1. Introduction
The class of strongly quasi-nonexpansive mappings plays a crucial role in convex optimizations on
spaces for
. In 1977, Bruck and Reich [
1] first introduced the concept of a strongly nonexpansive mapping for the first time, which generalizes the class of firmly nonexpansive mappings. Later, this mapping was generalized to a strongly quasi-nonexpansive mapping by Bruck [
2]. An intuitive example of strongly quasi-nonexpansive mappings is the metric projection
onto closed convex sets
C (see [
3,
4]), which is crucial for dealing with a convex feasibility problem. In 2012, Ba
ák, Searston and Sims [
5] proved that the sequence
-converges to a point in
whenever
and
are closed convex subsets of
spaces. Following that, Choi, Ji, and Lim [
6] proved the same result for any
spaces with
. Another important example of a strongly quasi-nonexpansive map is the resolvent mapping of a proper lower semicontinuous convex function
f in
spaces (see Proposition 3.1 of [
3]), which was proposed by Jost [
7,
8] and Mayer [
9]. The resolvent mapping of
f with respect to
is defined by
for
. It was not until 2013 that this resolvent was applied to solve convex optimizations in complete
spaces by Ba
ák [
10]. This method is called a proximal algorithm and it has been one of the earliest and most successful approximation schemes for convex optimizations. Ba
ák [
11] improved the proximal algorithm to solve the convex optimization when the objective function is usually expressed as the sum of several loss functions, i.e.,
when, for all
,
is a proper lower semicontinuous convex function. Such a method is called a splitting proximal algorithm. This is a useful technique that can be used to find means and medians phylogenetics on tree spaces (see [
12]). Following that, the convex optimization studied on
spaces grew in popularity.
Among the vast developments, Kimura and Kohsaka [
13] introduced a new resolvent on
spaces by replacing
in (
1) with a function corresponding to the metric on the model spaces. The new resolvents still carry the most important properties of being strongly quasi-nonexpansive on
spaces (see [
13] (Theorem 4.6)). In the same year, Espínola and Nicolae [
14] developed proximal and splitting proximal algorithms based on these novel resolvents on
spaces. With a similar idea, Kajimura and Kimura [
15] constructed a new resolvent on
spaces. In [
16], the authors subsequently used this resolvent to study splitting methods on
spaces and applied their results to solve convex feasibility problems, centroid problems, and, particularly, the Karcher means. Note that all preceding studies were investigated and verified using specific technical approaches to handle their results, while the main goal of this research is focused on a broader perspective—viewing the problem through the lens of fixed point theory.
In 2014, Ariza, López and Nicolae [
17] achieved new and more broader convergence results to approximate a common fixed point of a finite family of firmly nonexpansive mappings in
spaces. They studied the asymptotic behavior of the sequence constructed by the composition of firmly nonexpansive mappings on
p-uniformly convex spaces, and then the concept of asymptotic regularity was used to investigate the common fixed points of the finite number of firmly nonexpansive mappings. Later, Reich and Salinas [
18] generalized the Ariza–López–Nicolae results for
spaces. They proved a
-convergence theorem of infinite products of strongly nonexpansive mappings and they also used their results to solve the convex feasibility problems on complete
spaces. These results were obtained as consequences of the
-convergence of iterations for a sequence formed by strongly quasi-nonexpansive mappings on
spaces (see [
19]). Motivated by the above studies, we show, in this paper, the
-convergence of the infinite products of strongly quasi-nonexpansive mappings in the setting of a complete admissible
space and also the applications to convex feasibility and common minimizer problems. The main results and applications in this paper extend the studies of the Choi–Ji–Lim [
6], Ariza–López–Nicolae [
17] for
spaces, and Espínola–Nicolae [
14] if the intersection of
is nonempty.
Our paper is organized as follows. In
Section 2, we collect background materials which are required for our main results in subsequent sections and present a new definition of orbital
-demiclosedness. In
Section 3, we first prove several useful properties for our main results and then conclude with the main results concerning
-convergence of the infinite products for strongly quasi-nonexpansive orbital
-demiclosed mappings in complete admissible
spaces. In
Section 4, we discuss some possible applications of our main results to solve the convex feasibility and common minimizer problems in
spaces. In the final section,
Section 5, we demonstrate a numerical implementation applied to solve convex feasibility problems.
