Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
THE
DYNAMIC
STRING-AVERAGING
ALGORITHM
FOR
INVERSE
STRONGLY-MONOTONE
MAPPINGS
∗
Alexander
J.
Zaslavski
†
DOI 10.56082/annalsarscimath.2026.4.71
Abstract
In
the
work
by
W.
Takahashi
and
M.
Toyoda
(2003)
it
was
es-
tablished
the
weak
convergence
of
an
iteration
process
of
solving
a
variational
inequality
problem.
This
variational
inequality
is
associated
with
an
inverse
strongly-monotone
mapping.
In
our
recent
research
we
showed
that
most
of
exact
iterates
of
the
same
iterative
process
are
approximate
solutions
of
the
variational
inequality.
In
this
paper
we
apply
the
dynamic
string-averaging
algorithm
in
order
to
find
a
common
solution
of
a
finite
family
of
variational
inequality
problems,
generated
by
inverse
strongly-monotone
mappings,
and
a
finite
family
of
fixed
point
problems.
Keywords:
Hilbert
space,
iteration,
nonexpansive
mapping,
variational
inequality.
MSC:
47H05,
47H14.
1
Introduction
W.
Takahashi
and
M.
Toyoda
[
11
]
introduced
and
analyzed
a
method
which
finds
a
fixed
point
of
a
nonexpansive
mapping
which
is
also
a
solution
of
a
∗
Accepted
for
publication
on
July
02,
2026
†
ajzasl@technion.ac.il
,
Department
of
Mathematics,
Technion–Israel
Institute
of
Technology,
Haifa,
Israel
71
The
dynamic
string-averaging
algorithm
72
variational
inequality
problem
generated
by
an
inverse
strongly-monotone
map.
The
authors
of
[
11
]
established
the
weak
convergence
of
their
algorithm.
It
should
be
mentioned
that
in
practice
only
a
finite
number
of
iterates
can
be
done.
Thus
it
is
important
to
know
that
the
method
generates
approximate
solutions
and
the
number
of
iterations,
which
should
be
done
needed
in
order
to
reach
such
approximate
solutions.
This
goal
was
achieved
in
[
12
].
We
showed
in
that
work
that
for
the
method
of
[
11
]
most
of
its
exact
iterates
are
approximate
common
solutions
of
a
variational
inequality
associated
with
an
inverse
strongly-monotone
operator.
In
the
present
paper
we
study,
in
the
setting
of
a
Hilbert
space,
the
behavior
of
the
sequences
generated
by
the
dynamic
string-averaging
algorithm
for
finding
a
common
solution
of
a
finite
family
of
variational
inequality
problems
and
a
finite
family
of
fixed
point
problems.
We
show
that
most
exact
iterates
of
our
algorithm
are
approximate
common
solutions
of
the
family
of
variational
inequalities
and
approximate
common
fixed
points
for
the
family
of
mappings.
Assume
that
(
H
·
·
)
is
a
real
Hilbert
space
equipped
with
an
inner
product
which
induces
the
complete
Euclidean
norm
x
=
x
x
1
2
x
∈
H
U
⊂
H
is
a
nonempty,
convex
and
closed.
For
each
z
∈
H
and
each
r
>
0
set
B
(
z
r
)
=
{
u
∈
H
:
u
−
z
≤
r
}
Set
I
(
ξ
)
=
ξ
,
ξ
∈
H
.
For
each
set
D
⊂
H
and
each
x
∈
H
put
d
(
x
D
)
=
inf
{
x
−
y
:
y
∈
D
}
In
the
sequel
we
use
the
following
auxiliary
result
[
1
,
6
–
8
].
Lemma
1.
Suppose
that
D
⊂
H
is
a
nonempty,
convex
and
closed
set.
Then
for
each
ξ
∈
H
there
exists
a
unique
element
P
D
(
ξ
)
∈
D
satisfying
ξ
−
P
D
(
ξ
)
=
inf
{
ξ
−
η
:
η
∈
D
}
and
the
following
assertions
are
valid
true.
