteartive process was
investigated there but the framework was more general due to the kind of
operators (nonexpansive) and the size of the pool of opeartors (a semigroup,
not limited to the countable case).

This is a theoretical work in the spirit of the theoretical developments in
fixed point theory presented in many of the earlier referenced papers. The
string-averaging approach is actually not a single algorithm but an
“algorithmic scheme” so that every individual choice of lengths and
assignments of strings will give rise to a different algorithm. In this way
our string-averaging algorithmic scheme generalizes earlier algorithms that
become special cases of it. It is not practical to conduct a numerical
experiment without having a specific scientific or real-world problem in hand
but it is expected that researchers who need to solve the best approximation
problem to common fixed point sets will find here valuable algorithmic
information.

The paper is structured as follows. After preliminaries in Section 3, the work
is divided into two main parts: A first part where the given family of firmly
nonexpansive operators is finite, in Sections 5 and 6, and a second part where
the given family of firmly nonexpansive operators is a countable family, in
Section 7. The static, the quasi-dynamic and the simultaneous string-averaging
methods can be found in Subsections 5.1, 5.2 and 5.3, respectively. In
Subsection 6, we show for which choices of pairs ($\varOmega,w)$, the
simultaneous version of the Halpern-Lions-Wittman-Baucschke algorithm and the
Halpern-Wittman algorithm are special cases of our static string-averaging
approach. In Subsection 7.1, we propose our static string-averaging method for
solving the BAP in the infinite case, from which the well-known Combettes
algorithm follows.

##  2 Previous related works

An early approach, based on projection operators, is John von Neumann’s
alternating projection method [34], for solving the BAP with two closed linear
subspaces. It has been widely studied and generalized by many authors, see,
e.g., Bauschke and Borwein [5], Deutsch [23], Kopecká and Reich [31] and
Deutsch and Hundal [25]. For the general case of arbitrary convex sets, the
Dykstra’s algorithm is a suitable modification of the alternating projections
method to solve the BAP. This algorithm was first introduced by Dykstra in
[28] for closed and convex cones in finite-dimensional Euclidean spaces, and
later extended by Boyle and Dykstra in [11] for closed and convex sets in a
Hilbert space. Additional projections approaches can be found in Aragón
Artacho and Campoy [2] and Bergman, Censor, Reich, and Zepkowitz-Malachi [9].

For a recent bibliography of papers and monographs on projection methods see
Censor and Cegielski [15]. A well-known method for solving the more general
BAP is Halpern’s algorithm [29], whose strong convergence to the solution,
under various sets of assumptions on the parameters, has been proved by
several authors. The main contributions are due to Lions, to Wittman and to
Bauschke, see Bauschke’s paper [4]. Additional literature background of the
BAP and related problems appear in the excellent literature review of [2],
from which we adapted some of the above.

The literature on string-averaging algorithms has also expanded since its
first presentation in [16] and further related work has been published. In
Crombez [22], a string-averaging algorithm for solving the common fixed point
problem for a finite family of paracontracting operators is proposed. Censor
and Segal in [17] suggested a string-averaging solution for the common fixed
point problem for a finite family of sparse operators. In Censor and Tom [18],
a study on the behavior of a string-averaging algorithm for inconsistent
convex feasibility problems is done.

A generalized notion of string-averaging is the notion of dynamic string-
averaging, in which one is able to use in the iterative process pairs
$(\varOmega,w)$ from a pre-defined set, denoted by
$\mathcal{M}_{*}\subset\mathcal{M}$ (see, e.g., [20]). In every
$(\varOmega,w)\in\mathcal{M}_{*}$, the length of every $t\in\varOmega$ is
bounded and $w(t)$ is bounded away from zero. An example of a dynamic string-
averaging algorithm can be found in [19]. That iterative process generates a
convergent sequence whose limit is a point in the intersection of a finite
family of closed convex sets, namely, a solution to the CFP. Recently, Censor,
Brooke and Gibali in [12] were able to construct a dynamic string-averaging
method for solving the multiple-operator split common fixed point problem
(see, e.g., [12, Problem 1]) for families of cutters in Hilbert spaces.

##  3 Preliminaries

Throughout our work we denote by $H$ a real Hilbert space with the inner
product $\left\langle\cdot\,,\,\cdot\right\rangle$ and induced norm
$\|\cdot\|$, and by $\mathbb{N}$ the set of all natural numbers including
zero. Let $D\subseteq H$ be a nonempty set and let $T:D\rightarrow H$ be an
operator. A vector $x\in D$ is a fixed point of $T$ if it satisfies $T(x)=x.$
The set of all fixed points of $T$ is denoted by $Fix(T):=\left\\{x\in
D\,|\,T(x)=x\right\\}.$ If $D$ is a closed convex set, then for every $u\in H$
there exists a unique point $y\in D$ such that

$\|u-y\|=\text{inf}\left\\{\|u-d\|\,|\,d\in D\right\\}.$ (1)

Such a point $y$ is called the projection of $u$ onto $D$ and denoted by
$P_{D}(u).$

We now recall some definitions regarding various classes of operators.

######  Definition 3.1.

Let $D$ be a nonempty subset of $H$ and let $T:D\rightarrow H$. Then $T$ is:
(i) Nonexpansive (NE) if

$\|T(x)-T(y)\|\leq\|x-y\|,\>\forall x,y\in D.$ (2)

(ii) Firmly nonexpansive (FNE) if

$\|T(x)-T(y)\|^{2}\leq\left\langle x-y,T(x)-T(y)\right\rangle,\>\forall x,y\in
D.$ (3)

(iii) Quasi-nonexpansive (QNE) if

$\|T(x)-y\|\leq\|x-y\|,\>\forall x\in D,\>y\in Fix(T).$ (4)

(iv) Strictly quasi-nonexpansive (sQNE) if

$\|T(x)-y\|<\|x-y\|,\>\forall x\in D\setminus Fix(T),\>y\in Fix(T).$ (5)

(v) $C$-strictly quasi-nonexpansive ($C$-sQNE) if $T$ is quasi-nonexpansive
and

$\|T(x)-y\|<\|x-y\|,\>\forall x\in D\setminus Fix(T),\>y\in C,$ (6)

where $C\neq\emptyset$ and $C\subseteq Fix(T)$.

A useful fact which can be found, e.g., in [13, Proposition 2.1.11], is that
the fixed point set of a nonexpansive operator is a closed convex set. Since
an intersection of any collection of closed convex sets is a closed convex set
(see, e.g., Lemma 1.13 and Example 2.3 in Deutsch [24]) it follows that, the
set of common fixed point sets of a family of nonexpansive operators of any
cardinality is a closed convex set and, thus, the projection of any given
point $u$ onto this set is well-defined, provided that the intersection is
nonempty.

All our string-averaging methods use sequences of real numbers, called
steering sequences.

