1. Introduction
In 1975, Baillon proved the first nonlinear ergodic theorem in a Hilbert space. In 1978, Reich obtained the almost convergence and nonlinear ergodic theorems. In 2010, Kocourek et al. [
1] brought in
-generalized hybrid mappings in a Hilbert space for the first time. They also proved a mean convergence theorem for a generalized hybrid mapping that generalizes Baillon’s nonlinear ergodic theorem. Let
K be a nonempty subset of a real Hilbert space
X. A mapping
is called (
)-generalized hybrid if there exist
, such that
for all
.
f is said to be nonexpansive if
f is
-generalized hybrid;
f is said to be nonspreading if
f is
-generalized hybrid [
2];
f is said to be hybrid if
f is
-generalized hybrid [
3]. It can be observed that the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings are all included in
-generalized hybrid mappings.
The set of attractive points was proposed by Takahashi et al. [
4] in 2011. That is,
They also obtained some fundamental properties for attractive points in a real Hilbert space. Using these properties, they proved a mean convergence theorem without convexity for finding an attractive point of a generalized hybrid mapping. Moreover, Takahashi et al. [
5] gave the definition of a more general class of mappings, called
-normally generalized hybrid.
Definition 1 ([
5])
. A mapping is called -normally generalized hybrid if there exist , such that The theory of multi-valued mappings is widely applied in many fields, such as control theory, convex optimization, differential equations, economics, and so on [
6,
7,
8,
9,
10,
11,
12,
13]. In recent years, there is a growing interest in developing an approximation method for fixed points and attractive points of multi-valued mappings. In 2017, Lili Chen et al. [
14] raised the definitions of (
)-generalized hybrid multi-valued mappings in Banach spaces. By the way, they also gave the concepts of attractive points and strongly attractive points of (
)-generalized hybrid multi-valued mappings. In 2019, Lili Chen et al. [
15] introduced the concepts of
-generalized hybrid multi-valued mappings and the corresponding definitions of common attractive points and common strongly attractive points in Hilbert spaces.
In this work, we firstly extend the definitions of -generalized hybrid multi-valued mappings, the set of common attractive points () and the set of common strongly attractive points () of multi-valued mappings f and g to CAT(0) spaces. Furthermore, we present some essential properties in relation to the sets , and for the multi-valued mappings f and g in a complete CAT(0) space. In addition, we obtain a weak convergence theorem of common attractive points for two -generalized hybrid multi-valued mappings in the above space by means of Banach limits technique and Ishikawa iteration respectively. Moreover, we give a strong convergence theorem of two -generalized hybrid multi-valued mappings by the use of a new viscosity approximation method in CAT(0) spaces, which also resolves a kind of variational inequality problem.
2. Preliminaries
Let be a metric space. A geodesic path (or shortly a geodesic) joining a to b in E is a map , such that , and for all . The image of c is called a geodesic segment joining a and b when it is unique and denoted by . We denote the unique point , such that and by , where .
The metric space is called a geodesic space if any two points of E are joined by a geodesic, and E is said to be uniquely geodesic if there is exactly one geodesic joining a and b for each .
A geodesic triangle in a geodesic space consists of three points in E (the vertices of ) and a geodesic segment between each pair of points (the edges of ). A comparison triangle for in is a triangle in the Euclidean plane , such that for all .
A geodesic space E is called a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom:
Let
be a geodesic triangle in
E and
be a comparison triangle in
. Subsequently, the triangle is said to satisfy the CAT(0) inequality if
for all
and all comparison points
.
If
are points in a CAT(0) space and if
h is the midpoint of the segment
, then the CAT(0) inequality implies the so-called (CN) inequality, i.e.,
Moreover, a uniquely geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [
16].
Now, we collect some elementary facts about CAT(0) spaces.
Lemma 1 ([
16])
. Let E be a CAT(0) space, and . Afterwards, Suppose that
is a bounded sequence in a CAT(0) space
E. For
, put
The asymptotic radius
r(
) of
is given by
and the asymptotic center
of
is the set
It follows from [
17] that
is made up of one point in a CAT(0) space. A sequence
is said to be
-convergent to
if
for every subsequence
of
.
Lemma 2 ([
16])
. Every bounded sequence in a complete CAT(0) space always has a Δ
-convergent subsequence. Lemma 3 ([
16])
. If K is a closed convex subset of a complete CAT(0) space and if is a bounded sequence in K, then the asymptotic center of is in K. Subsequently, the definition of convergence and corresponding primary properties are presented below.
Let
K be a nonempty closed convex subset of a complete CAT(0) space
E. Afterwards, for any
, we know that there exists a unique nearest point
, such that
In this case, is the only nearest point of e in K.
Lemma 4 ([
18])
. Assume is a metric projection and . If for any , then converges strongly to some . In 2008, Berg et al. [
19] proposed the idea of quasilinearization in a metric space
E. Each pair
is denoted by
and called a vector. Subsequently, quasilinearization is a map
defined as
for all
. It can be observed easily that
,
and
, for all
. We say that
E satisfies Cauchy-Schwarz inequality if
The necessary and sufficient condition for geodesic connected metric space to be CAT(0) space is that geodesic connected metric space satisfies Cauchy-Schwarz inequality [
20].
In 2013, Dehghan and Rooin [
21] presented a characterization of a metric projection in CAT(0) spaces by using the concept of quasilinearization.
Lemma 5 ([
21])
. Let K be a nonempty convex subset of a complete CAT(0) space E, and . Subsequently,for all . Lemma 6 ([
22])
. Let E be a CAT(0) space and . For any , we set . Afterwards, for each , we have- (1)
;
- (2)
and .