2. Preliminaries
In this section, some basic notions and useful lemmas necessary for the subsequent results are given. Throughout this paper, the set of all positive integers is denoted by and the set of all real numbers is denoted by .
Let
be a metric space and
. A
geodesic path joining
x and
y is a mapping
such that
,
and
for any
. Let
D be a positive real number. A metric space
X is said to be (uniquely)
geodesic if for each
with
, there exists a (unique) geodesic joining
x and
y. We say that
X is a
(uniquely) geodesic metric space if any two elements
are connected by a (unique) geodesic. If
X is uniquely geodesic,
, and
is the geodesic path joining
x to
y, then we write
to denote the
geodesic segment of
. In this case, we also use the notation
. In addition, recall that the
geodesic triangle with vertices
, denoted by
, is defined by
. Recall that the sphere
is defined by
where
for all
. The spherical metric
on
is defined by
for all
. From [
20], if
,
and
, then the unique geodesic
from
x to
y is given by
for all
. For a triangle
in a uniquely
geodesic space
X satisfying
, we can find the comparison triangle
in
such that
for all
. Let
be the geodesic joining
to
and
for some
. Then, a point
is called the
comparison point of
u, where
is the geodesic joining
to
. If
X is a uniquely
geodesic metric space,
,
and every two points
in
and their comparison points
in
satisfy that,
then
X is called a
space. A
space
X is called
admissible if
for all
.
Remark 1 ([
20])
. In general, if , then is a space if and only if is a space. Let
be a uniquely geodesic metric space. A subset
is called
convex if
for all
. Let
be given. Recall that the
effective domain of
f is defined by
. If
, we say that
f is proper. In addition,
f is said to be
convex if
holds for all
and
.
Let
be a
space and
be a sequence of
X. The asymptotic center
of a sequence
is defined by
The sequence
is said to be
-convergent to an element
z of
X if
for each subsequence
of
. In this case, we say that
z is the
-limit of
. If
is convergent to
z, then it is
-convergent to
z. If
is
-convergent to
z, then it is bounded and every subsequence
of
is
-convergent to
z. Furthermore,
denotes the set of all
such that there exists a subsequence
of
which is
-convergent to
. A subset
is called
-closed if
whenever
is a
-limit of some sequence
of
C. See [
21,
22,
23,
24] for more details about the
-convergence. A sequence
is said to be spherically bounded if
In particular, if , then the space X is admissible and every sequence in X is spherically bounded.
Let
be a
space and
T be a mapping from
X into itself.
denotes the set of all fixed points of mapping
T. Suppose that the set
. If
for all
and
, then we say that
T is a quasi-nonexpansive mapping. If
T is quasi-nonexpansive mapping, and for every sequence
in
X and
such that
and
it follows that
, then
T is called a strongly quasi-nonexpan-sive mapping. Moreover,
T is said to be
-demiclosed in the sense of [
4] if for any
-convergent sequence
in
X, its
-limit belongs to
whenever
.
Next, we will present two examples of strongly quasi-nonexpansive and -demiclosed mapping in complete admissible spaces.
Let
be a complete admissible
space and
C be a nonempty closed convex subset of
X. The metric projection
is defined by
for every
. Then for all
, there exists a unique
such that
. It is known that
and
for every
and
(see [
21] (Proposition 3.5)). In particular, the metric projection onto a nonempty closed and convex subset of a complete admissible
space is strongly quasi-nonexpansive
-demiclosed (see [
4]).
On the other hand, Kimura and Kohsaka [
13] introduced a new concept of the resolvent
of a proper lower semicontinuous convex function
f of a complete admissible
space
X into
with respect to
defined by
for all
. This mapping is well defined as a single-valued mapping (see [
13] (Theorem 4.2)), and it is also a strongly quasi-nonexpansive mapping under the admissibility condition of the ambient space
X such that
(see [
13] (Theorem 4.6)). In addition, we know that the resolvent
is a
-demiclosed mapping (see [
25]).
Now, we propose a new generalization of
-demiclosedness.
Definition 1. Let be a complete admissible space and let T be a mapping from X to X. The mapping T is called orbital Δ-demiclosed if whenever is a sequence in X satisfying the following conditions:
- 1.
;
- 2.