1.
P
D
(
x
1
)
−
P
D
(
x
1
)
≤
x
1
−
x
2
for
each
pair
of
points
x
1
x
2
∈
H
and
for
any
ξ
∈
H
and
every
z
∈
D
,
z
−
P
D
(
ξ
)
ξ
−
P
D
(
ξ
)
≤
0
A.J.
Zaslavski
73
z
−
P
D
(
ξ
)
2
+
ξ
−
P
D
(
ξ
)
2
≤
z
−
ξ
2
2.
Let
x
∈
H
,
ξ
∈
D
and
let
for
any
element
z
∈
D
,
we
have
z
−
ξ
x
−
ξ
≤
0
Then
ξ
=
P
D
(
x
)
.
3.
For
each
ξ
η
∈
H
,
P
D
(
ξ
)
−
P
D
(
η
)
2
+
(
I
−
P
D
)(
ξ
)
−
(
I
−
P
D
)(
η
)
2
≤
ξ
−
η
2
P
D
(
ξ
)
−
P
D
(
η
)
2
≤
P
D
(
ξ
)
−
P
D
(
η
)
ξ
−
η
The
operator
P
D
:
H
→
D
is
the
metric
projection
on
the
convex
set
D
.
For
each
set
D
⊂
H
and
x
∈
X
,
set
d
(
x
D
)
=
inf
{
x
−
y
:
y
∈
D
}
Let
L
1
be
a
finite
set
of
pairs
(
F
C
)
where
C
is
a
nonempty
closed
convex
subset
of
U
,
and
F
:
U
→
H
and
L
2
be
a
finite
set
of
mappings
T
:
U
→
U
.
We
suppose
that
the
set
L
1
∪
L
2
is
nonempty.
(Note
that
one
of
the
sets
L
1
or
L
2
may
be
empty.)
We
suppose
that
for
each
(
F
C
)
∈
L
1
each
u
v
∈
U
,
F
(
u
)
−
F
(
v
)
u
−
v
≥
α
F
(
u
)
−
F
(
v
)
2
(1)
and
set
for
each
≥
0,
S
(
F
C
)
=
{
z
∈
C
:
F
(
z
)
u
−
z
≥
0
u
∈
C
}
(2)
S
(
F
C
)
=
{
z
∈
U
B
(
z
)
∩
C
=
∅
F
(
z
)
ξ
−
z
≥
−
ξ
−
z
−
for
all
ξ
∈
C
}
(3)
If
(
F
K
)
∈
L
1
,
then
the
operator
F
is
called
α
-inverse
strongly
monotone
[
3
,
5
,
9
,
11
].
Some
examples
of
inverse
strongly-monotone
mappings
were
considered
in
[
11
].
In
particular,
if
K
is
a
closed,
convex
subset
of
U
and
an
operator
T
:
K
→
K
is
nonexpansive,
then
the
operator
A
=
I
−
T
is
(1/2)
inverse
strongly
monotone
and
S
(
F
K
)
is
the
collection
of
fixed
points
of
T
.
We
suppose
that
for
every
mapping
T
∈
L
2
the
set
Fix(
T
)
:=
{
z
∈
U
:
T
(
z
)
=
z
}
=
∅
and
z
−
x
2
≥
z
−
T
(
x
)
2
+
¯
c
x
−
T
(
x
)
2
The
dynamic
string-averaging
algorithm
74
for
all
x
∈
U
and
all
z
∈
Fix(
T
)
(4)
Fix
0
<
a
≤
b
<
2
α
(5)
Set
l
=
Card(
L
1
∪
L
1
)
(6)
For
every
T
∈
L
2
and
every
≥
0
set
Fix
(
T
)
:=
{
z
∈
U
:
T
(
z
)
−
z
≤
}
(7)
For
each
λ
∈
[
a
b
]
and
each
(
F
C
)
∈
L
1
set
P
FCλ
=
P
C
(
I
−
λF
)
(8)
where
I
:
H
→
H
is
the
identity
operator.