######  Definition 3.2.

(steering sequences). A real sequence $(\lambda_{k})_{k\in\mathbb{N}}$ is
called a steering sequence if it has the following properties:

$\lambda_{k}\in[0,1]\enspace\text{for}\enspace\text{all}\enspace\,k\geq
0,\enspace\text{and}\enspace\underset{k\rightarrow\infty}{lim}\lambda_{k}=0,$
(7)
$\sum_{k=0}^{\infty}\lambda_{k}=+\infty\,\text{(\text{or,}\enspace\text{equivalently,
}}\prod_{k=0}^{\infty}(1-\lambda_{k})=0),$ (8)
$\sum_{k=0}^{\infty}|\lambda_{k+1}-\lambda_{k}|<\infty.$ (9)

Observe that although $\lambda_{k}\in[0,1]$ the definition rules out the
option of choosing all $\lambda_{k}$ equal to zero or all equal to one because
of contradictions with the other properties. Infinitely many zeros are
possible only if the remaining nonzero elements obey all properties. The third
property in (9) was introduced by Wittmann, see, e.g., the recent review paper
of López, Martin-Márquez and Xu [32].

Lemma 3.3 below is composed of several known claims which will be used in the
sequel. We supply pointers to the proof of the lemma for completeness.

######  Lemma 3.3.

Let $D$ be a nonempty closed convex subset of $H$ and let $T:D\rightarrow D$
be FNE. Then
(i) T is NE.
(ii) If $Fix(T)\neq\emptyset$ then T is QNE.
(iii) If $Fix(T)\neq\emptyset$ then T is sQNE.

###### Proof.

For (i) and (ii) see [13, Theorem 2.2.4] and [13, Lemma 2.1.20], respectively.
(iii) follows from the statement on page 70 of [7]. ∎

##  4 String-averaging methods for best approximation to the common fixed
points set of a family of firmly nonexpansive operators: General

In our present work we consider the best approximation problem with respect to
the common fixed points set of a family of FNEs. Our overall aim is to develop
and investigate string-averaging algorithms for this problem. In the string-
averaging algorithmic scheme, one constructs from a given family of operators,
a family of, so-called, string-operators which are certain compositions of
some of the operators from the given family.

According to these the string-averaging algorithm proceeds in its iterative
process. We show that such string-averaging algorithmic schemes converge to
the projection of a given point (commonly called the “anchor”) to the common
fixed point set of the given family.

We are able to ensure convergence of our string-averaging methods by demanding
that the operators of the given family be FNEs. There are well-known links
between the classes of FNEs, NEs, sQNEs and QNEs, defined above, that help us
in our analysis. To take advantage of these links we use Corollary 4.50 and
Proposition 4.47 in Bauschke and Combettes’ book [7], which are proved for a
finite family of sQNEs and a finite family of QNEs, respectively. We extend
the usage of [7, Proposition 4.47] to the countable case in order to
determine, in Section 7, the point to which our string-averaging algorithm
converges to.

##  5 String-averaging methods for best approximation to the common fixed
points set of a family of firmly nonexpansive operators: The finite case

In our work we develop string-averaging algorithms for two distinct
situations. One is the finite case wherein the family of given FNEs is finite.
The other is when the family of given FNEs is countably infinite. In this
section we consider the finite case. We start by defining the terms which we
use throughout this section.

######  Definition 5.1.

Let $D$ be a nonempty closed convex subset of $H$, let $(T_{i})_{i=1}^{m}$ be
a finite family of self-mapping operators on $D.$ An index vector is a vector
of the form $t=(t_{1},t_{2},\ldots,t_{p})$ such that
$t_{\ell}\in\\{1,2,\ldots,m\\}$ for all $\ell\in\\{1,2,\ldots,p\\}$. For a
given index vector $t=(t_{1},t_{2},\ldots,t_{q})$ we denote its length (i.e.,
the number of its components) by $\gamma(t)=q$, and define the operator $T[t]$
as the composition of the operators $T_{i}$ whose indices appear in the index
vector $t$, namely,

$T[t]:=T_{t_{q}}T_{t_{q-1}}\cdots T_{t_{1}},$ (10)

and call it a string operator. A finite set $\varOmega$ of index vectors is
called fit if for each $i\in\left\\{1,2,\ldots,m\right\\}$, there exists a
vector $t=(t_{1},t_{2},\ldots,t_{p})\in\varOmega$ such that $t_{\ell}=i$ for
some $\ell\in\left\\{1,2,\ldots,p\right\\}$. As in [19], we denote by
$\mathcal{M}$ the collection of all pairs $(\varOmega,w)$, where $\Omega$ is a
fit finite set of index vectors and $w:\Omega\rightarrow(0,1]$ is such that
$\sum_{t\in\varOmega}w(t)=1$.

As mentioned above, [7, Corollary 4.50] and [7, Proposition 4.47] are
cornerstones to our proofs of convergence. The following are slight rephrased
versions of them, respectively, adapted to our notations and needs.

######  Proposition 5.2.

Let $D$ be a nonempty subset of $H$ and let $(T_{i})_{i=1}^{m}$ be a finite
family of self sQNEs on $D$ such that $\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$
and set $T=T_{1}T_{2}\cdots T_{m}$. Then $T$ is sQNE and
$Fix(T)=\cap_{i=1}^{m}Fix(T_{i})$.

######  Proposition 5.3.

Let $D$ be a nonempty subset of $H$, let $(T_{i})_{i=1}^{m}$ be a finite
family of self QNEs on $D$ such that $\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$
and let $(w_{i})_{i=1}^{m}$ be a sequence of strictly positive real numbers
such that $\sum_{i=1}^{m}w_{i}=1$. Then
$Fix(\sum_{i=1}^{m}w_{i}T_{i})=\cap_{i=1}^{m}Fix(T_{i})$.

The following lemma combines all of the above into a useful auxiliary tool
which is used repeatedly. We consider the operator
$T:=\sum_{t\in\Omega}w(t)T[t]$ which is called a string-averaging operator.

######  Lemma 5.4.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i=1}^{m}$
be a finite family of self FNEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$. Let
$(\varOmega,w)\in\mathcal{\mathcal{M}}$ and let $T=\sum_{t\in\Omega}w(t)T[t]$.
Then $Fix(T)=\cap_{i=1}^{m}Fix(T_{i})$.

###### Proof.

Note that $F\neq\emptyset$ and (10) imply that for every $t\in\varOmega$ and
for every $x\in F$, $T[t](x)=x$ (since $T[t]$ is a composition). Hence,

$F\subseteq Fix(T[t]).$ (11)

Therefore,

$F\subseteq\cap_{t\in\Omega}Fix(T[t]).$ (12)

We want to use Proposition 5.3 for the family $(T[t])_{t\in\varOmega}$. Since
Every FNE is NE (see Lemma 3.3(i)), for every $t\in\varOmega$ we apply [13,
Lemma 2.1.12(ii)] to the family $(T_{t_{\ell}})_{\ell=1}^{q}$ and conclude
that the string operator $T[t]=T_{t_{q}}T_{t_{q-1}}\cdots T_{t_{1}}$ is NE.
Thus, from (11) and due to the fact that every NE with a fixed point is QNE,
the family of operators $(T[t])_{t\in\varOmega}$ is a finite family of QNEs.
This, together with (12), yields, according to Proposition 5.3, that

$Fix(T)=\cap_{t\in\varOmega}Fix(T[t]).$ (13)

$F\neq\emptyset$ implies that $Fix(T_{i})\neq\emptyset$, thus, by Lemma
3.3(iii), $(T_{i})_{i=1}^{m}$ is a family of sQNEs and, so, for every
$t\in\varOmega$, applying Proposition 5.2 with the family
$(T_{t_{\ell}})_{\ell=1}^{q}$ yields that the string operator $T[t]$ satisfies

$Fix(T[t])=\cap_{\ell=1}^{q}Fix(T_{t_{\ell}}).$ (14)

From the fitness of $\varOmega$ and from (14), we get

$\cap_{t\in\varOmega}Fix(T[t])=\cap_{t\in\varOmega}(\cap_{\ell=1}^{q}Fix(T_{t_{\ell}}))=\cap_{i=1}^{m}Fix(T_{i}),$
(15)

and, overall, from (13)

$Fix(T)=\cap_{i=1}^{m}Fix(T_{i}).$ (16)

∎

We are now ready to propose the new string-averaging methods for solving the
best approximation problem and prove their convergence.