;
- 3.
is a subsequence of for some .
The following lemmas are important for our main results.
Lemma 1 ([
21]).
Let be a complete admissible space. Suppose that the sequence is a spherically bounded sequence in X. Then is a singleton and has a Δ-convergent subseqeunce. Lemma 2 ([
26]).
Let be a complete admissible space. Suppose that the sequence is a spherically bounded sequence in X and is convergent for every z in . Then is Δ-convergent to an element . 3. Main Results
As the main theorems of this paper, we show that the product of strongly quasi-nonexpansive
-demiclosed mappings is a strongly quasi-nonexpansive and orbital
-demiclosed mapping. Next, we prove the
-convergence of the sequence, which is constructed by a strongly quasi-nonexpansive orbital
-demiclosed mapping in the setting of complete admissible
spaces. Motivated by Reich and Salinas [
18], we start with the following lemmas, which are some of the most important tools in determining our results (see also [
27]).
Lemma 3. Let be a complete admissible space. If is a strongly quasi-nonexpansive mapping for all and . Then, .
Proof. Firstly, we will show that .
Let
. Then, we obtain
From the above equation, we have
.
Therefore, .
Next, we will show that
Suppose first that
. Let
and
, then
. Since
and
are quasi-nonexpansive, for any
, we will obtain
Hence,
, and we have
Since
for all
and
is strongly quasi-nonexpansive, we have
Then, we see that
, which means that
z belongs to
and
. Therefore,
.
Finally, we suppose that (
4) holds for
. Let
and
. Then,
. Since
are quasi-nonexpansive, for any
, we have
Hence,
. This implies that
Since
for all
, and
is strongly quasi-nonexpansive, we have
and
which means that
z belongs to
and
. By the inductive hypothesis holds for
, we have
. Therefore,
. □
Lemma 4. Let be a complete admissible space. If is a strongly quasi-nonexpansive mapping for all and Fix, then is also a strongly quasi-nonexpansive mapping.
Proof. We prove this lemma by using mathematical induction.
Suppose first that
. Assume that
and
are strongly quasi-nonexpansive mappings such that
. Let
and
such that
and
for all
.
(i) We will show that T is quasi-nonexpansive.
Take
and
. By Lemma 3,
, we have
Then
T is quasi-nonexpansive.
(ii) We will show that T is strongly quasi-nonexpansive ().
Since
and
are quasi-nonexpansive, we obtain
this implies that
Then we obtain
and from (
5), we have
Since
is strongly quasi-nonexpansive and due to (
7), we obtain
From (
6), we have
and
Combining (
8) and (
9) with
is strongly quasi-nonexpansive, and we obtain
By the triangle inequality, we have
Thus, it follows that
, which proves that
is strongly quasi-nonexpansive.
Now, suppose that is strongly quasi-nonexpansive for .
Assume that
are strongly quasi-nonexpansive such that
and
is strongly quasi-nonexpansive. Let
and
such that
and
for all
.
(i) We will show that T is quasi-nonexpansive.
Take
and
. By Lemma 3,
, we have
Then,
T is quasi-nonexpansive.
(ii) We will show that T is strongly quasi-nonexpansive as .
Since
is a quasi-nonexpansive for all
, we obtain
This implies that
By dividing both sides by
in the above inequality, we have
and from (
10), we obtain
Since
is strongly quasi-nonexpansive and (
13), we have
and from (
11), we know that
Combining this information with (
10) and (
12), we obtain
From (
14) and (
15) and as
is strongly quasi-nonexpansive, we have
By the triangle inequality, we obtain
Thus, it follows that
, which proves that
is strongly quasi-nonexpansive. □
Now, we show the main theorems.
Theorem 1. Let be a complete admissible space. If is a strongly quasi-nonexpansive Δ-demiclosed mapping for all and Fix, then is a strongly quasi-nonexpansive orbital Δ-demiclosed mapping.
Proof. Suppose that are strongly quasi-nonexpansive -demiclosed mapping for all and . By Lemma 4, we have that T is strongly quasi-nonexpansive. Then we will prove that only T is orbital -demiclosed.
Let be a sequence in X and an element of X such that
- (i)
;
- (ii)
;
- (iii)
is a subsequence for some .
We will show that
to conclude
T is orbital
-demiclosed. Since
are strongly quasi-nonexpansive for all
, by Lemma 3 we have
Take
.