Set
S
:=
[
∩
(
FC
)
∈L
1
S
(
F
C
)]
∩
[
∩
T
∈L
2
Fix(
T
)]
(9)
We
apply
a
dynamic
string-averaging
method
with
variable
strings
and
weights
in
order
to
obtain
a
good
approximative
solution
of
the
common
fixed
point
problem.
It
should
be
mentioned
that
string-averaging
algorithms
were
first
introduced
by
Y.
Censor,
T.
Elfving
and
G.
T.
Herman
in
[
4
]
for
solving
a
convex
feasibility
problem,
when
a
given
collection
of
sets
is
divided
into
blocks
and
the
algorithms
operate
in
such
a
manner
that
all
the
blocks
are
processed
in
parallel
[
2
,
10
].
Next
we
describe
the
dynamic
string-averaging
method
with
variable
strings
and
weights.
By
an
mapping
vector,
we
a
mean
a
vector
T
=
(
T
1
T
p
)
such
that
for
all
i
=
1
p
,
T
i
∈
L
2
∪
{
P
FCλ
:
(
F
C
)
∈
L
1
λ
∈
[
a
b
]
}
(10)
For
an
mapping
vector
T
=
(
T
1
T
q
)
set
p
(
T
)
=
q
(11)
Denote
by
M
the
collection
of
all
pairs
(Ω
w
),
where
Ω
is
a
finite
set
of
mapping
vectors
and
w
:
Ω
→
(0
∞
)
satisfies
T
∈
Ω
w
(
T
)
=
1
(12)
A.J.
Zaslavski
75
The
dynamic
string-averaging
method
with
variable
strings
and
variable
weights
can
now
be
described
by
the
following
algorithm.
Initialization:
select
an
arbitrary
point
x
0
∈
C
.
Iterative
step:
given
a
current
iteration
vector
x
k
pick
(Ω
k
+1
w
k
+1
)
∈
M
calculate
the
next
iteration
vector
x
k
+1
by
x
k
+1
=
T
=(
T
1
T
p
(
T
)
)
∈
Ω
k
+1
w
k
+1
(
T
)
T
p
(
T
)
·
·
·
T
1
(
x
k
)
Fix
∆
∈
(0
l
−
1
)
(13)
a
natural
number
¯
N
and
an
integer
¯
q
≥
l
(14)
Denote
by
M
∗
the
set
of
all
(Ω
w
)
∈
M
such
that
p
(
T
)
≤
¯
q
for
all
T
∈
Ω
w
(
T
)
≥
∆
for
all
T
∈
Ω
(15)
Denote
by
R
the
set
of
all
sequences
{
(Ω
i
w
i
)
}
∞
i
=1
⊂
M
∗
such
that
the
following
properties
hold:
(Q1)
for
every
integer
j
≥
1
and
every
mapping
S
∈
L
2
there
exist
an
integer
k
∈
{
j
j
+
¯
N
−
1
}
and
T
=
(
T
1
T
p
(
T
)
)
∈
Ω
k
such
that
S
∈
{
T
1
T
p
(
T
)
}
;
(Q2)
for
each
integer
j
≥
0
and
every
pair
(
FC
)
∈
L
1
there
exist
k
∈
{
j
j
+
¯
N
−
1
}
,
T
=
(
T
1
T
p
(
T
)
)
∈
Ω
k
,
λ
∈
[
a
b
]
such
that
P
FCλ
∈
{
T
1
T
p
(
T
)
}
.
We
use
the
following
definitions
which
are
needed
in
order
to
study
our
algorithm.
Let
x
∈
U
and
let
T
=
(
T
1
Tt
p
(
T
)
)
be
an
mapping
vector.