###  5.1 The static string-averaging method for the finite case

First we discuss string-averaging methods in which a single pair
$(\varOmega,w)\in\mathcal{\mathcal{\mathcal{M}}}$ is picked at the outset and
kept fixed throughout the iterative process. Such string-averaging methods
will be termed “static string-averaging methods”. We will make use of the
convergence theorem in [29]. Halpern’s algorithm is a sequential algorithm
which generates a sequence via the iterative process

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})S(x^{k}),$ (17)

where $S:D\rightarrow D$ is NE with $Fix(S)\neq\emptyset,$ the anchor point
$u\in D$ is given and fixed, the initialization $x^{0}\in D$ is an arbitrary
point, and the sequence $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering
sequence as in Definition 3.2. Its proof of convergence can be found also in
[7, Theorem 30.1]. We present a slightly rephrased version of this theorem
without a proof.

######  Theorem 5.5.

Let $D$ be a nonempty closed convex subset of $H$ and let $S$ be a self NE on
$D$ such that $Fix(S)\neq\emptyset$. Let $(\lambda_{k})_{k\in\mathbb{N}}$ be a
steering sequence and let $u,x^{0}\in D$. Then any sequence generated by (17)
converges strongly to $P_{Fix(S)}(u)$.

Our new static string-averaging algorithm for a finite family of FNEs is as
follows.
Algorithm 1. The static string-averaging algorithm for solving the best
approximation problem in the finite case.
Initialization: Choose a single pair $(\varOmega,w)\in\mathcal{M}$ and an
arbitrary $x^{0}\in D.$
Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{t\in\Omega}w(t)T\left[t\right]\left(x^{k}\right),$
(18)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $w(t)$ and $T[t]$ are as in Definition 5.1.

The convergence proof of Algorithm 1 follows.

######  Theorem 5.6.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i=1}^{m}$
be a finite family of self FNEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$. Let
$(\varOmega,w)\in\mathcal{\mathcal{\mathcal{M}}}$ be fixed, let
$(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence and let $u,x^{0}\in
D$. Then any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 1,
converges strongly to $P_{F}(u)$.

###### Proof.

For $(T_{i})_{i=1}^{m}$ and $(\varOmega,w)\in\mathcal{\mathcal{M}}$ consider
the family of operators $(T[t])_{t\in\varOmega}$ and define the string-
averaging operator $T:=\sum_{t\in\varOmega}w(t)T\left[t\right]$. We show first
that the operator $T$ is NE and that $Fix(T)=\cap_{i=1}^{m}Fix(T_{i})$. From
the proof of Lemma 3.3 we conclude that $T$ is a convex combination of NEs
and, thus, since a convex combination of NEs is a NE (see, e.g., [13, Lemma
2.1.12(i)]), $T$ is NE. Moreover, $Fix(T)$ is not empty since it contains $F.$
Applying Halpern’s Theorem 5.5 with $T$ in the role of $S$, any sequence
$(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 1, converges strongly to
$P_{Fix(T)}(u)$. Now, applying Lemma 5.4 on $(T_{i})_{i=1}^{m}$ together with
$(\varOmega,w)$, results in

$Fix(T)=\cap_{i=1}^{m}Fix(T_{i}),$ (19)

and, therefore,

$x^{k}\rightarrow P_{F}(u).$ (20)

∎

This concludes our treatment of the static string-averaging algorithm for
solving the best approximation problem in the finite case. We make the
following remark about it.

######  Remark 5.7.

Theorem 5.6 is related to two important results that appear in [7]. It
generalizes Corollaries 30.2 and 30.3 in that book, from the algorithmic
structural point of view, because the algorithms there are fully-simultaneous
and fully-sequential, respectively. These two algorithmic options are special
cases of the static string-averaging algorithm that are obtained by choosing
either strings of length one with every index $i=1,2,\ldots,m$ appearing
exactly in one string or by choosing to use a single string that includes all
indices $i=1,2,\ldots,m,$ respectively. However, our Theorem 5.6 cannot be
considered as a generalization of those corollaries because the corollaries
deal with NEs while we restrict our analysis of the string-averaging
algorithmic structure to only FNEs. The question whether or not our Theorem
5.6 can be proven for NEs remains open.

Next we expand our results to a non-static case.

###  5.2 The quasi-dynamic string-averaging method for the finite case

The key adjustment which we did in the construction of Algorithm 1 in order to
prove its convergence with the aid of Halpern’s algorithm, was the repeated
use of the same single fixed pair $(\varOmega,w)$ in all iterations $k\geq 1$.
This reuse of a fixed $(\varOmega,w)$ throughout the whole iterative process
of Algorithm 1 is a special case of a more general method, mentioned briefly
already in Section 2, called the dynamic string-averaging method. The dynamic
string-averaging algorithmic scheme allows to pick and use in every step of
the iterative process any pair $(\varOmega,w)$ from a pre-defined set
$\mathcal{M}_{*}\subset\mathcal{M}$. The set $\mathcal{M}_{*}$ was defined in
[19, Equation (21)] as follows.

######  Definition 5.8.

Fix a number $\Delta\in(0,\,1/m)$ and an integer $\bar{q}\geq m$ and denote by
$\mathcal{M}_{*}\equiv\mathcal{M}_{*}(\Delta,\bar{q})$ the set of all
$(\varOmega,w)\in\mathcal{M}$ such that the lengths of the strings are bounded
and the weights are bounded away from zero, namely,

$\mathcal{M}_{*}:=\,\left\\{(\varOmega,w)\in\mathcal{M}\,|\,\gamma(t)\leq\bar{q}\>\text{and}\,\Delta\leq
w(t),\,\text{for}\,\text{all}\,t\in\varOmega\right\\}.$ (21)

The set $\mathcal{M}_{*}$ is an infinite subset of $\mathcal{M}$ but in our
quasi-dynamic string-averaging method, proposed below, we must confine
ourselves to a finite subset of $\mathcal{\mathcal{M}}.$ So, instead of
choosing the pairs $(\varOmega,w)$ from $\mathcal{M}_{*}$ we choose them from
a finite-cardinality subset, denoted by
$\mathcal{M}^{\prime}$$\subset\mathcal{\mathcal{\mathcal{M}}}$, and use them
in a cyclic manner. The finiteness of $\mathcal{M}^{\prime}$ guarantees that
it is actually a subset of $\mathcal{M}_{*}$. As one can tell, such an
algorithm is indeed not as “dynamic” as Algorithm 6 of [19] or Algorithm 3 of
[20] but, nevertheless, it is not static as Algorithm 1. Therefore, we name it
quasi-dynamic string-averaging method.