Since
T is strongly quasi-nonexpansive, we obtain
This implies that
By condition (ii), we have
Using the condition (iii), there exists
such that
, which further gives
Combining (
17) and (
18), there exists a subsequence
of
satisfying
Since
T is strongly quasi-nonexpansive, and from (
16), we have
This implies that
According to (
16), we can show that
to conclude that
.
Step 1 First, we will show that .
By (
19), we obtain
Then
this implies that
Since
is strongly quasi-nonexpansive, we obtain
and since
is
-demiclosed, the
-limit of
belongs to
. Then
.
Step 2 We next show that .
By (
19), we obtain
Then
this implies that
Since
is strongly quasi-nonexpansive, we obtain
and as
is
-demiclosed, the
-limit of
belongs to
.
Next, we will show that -convergent to .
Let
be an arbitrary subsequence of
.
By (20), we have
which means that
for each subsequence
of
. This implies that
-convergent to
. Thus,
belongs to
.
We now show that
for the remaining
. We suppose that
step i holds for
and we will show that
. By (
19), we obtain
Then
This implies that
Since
is strongly quasi-nonexpansive, we obtain
and as
is
-demiclosed, the
-limit of
belongs to
.
Next, we will show that -convergent to .
Let
be an arbitrary subsequence of
.
By the assumption holds for
, we have
Then, we obtain
which means that
for each subsequence
of
. This implies that the sequence
-convergent to
. Thus,
belongs to
. From
step 1 to
step m and (
16), then
; this means that
. Therefore,
T is orbital
-demiclosed. □
Theorem 2. Let be a complete admissible space, is a strongly quasi-nonexpansive orbital Δ-demiclosed mapping, and Fix. Then, for any initial point , the sequence defined for each byis Δ-convergent to an element in . Proof. Given
, consider the sequence
that is defined by
Let
. Since
T is quasi-nonexpansive, we obtain
Observe that the sequence
is bounded and nonincreasing for each
. Then the sequence
is convergent to an element of
, for each
. We will have
This implies that
then we obtain
Combining this information with the strong quasi-nonexpansivity of
T, we have
Next, we will show that
is spherically bounded. Since
then the sequence
is spherically bounded. By Lemma 1, the sequence
has a
-convergent subsquence. Take
such that
is
-convergent to
, for some subsequence
of
. We know that
from definition of
. This means that
is a subsequence for some orbit. By (
21), we have
Combining this information with
T as a orbital
-demiclosed mapping, we obtain that
belongs to
. Thus,
. This implies that the sequence
is convergent to an element of
, for all
. By Lemma 2, the sequence
is
-convergent to an element of
X, since we know that
. Therefore, the sequence
is
-convergent to an element of
. □
4. Applications
In this section, we consider some viable applications of our main results. Particularly, we shall apply our results from the previous section to solve the convex optimizations on spaces.
Given a complete admissible
space
X, from
Section 2, we know that the resolvent mapping
of a proper lsc convex function
f of
X with respect to
, is a strongly quasi-nonexpansive
-demiclosed mapping and
. Then, we can apply our results from the previous section to optimize the function
f. Moreover, the convex feasibility problem seeks a common point
in the intersection
, where
is a nonempty closed convex subset of
X for all
. The metric projection is crucial for dealing with a convex feasibility problem. According to
Section 2, we know that the metric projection
from
X onto
is a strongly quasi-nonexpansive
-demiclosed mapping and
for all
. If
is nonempty, we can use the alternating projection mapping for finding the solution of this problem.
4.1. Convex Minimization Problems
Firstly, we will present the application for solving the convex minimization problem of proper lsc convex functions in a complete admissible
space
X. Let
be a proper lsc convex function and
for all
. According to [
13] (Theorem 4.6), we know that the resolvent mapping
is a strongly quasi-nonexpansive
-demiclosed mapping such that
is equal to
for all
. Then, we obtain the following corollary.