Define
A
0
(
x
T
)
=
{
(
y
λ
)
∈
U
×
R
1
:
there
exist
a
sequence
{
y
i
}
p
(
T
)
i
=0
⊂
U
such
that
y
0
=
x
The
dynamic
string-averaging
algorithm
76
and
for
all
i
∈
{
1
p
(
T
)
}
if
T
i
∈
L
2
,
then
y
i
=
T
i
(
y
i
−
1
)
and
if
T
i
=
P
FCλ
),
where
(
F
C
)
∈
L
1
λ
∈
[
a
b
]
then
y
i
=
P
C
(
y
i
−
1
)
y
=
y
p
(
T
)
λ
=
max
{
y
i
−
y
i
−
1
:
i
=
1
p
(
T
)
}}
(16)
Let
x
∈
U
and
let
(Ω
w
)
∈
M
.
Define
A
(
x
(Ω
w
))
=
{
(
y
λ
)
∈
U
×
R
1
:
there
exist
(
y
T
λ
T
)
∈
A
0
(
x
T
)
T
∈
Ω
such
that
y
=
T
∈
Ω
w
(
T
)
y
T
λ
=
max
{
λ
T
:
T
∈
Ω
}}
(17)
Our
goal
is,
for
a
given
>
0,
to
find
a
point
x
which
is
an
-approximate
common
solution
of
the
problems
associated
with
the
family
of
operators
L
1
∪
L
2
.
We
study
inexact
iterates
of
this
algorithm
with
summable
and
nonsummable
errors.
2
Auxiliary
results
In
the
sequel
we
use
the
following
results
of
[
12
].
Proposition
1.
Let
(
A
C
)
∈
L
1
,
τ
>
0
,
u
∈
K
.
Then
u
∈
S
(
A
C
)
if
and
only
if
P
ACτ
(
u
)
=
P
C
(
u
−
τA
(
u
))
=
u
Proposition
2.
Let
(
A
C
)
∈
L
1
,
u
1
u
2
∈
U
and
τ
>
0
.
Then
(
I
−
τA
)(
u
1
)
−
(
I
−
τA
)(
u
2
)
2
≤
u
1
−
u
2
2
+
τ
(
τ
−
2
α
)
A
(
u
1
)
−
A
(
u
2
)
2
If
τ
≤
2
α
,
then
I
−
λA
is
a
nonexpansive
operator.
Lemma
2.
Assume
that
(
A
C
)
∈
L
1
,
x
∈
U
τ
∈
[
a
b
]
u
∈
K
u
=
P
C
(
u
−
τA
(
u
))
y
=
P
C
(
x
−
τA
(
x
))
A.J.
Zaslavski
77
Then
y
−
u
2
≤
x
−
u
2
+
τ
(
τ
−
2
α
)
A
(
u
)
−
A
(
x
)
2
y
−
u
2
≤
x
−
u
2
−
y
−
x
+
τ
(
A
(
x
)
−
A
(
u
))
2
and
y
−
u
2
≤
x
−
u
2
−
4
−
1
y
−
x
2
min
{
1
b
−
2
a
(2
α
−
b
)
}
Lemma
3.
Let
(
F
K
)
∈
L
1
,
0
>
0
,
x
∈
U
,
λ
∈
[
a
b
]
and
let
x
−
P
K
(
x
−
λF
(
x
))
≤
0
Then
for
every
element
η
∈
K
,
we
have
Fx
η
−
x
≥
−
λ
−
1
0
η
−
x
−
λ
−
1
2
0
−
F
(
x
)
0
3
The
main
result
We
prove
the
following
result
which
shows
that
most
exact
iterates
of
our
algorithm
are
approximate
common
solutions
of
the
family
of
variational
inequalities
and
approximate
common
fixed
points
for
the
family
of
mappings
L
2
.
Theorem
1.