Next, we explain how the construction of our quasi-dynamic string-averaging
algorithm is done. Let us construct a sequence that is an ordered version of
$\mathcal{M}^{\prime}$ as follows: Let
$\sigma:\mathcal{M}^{\prime}\rightarrow\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$
be a one-to-one correspondence and denote, for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$,
$S_{r}:=(\varOmega_{r},w_{r})$. Let
$\left(S_{r}\right)_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ be a sequence
of all $(\varOmega,w)\in\mathcal{M}^{\prime}$, sorted by
$\sigma((\varOmega,w))$. Namely, $S_{1}$ is the pair $(\varOmega,w)$ such that
$\sigma((\varOmega,w))=1$, $S_{2}$ is the pair $(\varOmega,w)$ such that
$\sigma((\varOmega,w))=2$ and so forth until
$S_{\left|\mathcal{M}^{\prime}\right|}$ is the pair $(\varOmega,w)$ such that
$\sigma((\varOmega,w))=\left|\mathcal{M}^{\prime}\right|$.

The following is our quasi-dynamic string-averaging algorithm.

Algorithm 2. The quasi-dynamic string-averaging algorithm for solving the best
approximation problem in the finite case.
Initialization: $x^{0}\in D$ is arbitrary.

Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{t\in\Omega_{j(k)}}w_{j(k)}(t)T[t](x^{k}),$
(22)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point, $j(k)=k\,mod\left|\mathcal{M}^{\prime}\right|+1$ for all
$k\geq 0,$ is a cyclic control sequence (see, e.g., [17, Definition 4]) and
$\varOmega_{j(k)}$ and $w_{j(k)}$ are the elements of the pair
$S_{j(k)}=(\varOmega_{j(k)},w_{j(k)})$, respectively.

To prove the convergence of Algorithm 2 we make use of the following theorem,
which is a slightly rephrased version of Theorem 3.1 in [4].

######  Theorem 5.9.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i=1}^{m}$
be a finite family of self NEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset.$ Assume that

$\displaystyle F$ $\displaystyle=Fix(T_{m}T_{m-1}\cdots
T_{1})=Fix(T_{1}T_{m}\cdots T_{3}T_{2})$ (23)
$\displaystyle=\ldots=Fix(T_{m-1}T_{m-2}\cdots T_{1}T_{m}).$

Let $i(k)=k\,mod\,m+1$ be a cyclic control sequence, let
$(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence and let $u,x^{0}\in
D$. Then any sequence $(x^{k})_{k\in\mathbb{N}}$ generated by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})T_{i(k)}(x^{k}),$ (24)

converges strongly to $P_{F}(u)$.

With the aid of the sequence
$(S_{r})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ and Theorem 5.9 we present
in the next theorem a proof of convergence of Algorithm 2.

######  Theorem 5.10.

Let $D$ be a nonempty closed convex subset of H and let $(T_{i})_{i=1}^{m}$ be
a finite family of self FNEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$. Let $\mathcal{M}^{\prime}$ be a
finite subset of $\mathcal{\mathcal{\mathcal{M}}},$ let
$j(k)=k\,mod\left|\mathcal{M}^{\prime}\right|+1$ , for all $k\geq 0,$ be a
cyclic control sequence and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a steering
sequence. Let $u,x^{0}\in D.$ Then any sequence $(x^{k})_{k\in\mathbb{N}}$,
generated by Algorithm 2, converges strongly to $P_{F}(u)$.

###### Proof.

Define the finite family of operators
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ by
$T_{S_{r}}:=\sum_{t\in\varOmega_{r}}w_{r}(t)T[t]$ where for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$,
$S_{r}:=(\varOmega_{r},w_{r})$. We first show that for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$ the
operator $T_{S_{r}}$ is both NE and sQNE. By similar arguments to those made
for $T$ in the proof of Theorem 5.6, it follows that
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ is a family of NEs.
From the proof of Lemma 5.4 we deduce that $(T[t])_{t\in\varOmega_{r}}$ is a
family of QNEs such that

$\emptyset\neq F\subseteq\cap_{t\in\varOmega_{r}}Fix(T[t]),$ (25)

for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$.
Therefore, we are able to apply Proposition 5.3 to the family
$(T[t])_{t\in\varOmega_{r}}$ (in place of the family $(T_{i})_{i=1}^{m}$) for
every $r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$,
which results in

$Fix(T_{S_{r}})=\cap_{t\in\varOmega_{r}}Fix(T[t]).$ (26)

Therefore, by (25),

$Fix(T_{S_{r}})\neq\emptyset.$ (27)

Next we show that $T_{S_{r}}$ is sQNE. According to Lemma 3.3(iii), each
$T_{i}$ is sQNE and, hence, similarly to the analysis made in the proof of
Lemma 5.4, with the aid of [7, Corollary 4.50] instead of [13, Lemma
2.1.12(ii)], $T[t]$ is sQNE for every $t\in\varOmega_{r}$, and for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$. In
particular, the latter shows that for a given $r$ and any $t\in\varOmega_{r}$,
we have, due to (26), that for all $x\in D\setminus Fix(T[t])$ and $y\in
Fix(T_{S_{r}})$,

$\|T[t](x)-y\|<\|x-y\|.$ (28)

Thus, $T[t]$ is $C$-sQNE with $C:=Fix(T_{S_{r}}).$ Consequently, from (26),
(27) and [13, Theorem 2.1.26(i)], $T_{S_{r}}$is $C$-sQNE for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$ and, so,
by the definition of $C$ and [13, page 47], $T_{S_{r}}$ is sQNE for every
$r\in\left\\{1,2,\ldots\,,\left|\mathcal{M}^{\prime}\right|\right\\}$.

Therefore, since $\emptyset\neq
F\subseteq\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}Fix(T_{S_{r}})$, we
can apply Proposition 5.2 to the family
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$, which yields that
(23) holds for $(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$. Hence,
since the family $(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ is a
family of NEs and it satisfies (23), we let the role of the family
$(T_{i})_{i=1}^{m}$ in Theorem 5.9 to be played by the family
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$. Moreover, by taking
the sequence $(i(k))_{k\in\mathbb{N}}$ in Theorem 5.9 to be
$(j(k))_{k\in\mathbb{N}}$, (22) turns out to be a special case of (24) and we
deduce that any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by (22),
converges strongly to
$P_{\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}Fix(T_{S_{r}})}(u)$.

Now, (26), (14) and the fitness of $\varOmega$ (recall Definition 5.1) imply
that

$\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}Fix(T_{S_{r}})=\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}(\cap_{t\in\varOmega_{r}}Fix(T[t]))=\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}(\cap_{i=1}^{m}Fix(T_{i}))=F,$
(29)

and, in conclusion,

$x^{k}\rightarrow P_{F}(u).$ (30)

∎

This concludes our treatment of the quasi-dynamic string-averaging algorithm
for solving the best approximation problem in the finite case.