Corollary 1. Let be a complete admissible space and be a proper lower semicontinuous convex function. Let be a resolvent mapping of f with respect to for all . If is nonempty, then for any initial point , the sequence defined for each byis Δ-convergent to an element in . Proof. Since is a strongly quasi-nonexpansive -demiclosed mapping for all such that , then we have that is a strongly quasi-nonexpansive orbital -demiclosed mapping by using Lemma 1. Let . Define a sequence by . By using Theorem 2, the sequence is -convergent to some point in . Then, the point . □
In the next application, we consider the sum of several loss functions, given that the objective function
f is defined by
where
is a proper lsc convex function for all
. According to [
13] (Theorem 4.6), we know that the resolvent mapping
is a strongly quasi-nonexpansive
-demiclosed mapping such that
is equal to
for all
. By Lemma 3, we have that
and we know that
whenever
is nonempty. Then, we can apply the next corollary for finding the solutions of the objective function.
Corollary 2. Let be a complete admissible space and defined by where is a proper lower semicontinuous convex function for all . Let be a resolvent mapping of with respect to for all . If is nonempty, then for any initial point , the sequence defined for each by the fact thatis Δ-convergent to an element in . Proof. Since is a strongly quasi-nonexpansive -demiclosed mapping for all such that , then is a strongly quasi-nonexpansive orbital -demiclosed mapping by using Lemma 1. Let . Define a sequence by . By using Theorem 2, we obtain that the sequence is -convergent to some point in . Then, the point . □
4.2. Convex Feasibility Problem
Finally, we consider the feasibility problem with each
being closed and convex. We call this particular case the convex feasibility problem. Let
be a complete admissible CAT
space and let
be nonempty closed convex subsets of
X. The convex feasibility problem is to find some point
when this intersection is nonempty.
Now, we will prove the -convergence of alternating projection on complete admissible CAT space.
Corollary 3. Let be a complete admissible space. If is the metric projection of X onto closed and convex subset for all with , then for any initial point , the sequence defined for each byis Δ-convergent to an element . Proof. Since each is a strongly quasi-nonexpansive and -demiclosed mapping for all , respectively, according to Lemma 1, the products are strongly quasi-nonexpansive and orbital -demiclosed mapping. By Theorem 2, for each , there exists a point such that the sequence is -convergent to . The fact that follows from Lemma 4. □
Next, we prove the strong convergence alternating projection from the previous corollary by letting be a compact set for some .
Corollary 4. Let be a complete admissible space. If is the metric projection of X onto a nonempty closed and convex subset for all with and is a compact set for some , then for any initial point , the sequence defined for each byis convergent to an element . Proof. We will prove this by letting be a compact set for some .
Case I Suppose that is a compact set.
Let the starting point for some element and for all . By Corollary 3, we have that the sequence is -convergent to for some . Then, every subsequence of is -convergent to . Since is a compact set and , then for every subsequence of there exists such that ; this implies that .
Case II Suppose that is a compact set for some .
Define
and
. By
case I, we have
for any
. Observe that
and then
as
. Therefore, we can conclude that
. □
5. Numerical Implementations
In this section, we implement the proposed alternating projection algorithm to approximate the convex feasibility problem of given closed convex sets which are fitted to
,
X defined by
where
,
, and
is the spherical metric. Recall that
X is a complete admissible
space when equipped with the spherical metric
.
Given that
are closed balls on
X,
where
and
for all
. Then
are closed convex subsets of
X and
. We can define the metric projection
from
X onto
for all
by
where
is defined in (
2).
In the numerical implementations, we set
where
, and we use three points,
, which are elements in
X to generate closed convex subsets
with
for all
(see
Figure 1).
In the following numerical implementations of Corollary 4, we will show the convergence of alternating sequences by using two distinct starting points
in
X. Since the dataset is quite large, we only report the implementation results as presented in
Figure 2. Therein, the LHS figures represent the center points of
(green ‘
•’ marks), the elements of intersection of
(blue ‘
•’ marks), and the initial point
(black ‘x’ marks), the alternating sequence
(black ‘*’ marks), together with approximated solution of CFP (red ‘
x’ marks) obtained from the alternating method. The RHS figures show plots of errors after the iteration of
N. Note that the errors presented thereby are computed by
. The accepted tolerance in these illustrations is set at
. The calculation was performed using Python in Google Colab.
The features of the alternating sequence’s convergence will vary depending on the chosen initial point, as demonstrated in the example above. Furthermore, we can see that errors fall quickly in the early phases of the computation, but then slow down and eventually halt when errors are fewer than the tol= value we set.