Let
∈
(0
1]
,
M
0
≥
1
be
such
that
{
u
∈
S
∩
B
(0
M
0
)
:
F
(
u
)
∈
B
(0
M
0
)
(
F
C
)
∈
L
1
}
=
∅
0
>
0
satisfy
0
≤
min
{
4
−
1
a
4
−
1
(
M
0
+
1)
−
1
(1
+
4
α
−
1
)
−
1
¯
q
−
1
(
¯
N
+
1)
−
1
}
Assume
that
{
(Ω
i
w
i
)
}
∞
i
=1
∈
R
{
x
i
}
∞
i
=1
⊂
U
{
λ
i
}
∞
i
=1
⊂
[0
∞
)
x
0
∈
B
(0
M
0
)
and
that
for
each
integer
i
≥
1
,
(
x
i
λ
i
)
∈
A
(
x
i
−
1
(Ω
i
w
i
)
0)
Then
x
i
≤
3
M
0
i
=
0
1
The
dynamic
string-averaging
algorithm
78
and
Card
(
{
k
∈
{
0
1
}
:
there
is
i
∈
{
k
¯
N
−
1
}
such
that
λ
i
+1
>
0
}
)
≤
16
M
2
0
¯
N
∆
−
1
−
2
0
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
−
1
Moreover,
if
an
integer
q
≥
0
and
λ
i
+1
≤
γ
0
i
=
q
q
+
¯
N
−
1
then
for
each
k
∈
{
q
q
+
¯
N
}
,
for
each
each
T
∈
L
2
and
each
(
F
C
)
∈
L
1
,
d
(
x
k
∈
Fix
(
T
))
≤

d
(
x
i
S
(
F
C
))
≤

Proof.
In
view
of
our
assumptions,
there
exists
u
∗
∈
B
(0
M
0
)
∩
S
(18)
such
that
F
(
u
∗
)
≤
M
0
(
F
C
)
∈
L
1
(19)
By
our
assumptions
and
(18),
x
0
−
u
∗
≤
2
M
0
(20)
Let
k
≥
0
be
an
integer.
We
have
(
x
k
+1
λ
k
+1
)
∈
A
(
x
k
(Ω
k
+1
w
k
+1
)
0)
(21)
Equations
(17)
and
(21)
imply
that
there
exist
(
y
kT
λ
kT
)
∈
A
0
(
x
k
T
)
T
∈
Ω
k
+1
(22)
such
that
x
k
+1
=
T
∈
Ω
k
+1
w
k
+1
(
T
)
y
kT
(23)
λ
k
+1
=
max
{
λ
kT
:
T
∈
Ω
k
+1
}
(24)
Let
T
=
(
T
1
T
p
(
T
)
∈
Ω
k
+1
By
(16)
and
(28),
there
exists
a
sequence
{
y
(
kT
)
i
}
p
(
T
)
i
=0
⊂
U
such
that
y
(
kT
)
0
=
x
k
y
kT
=
y
(
kT
)
p
(
T
)
(25)
A.J.
Zaslavski
79
and
for
each
i
∈
{
1
p
(
T
)
}
if
T
i
∈
L
2
then
y
(
kT
)
i
=
T
i
(
y
(
kT
)
i
−
1
)
(26)
and
if
T
i
=
P
FCτ
,
where
(
F
C
)
∈
L
1
τ
∈
[
a
b
]
then
y
(
kT
)
i
=
P
C
(
y
(
kT
)
i
−
1
−
τF
(
y
(
kT
)
i
−
1
))
(27)
λ
kT
=
max
{
y
(
kT
)
i
−
y
(
kT
)
i
−
1
:
i
=
1
p
(
T
)
}
(28)
Let
k
≥
0
be
an
integer,
T
∈
Ω
k
+1
,
i
∈
{
1
p
(
T
)
}
and
y
(
kT
)
i
−
1
−
u
∗
≤
2
M
0
(29)
Assume
that
T
i
∈
L
2
.