###  5.3 Simultaneous string-averaging methods

One possibility to define a simultaneous string-averaging method was discussed
in Remark 5.7 above and termed “fully simultaneous”. By using a family of
string-averaging operators
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$, as in Subsection 5.2,
and employing an additional weight sequence
$(\hat{w_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ of strictly positive
real numbers such that
$\sum_{r=1}^{\left|\mathcal{M}^{\prime}\right|}\hat{w_{r}}=1$, we can
construct yet another algorithm of a simultaneous nature for solving the best
approximation problem to common fixed point sets of operators in the finite
case. This algorithm convexly-combines via
$(\hat{w_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ the points
$T_{S_{r}}(x^{k})$ which amounts to string-averaging the end-points of the
string operators $T[t]$ for every $t\in\varOmega_{r}$, and for every
$r\in\left\\{1,2,\ldots,\left|\mathcal{M}^{\prime}\right|\right\\}$. The
scheme is as follows.

Algorithm 3. The simultaneous string-averaging algorithm for solving the best
approximation problem in the finite case.
Initialization: $x^{0}\in D.$
Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{r=1}^{\left|\mathcal{M}^{\prime}\right|}\hat{w_{r}}T_{S_{r}}(x^{k})=\lambda_{k}u+(1-\lambda_{k})\sum_{r=1}^{\left|\mathcal{M}^{\prime}\right|}\hat{w_{r}}(\sum_{t\in\varOmega_{r}}w_{r}(t)T[t](x^{k})),$
(31)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and
$(\hat{w_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ are user-chosen
strictly positive real numbers such that
$\sum_{r=1}^{\left|\mathcal{M}^{\prime}\right|}\hat{w_{r}}=1$.

It is possible to obtain the convergence of Algorithm 3 from Theorem 5.6 about
our static string-averaging Algorithm 1\. But this would limit the scope to a
family $(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ of FNEs only.
Therefore, we derive the convergence of Algorithm 3 from Corollary 30.2 in [7]
which holds for NEs. We do this next. Let $D$ be a nonempty closed convex
subset of $H$ and let $(T_{i})_{i=1}^{m}$ be a finite family of self NEs on
$D.$

Algorithm 4. The fully-simultaneous algorithm for the best approximation
problem in Corollary 30.2 of [7].
Initialization: $x^{0}\in D.$

Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{i=1}^{m}w_{i}T_{i}(x^{k}),$ (32)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $(w_{i})_{i=1}^{m}$ is a sequence of user-chosen
strictly positive numbers such that $\sum_{i=1}^{m}w_{i}=1.$

A slightly rephrased version of Corollary 30.2 of [7] is as follows.

######  Theorem 5.11.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i=1}^{m}$
be a finite family of self NEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$. Let $(w_{i})_{i=1}^{m}$ be a
sequence of strictly positive real numbers such that $\sum_{i\in I}w_{i}=1$
and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence. Let
$u,x^{0}\in D$. Then any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by
Algorithm 4, converges strongly to $P_{F}(u)$.

Algorithm 3 is now a special case of Algorithm 4 and its convergence follows
from the above theorem.

######  Theorem 5.12.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i=1}^{m}$
be a finite family of self FNEs on $D$ such that
$F:=\cap_{i=1}^{m}Fix(T_{i})\neq\emptyset$. Let
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ be as in Theorem 5.10,
let $(\hat{w_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ be a sequence of
strictly positive real numbers such that
$\sum_{r=1}^{\left|\mathcal{M}^{\prime}\right|}\hat{w_{r}}=1$, and let
$(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence. Let $u,x^{0}\in D.$
Then any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 3,
converges strongly to $P_{F}(u)$.

###### Proof.

In the proof of Theorem 5.10 we showed that the family
$(T_{S_{r}})_{r=1}^{\left|\mathcal{M}^{\prime}\right|}$ is a family of NEs
with a nonempty common fixed points set and, so, Theorem 5.11 implies that any
sequence $(x^{k})_{k\in\mathbb{N}},$ generated by Algorithm 3, converges
strongly to
$P_{\cap_{r=1}^{\left|\mathcal{M}^{\prime}\right|}Fix(T_{S_{r}})}(u)$. Now,
from (29), we conclude that

$x^{k}\rightarrow P_{F}(u).$ (33)

∎

##  6 String-averaging with orthogonal projections for the best approximation
to a finite family of closed convex sets

In this section we specialize our results, on string-averaging methods for
solving the best approximation problem, to orthogonal projections since they
are known to be FNEs (see, e.g., [6, Facts 1.5(i)]). We start by looking at
the static string-averaging method with orthogonal projections, which is a
special case of our Algorithm 1. We show that the simultaneous version of the
Halpern-Lions-Wittman-Bauschke (HLWB) algorithm, used in [14, Algorithm 5],
and the sequential Halpern-Wittman algorithm (see, e.g., Bauschke and Koch [8,
Algorithm 4.1]) are special cases of the string-averaging methods.

###  6.1 The static string-averaging method for a finite family of closed
convex sets

We write down formally the static string-averaging method for a finite family
of orthogonal projections in order to make sure that our string operators in
such a case remain well-defined. Let $D\subseteq H$ be a subset of $H$, let
$(C_{i})_{i=1}^{m}$ be a finite family of closed convex sets $C_{i}\subseteq
D$ for every $i\in\left\\{1,2,\ldots.m\right\\}$. We denote by
$P_{C_{i}}:D\rightarrow C_{i}$ the orthogonal projection onto the closed
convex set $C_{i}$. We now present the static string-averaging method for a
finite family of closed convex sets.

Algorithm 5. The static string-averaging algorithm for the best approximation
to a finite family of closed convex sets.
Initialization: Choose a single pair $(\varOmega,w)\in\mathcal{M}$ and an
arbitrary $x^{0}\in D.$

Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{t\in\Omega}w(t)P[t](x^{k}),$ (34)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $w(t)$ and $P[t]$ are as in Definition 5.1 with
$(T_{i})_{i=1}^{m}=(P_{C_{i}})_{i=1}^{m}$ and $T[t]=P[t]$.

The proof of convergence of Algorithm 5 follows.

######  Theorem 6.1.

Let $D$ be a nonempty closed convex subset of $H$, let $(C_{i})_{i=1}^{m}$ be
a finite family of closed convex sets $C_{i}\subseteq D$ for every
$i\in\left\\{1,2,\ldots.m\right\\}$ such that
$C:=\cap_{i=1}^{m}C_{i}\neq\emptyset$. Let $(\varOmega,w)\in\mathcal{M}$, let
$(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence and let $u,x^{0}\in
D$. Then any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 5,
converges strongly to $P_{C}(u)$.

###### Proof.

This is a straightforward consequence of Theorem 5.6. Set the finite family of
FNEs in Theorem 5.6 to be $(P_{C_{i}})_{i=1}^{m}$. Then (34) turns out to be
special case of (17) and, so, by Theorem 5.6, any sequence
$(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 5, converges strongly to
$P_{C}(u).$ ∎

###  6.2 The simultaneous Halpern-Lions-Wittman-Bauschke algorithm as a
special case

In [21] a parallel version of Halpern’s algorithm leads to a simultaneous
Halpern-Lions-Wittman-Bauschke (HLWB) algorithm for a countable family of
FNEs, see also Deutsch and Yamada [26]. Here we present a simultaneous HLWB
algorithm for the case of a finite family of closed convex sets. The
convergence of this algorithm follows directly by choosing strings that are
singletons such that each index $i\in\left\\{1,2,\ldots.m\right\\}$ appears in
one string. Thus, this algorithm is not only a consequence of the above
mentioned work of Combettes but also a consequence of our work here. Again,
let $D$ be a nonempty closed convex subset of $H$, let $(C_{i})_{i=1}^{m}$ be
a finite family of closed convex sets $C_{i}\subseteq D$ for every
$i\in\left\\{1,2,\ldots.m\right\\}$ such that
$C:=\cap_{i=1}^{m}C_{i}\neq\emptyset$.

Algorithm 6. The simultaneous HLWB algorithm with orthogonal projections for
the best approximation to a finite family of closed convex sets.
Initialization: $x^{0}=u.$

Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{i=1}^{m}w_{i}P_{i}(x^{k}),$ (35)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $(w_{i})_{i=1}^{m}$ is a sequence of user-chosen
strictly positive real numbers such that $\sum_{i\in I}w_{i}=1$.

###  6.3 The Halpern-Wittman algorithm as a special case

Following the results in [29], Wittman [35] showed the convergence of an
algorithm that is presented in [8, Algorithm 4.1] and is named there the
Halpern-Wittman algorithm. It is designed for a finite family of orthogonal
projections and a specific steering sequence. This algorithm is presented
below as Algorithm 7 and its convergence follows directly from the convergence
of Algorithm 5 by putting all indices of $i\in\left\\{1,2,\ldots.m\right\\}$
into a single string. Again, let $D$ be a nonempty closed convex subset of
$H$, let $(C_{i})_{i=1}^{m}$ be a finite family of closed convex sets
$C_{i}\subseteq D$ for every $i\in\left\\{1,2,\ldots.m\right\\}$ such that
$C:=\cap_{i=1}^{m}C_{i}\neq\emptyset$.

Algorithm 7. The Halpern-Wittman algorithm.
Initialization: $x^{0}=u.$

Iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\frac{1}{k+1}u+\frac{k}{k+1}P_{C_{m}}P_{C_{m-1}}\cdots
P_{C_{1}}(x^{k}),$ (36)

where $u$ is the given anchor point.

##  7 String-averaging methods for best approximation to the common fixed
points set of a family of firmly nonexpansive operators: The infinite case

In this section we propose a string-averaging method for solving the best
approximation problem to the common fixed point set of a countable family of
FNEs $(T_{i})_{i\in I},$ where $I$ is a countable set of positive integers. We
use similar terms to the ones that were used in Section 5 for the finite case,
extending Definition 5.1 to the countable case.

The following definition elaborates how this expansion is made.

######  Definition 7.1.

Let $D$ be a nonempty closed convex subset of $H$, let $(T_{i})_{i\in I}$ be a
countable family of self operators on $D.$ An index vector is a vector of the
form $t=(t_{1},t_{2},\ldots,t_{p})$ such that $t_{\ell}\in I$ for all
$\ell\in\left\\{1,2,\ldots,p\right\\}$. For a given index vector
$t=(t_{1},t_{2},\ldots,t_{q})$ we denote its length (i.e., the number of its
components) by $\gamma(t)=q$, and define the operator $T[t]$ as the (finite)
composition of the operators $T_{i}$ whose indices appear in the index vector
$t$, namely,

$T[t]:=T_{t_{q}}T_{t_{q-1}}\cdots T_{t_{1}},$ (37)

and call it a string operator. An infinite set $\varOmega$ of index vectors is
called fit if for each $i\in I$, there exist a vector
$t=(t_{1},t_{2},\ldots,t_{p})\in\varOmega$ such that $t_{\ell}=i$ for some
$\ell\in\left\\{1,2,\ldots,p\right\\}$. Denote by $\mathcal{M}$ the collection
of all pairs $(\varOmega,w)$, where $\varOmega$ is a fit countable set of
index vectors and $w:\varOmega\rightarrow(0,1)$ is such that
$\sum_{t\in\Omega}w(t)=1$.

Observe that in Definition 5.1 $w:\Omega\rightarrow(0,1]$ is permitted whereas
here we must have $w:\varOmega\rightarrow(0,1)$. This is due to the fact that
in the infinite case here it is impossible to put all operators $(T_{i})_{i\in
I}$ in a single string operator $T[t]$.

###  7.1 The static string-averaging method for the infinite case

Algorithm 1 handles a finite family of FNEs $(T_{i})_{i=1}^{m}$ and throughout
its iterative process, a finite number of string operators $T\left[t\right]$
is used to construct a single string-averaging operator
$T=\sum_{t\in\Omega}w(t)T\left[t\right]$. In the infinite case we allow an
infinite family $(T_{i})_{i\in I}$. In our extension to the infinite case
$\varOmega$ is countable (not finite as it was before) and we allow a
countable number of string operators $T\left[t\right]$ and a chosen fixed
(infinite dimensional) weight vector, making now the single string-averaging
operator $T:=\sum_{t\in\Omega}w(t)T\left[t\right]$ an infinite series. The
fact that only a single $T$ is used makes this a “static” string-averaging
method. Algorithm 8 below is our suggested method for this case.

Algorithm 8. The static string-averaging algorithm for solving the best
approximation problem in the countable case.
Initialization: Choose a single pair $(\varOmega,w)\in\mathcal{M}$ and an
arbitrary $x^{0}\in D.$

iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{t\in\Omega}w(t)T\left[t\right]\left(x^{k}\right),$
(38)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $w(t)$ and $T[t]$ are as in Definition 7.1.

Our aim is to use again Theorem 5.5 above to prove the convergence of
Algorithm 8 to the projection of the anchor point onto the common fixed point
set of the initial given family of FNEs. Recall that, when we did so earlier
for the finite case, we used Lemma 5.4, which in turn, depended on
Propositions 5.2 and 5.3. As we are now dealing with the countable case, we
first extend these propositions to the countable case. Note that since the
length of any index vector remains finite in our countable case, Proposition
5.2 remains applicable. Hence, we only slightly adjust Proposition 5.3 as
follows and present an elementary proof of it.

######  Proposition 7.2.

Let $D$ be a nonempty subset of $H$, let $(T_{i})_{i\in I}$ be a countable
family of self QNEs on $D$ such that $\cap_{i\in I}Fix(T_{i})\neq\emptyset$
and let $(w_{i})_{i\in I}$ be a countable sequence of strictly positive real
numbers such that $\sum_{i\in I}w_{i}=1$. Then $Fix(\sum_{i\in
I}w_{i}T_{i})=\cap_{i\in I}Fix(T_{i})$.