By
(4),
(18)
and
(26),
u
∗
−
y
(
kT
)
i
−
1
2
≥
u
∗
−
T
i
(
y
(
kT
)
i
−
1
)
2
+
¯
c
T
i
(
y
(
kT
)
i
−
1
)
−
y
(
kT
)
i
−
1
2
=
u
∗
−
y
(
kT
)
i
2
+
¯
c
y
(
kT
)
i
−
y
(
kT
)
i
−
1
2
(30)
Assume
that
T
i
=
P
FCτ
,
where
(
F
C
)
∈
L
1
τ
∈
[
a
b
]
In
view
of
Proposition
2
and
(18),
(27),
y
(
kT
)
i
−
u
∗
=
P
FCτ
(
y
(
kT
)
i
)
−
u
∗
≤
y
(
kT
)
i
−
u
∗
(31)
It
follows
from
(18),
(27),
Proposition
1
and
Lemma
2
applied
with
(
A
K
)
=
(
F
C
)
x
=
y
(
kT
)
i
−
1
y
=
y
(
kT
)
i
u
=
u
∗
λ
=
τ
that
y
(
kT
)
i
−
u
∗
2
≤
y
(
kT
)
i
−
1
−
u
∗
2
−
4
−
1
y
(
kT
)
i
−
1
−
y
(
kT
)
i
2
min
{
1
b
−
2
a
(2
α
−
b
)
}
(32)
In
both
cases
by
(30)
and
(32),
y
(
kT
)
i
−
1
−
u
∗
2
≥
y
(
kT
)
i
−
u
∗
2
+
y
(
kT
)
i
−
1
−
y
(
kT
)
i
2
min
{
¯
c
4
−
1
4
−
1
b
−
2
a
(2
α
−
b
)
}
(33)
The
dynamic
string-averaging
algorithm
80
Thus
we
showed
that
the
following
property
holds:
(a)
If
k
≥
0
is
an
integer,
T
∈
Ω
k
+1
,
i
∈
{
1
p
(
T
)
}
and
y
(
kT
)
i
−
1
−
u
∗
≤
2
M
0
then
(33)
is
true.
Applying
property
(a)
by
induction
we
conclude
that
the
following
prop-
erty
holds:
(b)
If
k
≥
0
is
an
integer,
T
∈
Ω
k
+1
and
x
k
−
u
∗
≤
2
M
0
then
for
each
i
∈
{
1
p
(
T
)
}
,
y
(
kT
)
i
−
u
∗
≤
2
M
0
and
(33)
is
true.
Lemma
4.
Let
k
≥
0
be
an
integer
and
x
k
−
u
∗
≤
2
M
0
(34)
The
for
each
T
∈
Ω
k
+1
,
u
∗
−
y
kT
2
≤
u
∗
−
x
k
2
−
4
−
1
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
λ
2
kT
u
∗
−
x
k
+1
2
≤
u
∗
−
x
k
2
−
4
−
1
∆
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
λ
2
k
+1
Proof.
Let
T
∈
Ω
k
+1
.
Properties
(a),
(b)
and
(34)
imply
that
for
each
i
∈
{
1
p
(
T
)
}
,
y
(
kT
)
i
−
u
∗
≤
2
M
0
and
(33)
is
true.
It
follows
from
(25),
(28)
and
(33)
that
u
∗
−
x
k
2
−
u
∗
−
y
kT
2
=
u
∗
−
y
(
kT
)
0
2
−
u
∗
−
y
(
kT
)
p
(
T
)
2
=
p
(
T
)
i
=1
(
u
∗
−
y
(
kT
)
i
−
1
2
−
u
∗
−
y
(
kT
)
i
2
)
≥
4
−
1
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
p
(
T
)
i
=1
y
(
kT
)
i
−
1
−
v
(
kT
)
i
2
≥
4
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
λ
2
kT
(35)
A.J.
Zaslavski
81
By
the
convexity
of
the
function
·
2
,
(12),
(23),
(24)
and
(35),
u
∗
−
x
k
+1
2
=
u
∗
−
T
∈
Ω
k
+1
w
k
+1
(
T
)
y
kT
2
≤
T
∈
Ω
k
+1
w
k
+1
(
T
)
u
∗
−
y
kT
2
≤
T
∈
Ω
k
+1
w
k
+1
(
T
)(
u
∗
−
x
k
2
−
4
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
λ
2
kT
)
≤
u
∗
−
x
k
2
−
4
−
1
∆
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
λ
2
k
+1
Lemma
4
is
proved.