###### Proof.

First, let us show that $\sum_{i\in I}w_{i}T_{i}$ is a well-defined operator.
That is, for every $x\in D$, we have to show that $\sum_{i\in I}w_{i}T_{i}(x)$
converges as an infinite series. Fix $x\in D$, let $f\in\cap_{i\in
I}Fix(T_{i})$ and let $T_{i}\in(T_{i})_{i\in I}$. First, we observe that, by
the quasi-nonexpansivity of $T_{i}$,

$\|T_{i}(x)\|=\|T_{i}(x)-f+f\|\leq\|T_{i}(x)-f\|+\|f\|\leq\|x-f\|+\|f\|,$ (39)

which shows that the series

$\sum_{i\in I}w_{i}(\|x-f\|+\|f\|)=\|x-f\|+\|f\|$ (40)

converges. Define the sequence of partial sums of $\sum_{i\in
I}w_{i}T_{i}(x)$, i.e., for every $n\in\mathbb{N},$
$S_{n}:=\sum_{i=1}^{n}w_{i}T_{i}(x)$.

In order to apply the Cauchy criterion for series convergence, we use (39) to
obtain

$\|S_{n+p}-S_{n}\|=\|\sum_{i=n+1}^{n+p}w_{i}T_{i}(x)\|\leq\sum_{i=n+1}^{n+p}w_{i}\|T_{i}(x)\|\leq\sum_{i=n+1}^{n+p}w_{i}(\|x-f\|+\|f\|).$
(41)

Since the series in (40) is convergent it follows that

$\sum_{i=n+1}^{n+p}w_{i}(\|x-f\|+\|f\|)<\varepsilon,$ (42)

which shows that $(S_{n})_{n\in\mathbb{N}}$ is a Cauchy sequence, thus, from
the completeness of $H$, $(S_{n})_{n\in\mathbb{N}}$ converges and, hence,
$\sum_{i\in I}w_{i}T_{i}(x)$ converges as well and $\sum_{i\in
I}w_{i}T_{i}(x$) is well-defined, as required.

The rest of the proof is similar to the proof of Corollary 4.48 in [7], which
is quoted above in our Proposition 5.3. Set $Q:=\sum_{i\in I}w_{i}T_{i}.$ It
is clear that $\cap_{i\in I}Fix(T_{i})\subseteq Fix(Q)$. To handle the
opposite inclusion, we first do the following calculation. Let $y\in\cap_{i\in
I}Fix(T_{i})$. Then, for every $i\in I$ and any $x\in D,$

$\displaystyle\|T_{i}(x)-y\|^{2}$ $\displaystyle=\|(T_{i}(x)-x)+(x-y)\|^{2}$
(43) $\displaystyle=\|T_{i}(x)-x\|^{2}+2\left\langle
T_{i}(x)-x,x-y\right\rangle+\|x-y\|^{2},$

which, together with the quasi-nonexpansivity of $T_{i},$ yields

$\displaystyle 2\left\langle T_{i}(x)-x,x-y\right\rangle$ (44)
$\displaystyle=\|T_{i}(x)-y\|^{2}-\|T_{i}(x)-x\|^{2}-\|x-y\|^{2}$
$\displaystyle\leq\|x-y\|^{2}-\|T_{i}(x)-x\|^{2}-\|x-y\|^{2}$
$\displaystyle=-\|T_{i}(x)-x\|^{2}.$

Now let us take a point $z\in Fix(Q)$ and observe that it can be rewritten as
$z=\sum_{i\in I}w_{i}z$. Then, from (44) it follows that

$0=2\left\langle Q(z)-z,z-y\right\rangle=2\sum_{i\in I}w_{i}\left\langle
T_{i}(z)-z,z-y\right\rangle\leq-\sum_{i\in I}w_{i}\|T_{i}(z)-z\|^{2},$ (45)

but only if the right-hand side series $-\sum_{i\in I}w_{i}\|T_{i}(z)-z\|^{2}$
is convergent. To prove this last claim we use (39) and the Cauchy-Schwarz
inequality, to reach

$\left\langle T_{i}(z),z\right\rangle\leq|\left\langle
T_{i}(z),z\right\rangle|\leq\|T_{i}(z)\|\|z\|\leq(\|z-f\|+\|f\|)\|z\|,$ (46)

for any $f\in\cap_{i\in I}Fix(T_{i})$. Hence, by (46), and, once more, by
(39), we have

$\sum_{i\in I}w_{i}\|T_{i}(z)-z\|^{2}=\sum_{i\in
I}w_{i}\|T_{i}(z)\|^{2}-2\sum_{i\in I}w_{i}\left\langle
T_{i}(z),z\right\rangle+\sum_{i\in I}w_{i}\|z\|^{2}.$ (47)

The first infinite sum on the right-hand side of (47) is a convergent series
due to (39), the middle series on the right-hand side of (46) is convergent
because of (46), and the last series converges trivially, thus, the series
$\sum_{i\in I}w_{i}\|T_{i}(z)-z\|^{2}$ is convergent and (45) guarantees that
$\sum_{i\in I}w_{i}\|T_{i}(z)-z\|^{2}=0.$ Since $w_{i}\neq 0$ for all $i\in
I,$ we obtain that $z\in\cap_{i\in I}Fix(T_{i})$. ∎

We are now ready to extend Lemma 5.4 to the countable case.

######  Lemma 7.3.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i\in I}$
be a countable family of self FNEs on $D$ such that $F:=\cap_{i\in
I}Fix(T_{i})\neq\emptyset$. Let the single pair
$(\varOmega,w)\in\mathcal{\mathcal{M}}$ and let $T:=\sum_{t\in\Omega}w(t)T[t]$
be the string-averaging operator. Then $Fix(T)=\cap_{i\in I}Fix(T_{i})$.

###### Proof.

The proof proceeds in several steps. First, the family
$(T[t])_{t\in\varOmega}$ is countable because $\varOmega$ is, and since
$T[t]=T_{t_{q}}T_{t_{q-1}}\cdots T_{t_{1}}$ is a finite composition we use
similar arguments to those in the proof of Lemma 5.4, to show that
$(T[t])_{t\in\varOmega}$ is a family of QNEs with nonempty common fixed points
set. Note that $F\neq\emptyset$ and (37) imply that for every $t\in\varOmega$
and for every $x\in F$, $T[t](x)=x$. Hence,

$F\subseteq Fix(T[t]).$ (48)

Therefore,

$F\subseteq\cap_{t\in\Omega}Fix(T[t]).$ (49)

Since every FNE is NE (see Lemma 3.3(i)), we apply, for every $t\in\varOmega,$
[13, Lemma 2.1.12(ii)] to the family $(T_{t_{\ell}})_{\ell=1}^{q}$ and
conclude that the string operator $T[t]:=T_{t_{q}}T_{t_{q-1}}\cdots
T_{t_{1}},$ is NE. Thus, from (48) which guarantees the non-emptiness of
$Fix(T[t])$ and due to the fact that every NE with a fixed point is QNE, the
family of operators $(T[t])_{t\in\varOmega}$ is a family of QNEs.