By
(199),
(20)
and
Lemma
4
applied
by
induction,
we
have
x
k
−
u
∗
≤
2
M
0
x
k
≤
3
M
0
k
=
0
1
(36)
Let
n
be
a
natural
number,
In
view
of
(36)
and
Lemma
4,
4
M
2
0
≥
x
0
−
u
∗
2
≥
x
0
−
u
∗
2
−
x
n
−
u
∗
2
=
n
−
1
i
=0
(
x
i
−
u
∗
2
−
x
i
+1
−
u
∗
2
)
≥
4
−
1
∆
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
n
−
1
i
=0
λ
2
i
+1
≥
4
−
1
∆
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
2
0
Card(
{
i
∈
{
0
n
−
1
}
:
λ
i
+1
≥
0
}
)
Since
the
relation
above
holds
for
any
natural
number
n
we
conclude
that
Card(
{
i
∈
{
0
1
}
:
λ
i
+1
≥
0
}
)
≤
16
M
2
0
∆
−
1
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
−
1
−
2
0
(37)
Set
E
=
{
k
∈
{
0
1
}
:
there
is
i
∈
{
k
¯
N
−
1
}
such
that
λ
i
+1
≥
0
}
(38)
By
(37)
and
(38),
Card(
E
)
≤
16
M
2
0
¯
N
∆
−
1
min
{
¯
c
b
−
2
a
(2
α
−
b
)
}
−
1
−
2
0
Assume
that
q
≥
0
is
an
integer
and
λ
i
+1
≤
0
i
=
q
q
−
1
+
¯
N
(39)
The
dynamic
string-averaging
algorithm
82
Let
j
∈
{
q
q
+
¯
N
−
1
}
,
T
=
(
T
1
T
p
(
T
)
)
∈
Ω
j
+1
and
each
i
∈
{
1
p
(
T
)
}
.
By
(24),
(28)
and
(29),
λ
jT
≤
0
y
(
jT
)
i
−
1
−
y
(
jT
)
i
≤
0
(40)
If
T
i
∈
L
2
,
then
in
view
of
(26)
and
(40),
0
≥
y
(
jT
)
i
−
1
−
y
(
jT
)
i
=
y
(
jT
)
i
−
1
−
T
i
(
y
(
jT
)
i
−
1
)
y
(
jT
)
i
−
1
∈
Fix
(
T
i
)
(41)
Assume
that
τ
∈
[
a
b
]
(
F
C
)
∈
L
1
T
i
=
P
FCτ
(42)
Equations
(14.120)
and
(14.122)
imply
that
0
≥
y
(
jT
)
i
−
1
−
y
(
jT
)
i
=
y
(
jT
)
i
−
1
−
P
C
(
y
(
jT
)
i
−
1
−
τF
(
y
(
jT
)
i
−
1
))
(43)
It
follows
from
(1),
(18),
(19)
and
(36)
imply
that
F
(
y
(
jT
)
i
−
1
)
≤
F
(
u
∗
)
+
F
(
y
(
jT
)
i
−
1
)
−
F
(
u
∗
)
≤
M
0
+
α
−
1
y
(
jT
)
i
−
1
−
u
∗
≤
M
0
+
4
M
0
α
−
1
(44)
By
(15),
(43),
(44),
Proposition
1
and
Lemma
3
applied
with
x
=
y
(
jT
)
i
−
1
,
λ
=
τ
(
A
K
)
=
(
F
C
),
for
each
ξ
∈
C
,
F
(
y
(
jT
)
i
−
1
)
ξ
−
y
(
jT
)
i
−
1
≥
−
τ
−
1
0
ξ
−
y
(
jT
)
i
−
1
−
τ
−
1
2
0
−
0
M
0
(1
+
4
α
−
1
)
≥
−
a
−
1
0
ξ
−
y
(
jT
)
i
−
1
−
a
−
1
2
0
−
0
M
0
(1
+
4
α
−
1
)
≥
−
ξ
−
y
(
jT
)
i
−
1
−

y
(
jT
)
i
−
1
∈
S
(
F
C
)
Thus
we
showed
that
the
following
property
holds:
(c)
if
T
i
∈
L
2
,
then
y
(
jT
)
i
−
1
∈
Fix
(
T
i
)
and
if
τ
∈
[
a
b
]
(
F
C
)
∈
L
1
T
i
=
P
FCτ
then
y
(
jT
)
i
−
1
∈
S
(
F
C
)
A.J.
Zaslavski
83
By
(25)
and
(40),
for
each
i
i
1
i
2
∈
{
1
p
(
T
)
}
,
y
(
jT
)
i
1
−
y
(
jT
)
i
2
1
≤
0
¯
q
(45)
x
j
−
y
(
jT
)
i
≤
0
¯
q
x
j
−
y
jT
≤
0
¯
q
(46)
By
(12),
(23),
(46)
and
the
convexity
of
the
norm,
x
j
−
x
j
+1
=
x
j
−
T
∈
Ω
j
+1
w
j
+1
(
T
)
y
jT
≤
T
∈
Ω
j
+1
w
j
+1
(
T
)
x
j
−
y
jT
≤
¯
q
0
(47)
It
follows
from
(47)
that
for
each
j
1
j
2
∈
{
q
q
+
¯
N
}
,
x
j
1
−
x
j
2
≤
0
¯
q
¯
N
(48)
Let
S
∈
L
2
.
Property
(Q1)
implies
that
there
exist
j
∈
{
q
q
+
¯
N
−
1
}
,
T
=
(
T
1
T
p
(
T
)
)
∈
Ω
j
+1
such
that
S
∈
{
T
1
T
p
(
T
)
}
There
exists
i
∈
{
1
p
(
T
)
}
such
that
S
=
T
i
Property
(c)
implies
that
y
(
jT
)
i
−
1
∈
Fix
(
T
i
)
In
view
(46),
d
(
x
j
Fix
(
S
))
≤
x
j
−
y
(
jT
)
i
−
1
≤
¯
q
0
(49)
It
follows
from
(15),
(48),
(49)
that
for
each
k
∈
{
q
q
+
¯
N
}
d
(
x
k
Fix
(
S
))
≤
x
k
−
x
j
+
d
(
x
j
Fix
(
S
))
≤
0
(
¯
N
+
1)¯
q
≤

Thus
d
(
x
k
Fix
(
S
))
≤
for
each
S
∈
L
2
.
Let
(
F
C
)
∈
L
1
.
Property
(Q2)
implies
that
there
exist
j
∈
{
q
q
+
¯
N
−
1
}
,
τ
∈
[
a
b
]
and
T
=
(
T
1
T
p
(
T
)
)
∈
Ω
j
+1
such
that
P
FCτ
∈
{
T
1
T
p
(
T
)
}
The
dynamic
string-averaging
algorithm
84
There
exists
i
∈
{
1
p
(
T
)
}
such
that
T
i
=
P
FCτ
(50)
Property
(c)
and
(50)
imply
that
y
(
jT
)
i
−
1
∈
S
(
F
C
))
(51)
By
(46)
and
(51),
d
(
x
j
S
(
F
C
))
≤
x
j
−
y
(
jT
)
i
−
1
≤
¯
q
0
(52)
In
view
of
(48)
and
(52),
for
each
k
∈
{
q
q
+
¯
N
}
d
(
x
k
S
(
F
C
))
≤
x
k
−
x
j
+
d
(
x
j
S
(
F
C
))
≤
0
¯
N
¯
q
+
¯
q
0
≤

Theorem
1
is
proved.
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Bauschke
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P.L.
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R.
Zalas,
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