Secondly, by similar arguments to those that appear in the first part of the
proof of Proposition 7.2, $T$ is a well-defined operator and, we can apply
Proposition 7.2 with the family $(T[t])_{t\in\varOmega}$ instead of the family
$(T_{i})_{i\in I}$ and with the family $(w(t))_{t\in\varOmega}$ instead of the
family $(w_{i})_{i\in I}$, used there. Hence, we get

$Fix(T)=\cap_{t\in\varOmega}Fix(T[t]).$ (50)

Now we apply Proposition 5.2 to the finite family
$(T_{t_{\ell}})_{\ell=1}^{q}.$ To do so we note that, by Lemma 3.3(iii), all
members of $(T_{i})_{i\in I}$ are sQNEs and, so, for every $t\in\varOmega$,
the Proposition 5.2 yields that the string operator $T[t]$ satisfies

$Fix(T[t])=\cap_{\ell=1}^{q}Fix(T_{t_{\ell}}).$ (51)

Finally, from the fitness of $\varOmega$, from (50) and from (51), we obtain
that

$Fix(T)=\cap_{t\in\varOmega}Fix(T[t])=\cap_{t\in\varOmega}(\cap_{\ell=1}^{q}Fix(T_{t_{\ell}}))=\cap_{i\in
I}Fix(T_{i}).$ (52)

∎

It is well-known that a convex combination of NEs is a NE (see, e.g, [13,
Lemma 2.1.12]). Since the string operators $T[t]$ in Algorithm 8 appear inside
an infinite series that resembles a convex combination, our next aim is to
show that, under certain assumptions, the string-averaging operator
$T=\sum_{t\in\Omega}w(t)T[t]$ is NE. The following lemma will show this.

######  Lemma 7.4.

Let $D$ be a nonempty subset of $H$ and let $(S_{i})_{i\in I}$ be a countable
family of NEs such that, for every $i\in I$, $S_{i}:D\rightarrow H$ and
$\cap_{i\in I}Fix(S_{i})\neq\emptyset.$ Then $S:=\sum_{i\in I}w_{i}S_{i}$,
where $(w_{i})_{i\in I}$ is a countable sequence of strictly positive real
numbers such that $\sum_{i\in I}w_{i}=1$, is NE.

###### Proof.

By using similar arguments to the ones that were used in Proposition 7.2 for
$T$, we deduce that $S$ is well-defined. Now, let $x,y\in D$. By the
nonexpansivity of every $S_{i}$ and since $\sum_{i\in I}w_{i}=1$, it follows
that

$\displaystyle||S(x)-S(y)||$ $\displaystyle=||\sum_{i\in
I}w_{i}S_{i}(x)-\sum_{i\in I}w_{i}S_{i}(y)||\leq\sum_{i\in
I}||w_{i}S_{i}(x)-w_{i}S_{i}(y)||$ (53) $\displaystyle\leq\sum_{i\in
I}w_{i}||S_{i}(x)-S_{i}(y)||\leq\sum_{i\in I}w_{i}||x-y||=||x-y||.$

Thus, by Definition 3.1(i), $S$ is NE. ∎

We are now ready to prove convergence of Algorithm 8.

######  Theorem 7.5.

Let $D$ be a nonempty closed convex subset of $H$ and let $(T_{i})_{i\in I}$
be a countable family of self FNEs on $D$ such that $F:=\cap_{i\in
I}Fix(T_{i})\neq\emptyset$. Let $(\Omega,w)\in\mathcal{\mathcal{M}}$, let
$(\lambda_{k})_{k\in\mathbb{N}}$ be a steering sequence and let $u,x^{0}\in
D$. Then any sequence $(x^{k})_{k\in\mathbb{N}}$, generated by Algorithm 8,
converges strongly to $P_{F}(u)$.

###### Proof.

For $(T_{i})_{i\in I}$ and $(\Omega,w)\in\mathcal{\mathcal{M}}$ consider the
family of operators $(T[t])_{t\in\varOmega}$ and define the string-averaging
operator $T:=\sum_{t\in\varOmega}w(t)T\left[t\right]$. By the proof of Lemma
5.4, we deduce that $(T[t])_{t\in\varOmega}$ is a countable family of NEs and,
thus, $T$ is a “convex combination” of countably many NEs. Since
$F\subseteq\cap_{t\in\varOmega}Fix(T[t])$, it follows that
$\cap_{t\in\varOmega}Fix(T[t])\neq\emptyset$ and, therefore, it can be shown
by using similar arguments to the ones that were used in Proposition 7.2, that
$T$ is a well-defined operator. Moreover, Lemma 7.4 yields that $T$ is NE, and
since $Fix(T)\supseteq F\neq\emptyset$, Theorem 5.5 is applicable to $T$ and
implies that every sequence $(x^{k})_{k\in\mathbb{N}}$ generated by Algorithm
8 converges strongly to $P_{Fix(T)}(u)$. Now, by applying Lemma 7.3 on
$(T_{i})_{i\in I}$ together with $(\varOmega,w)$, we obtain that $Fix(T)=F$,
thus, $x^{k}\rightarrow P_{F}(u).$ ∎

In [21] the following simultaneous algorithm for solving the best
approximation problem to the common fixed point set of a countable family of
FNEs with a nonempty common fixed points set is studied. This algorithm turns
out to be a special case of our Algorithm 8 when the pair $(\varOmega,w)$ is
chosen $\Omega:=\left\\{(1),(2),(3),\ldots\right\\}$ and for every $i\in I$,
$w((i)):=w_{i}$. The convergence of this algorithm then follows from our
Theorem 7.5.

Algorithm 9. Combettes simultaneous algorithm.
Initialization: Choose $x^{0}\in D.$

iterative step: Given the current iterate $x^{k},$ calculate the next iterate
$x^{k+1}$ by

$x^{k+1}=\lambda_{k}u+(1-\lambda_{k})\sum_{i\in I}w_{i}T_{i}(x^{k}),$ (54)

where $(\lambda_{k})_{k\in\mathbb{N}}$ is a steering sequence, $u$ is the
given anchor point and $(w_{i})_{i\in I}$ is a sequence of user-chosen
strictly positive real numbers such that $\sum_{i\in I}w_{i}=1$.

##  8 Concluding comments

In this work we expand the class of string-averaging methods by proposing new
string-averaging approaches for solving the BAP for the common fixed points
set of either a finite or an infinite family of FNEs. These methods vary from
the “static” format, in which one uses a single pair
$(\varOmega,w)\in\mathcal{\mathcal{M}},$ to quasi dynamic and simultaneous
formats, where more than a single pair is used. Nevertheless, these methods
are obtained with string operators which are defined as finite compositions of
operators, taken from the given family of operators, thus, the question for
infinitely many compositions, i.e., strings of infinite length, remains open,
as well as the question whether our methods can be extended to the fully
dynamic case.

Another question that remains open is whether the results of [1], where a
sequential algorithm for solving the BAP to the common fixed points set of a
semigroup of nonexpansive operators in Hilbert space was studied, can be
extended to encompass string-averaging algorithmic schemes.

##  9 Declarations

Availability of data and materials: Not applicable.

Competing interests: The authors declare that they have no competing
interests.

Funding: This work was supported by the ISF-NSFC joint research program grant
No. 2874/19.

Authors’ contributions: All authors contributed equally and significantly in
this research work. All authors read and approved the final manuscript.

Acknowledgements: We are grateful to Shoham Sabach and Alexander Zaslavski for
their very useful comments on an earlier version of this work. We thank the
Referee and the Editor for their comments.

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