In 2012, Kakavandi [
23] proposed the following results in a complete CAT(0) space.
Lemma 7 ([
23])
. A sequence in a complete CAT(0) space () Δ
-converges to if and only if for all , and w-converges to if for all . Definition 2 ([
23]).
We say that a complete CAT(0) space () satisfies the (S) property if for any there exists a point , such that . Obviously in metric spaces the strong convergence implies
w-convergence, and
w-convergence implies
-convergence, the Example 4.7 of [
23] shows that the converse is not valid.
Let
be the Banach space of bounded sequences with supremum norm [
14,
24,
25]. Let
be an element of
(the dual space of
). Subsequently, we denote by
the value of
at
. Sometimes, we denote by
the value
. A linear functional
on
is said to be a mean if
, where
. If
, a mean
is said to be a Banach limit on
. We know that there exists a Banach limit on
. If
is a Banach limit on
, then for
,
In particular, if and , then we obtain . A useful lemma would be given.
Lemma 8 ([
24])
. Let F be a Hilbert space, let be a bounded sequence in F and let μ be a mean on . Afterwards, there exists a unique point , such thatfor any . 3. Main Results
In this section, we shall prove a weak convergence theorem of common strongly attractive points for two -generalized hybrid multi-valued mappings in a complete CAT(0) space. Now, we present the following notions and lemmas in CAT(0) spaces which will be used in the sequel. Suppose E is a CAT(0) space and K is a nonempty subset of E, and let be a multi-valued mapping. Let be the set of all fixed points of the mapping f.
Let
H be the Hausdorff distance, as defined by
where
and
.
Definition 3. A mapping f defined on a CAT(0) space E is called -generalized hybrid multi-valued if there exist , such that Example 1. Let , and define for all and let H be the Hausdorff distance. It is not difficult to see that is also a complete CAT(0) space. Let , which is a closed convex subset of E, and let f be a multi-valued mapping on K defined by for each . Let , , , we will show that f is -generalized hybrid multi-valued, which is,Indeed, we haveandHence, we concludeTherefore, f is -generalized hybrid multi-valued. Definition 4. The set of all attractive points of the mapping f is defined as Definition 5. The set of all common attractive points of the multi-valued mappings f and g is defined as Definition 6. The set of all strongly attractive points of the mapping f is denoted by Definition 7. The set of all common strongly attractive points of the multi-valued mappings f and g is defined as Ishikawa iterative process for two mappings
f and
g in CAT(0) spaces is as follows:
where
and
,
.
We use to denote the family of all the closed convex subsets of E. We can observe the following results.
Proposition 1. Let E be a complete CAT(0) space and K be a nonempty closed convex subset of E. Let be two mappings. If , then . In particular, if , then .
Proof. Let
, then
and
. Thus, by the definition of metric projection there exists a unique element
, such that
Similarly, since
is a closed and convex subset of
K, there exists
such that
On the other hand,
implies that
for all
, especially,
Combining with Equations (
3)–(
5), we deduce that
Because of the uniqueness, we get , which implies that . Similarly, we can claim . Hence, and . □
Proposition 2. Let E be a complete CAT(0) space and let K be a nonempty subset of E. Let be a quasi-nonexpansive mapping(i.e. for each , holds for all ). Subsequently, .
Proof. First of all, it is not hard to see that
. Now, we will show that
. Let
, then, for any
, we have
which implies that
Because is closed, we obtain . □
Proposition 3. Let E be a complete CAT(0) space and let K be a nonempty subset of E. Suppose is a bounded sequence and is a multi-valued mapping, such that , . Then
- (1)
the sequences , and are bounded for all ;
- (2)
for any Banach limit μ on .
Proof. Suppose
,
. We deduce that the sequence
is bounded, since
is bounded. Combined with
and
, it follows that
is bounded. Moreover, the sequence
is bounded, since
. Subsequently, we have
and
Both sides of formulas (
6) and (
7) are applied to the Banach limit
, combined with
, we can get
□
Theorem 1. Let E be a complete CAT(0) space and K be a nonempty subset of E. Let be two multi-valued mappings. Suppose that . If the sequence is defined by (2), where , are sequences in (0,1) with , then the following conclusions hold: - (1)
the sequence is bounded;
- (2)
exists for each ;
- (3)
.
Proof. Let
. Then by (
2), we get
and
It follows that the limit
exists and the sequence
is bounded. Now we show the last conclusion holds. Because
E is a complete CAT(0) space, then
among
Since
, we obtain that
. Noticing that
, we get
which completes the proof. □
Now, we show the existence of common attractive points for two -generalized hybrid multi-valued mappings by Ishikawa iterative process in a CAT(0) space.
Theorem 2. Let E be a complete CAT(0) space satisfying the (S) property and K be a nonempty closed convex subset of E. Let be two -generalized hybrid multi-valued mappings satisfying , either or . Suppose . If is a sequence generated by (2) satisfying , where , are sequences in (0,1) with , then there is a subsequence of , such that —converges to . Proof. Because
g is a
-generalized hybrid multi-valued mapping, for any
, we have
Now, we consider the following two cases.
Case I. If
, then
where
Subsequently, for any
, we get
By the conclusion (3) of Theorem 1, there exists
, such that
Meanwhile, we notice that
On the other hand, we choose
, such that
. We can obtain that
Making use of Banach limit
and due to Proposition 3, we observe that
which implies
for all
. Since
and
, we obtain
Case II. If
, then
By a similar argument, for
, we have
Making use of Banach limit
, we can get
which implies that
holds for all
. Since
and
, we get
Therefore, we deduce
for any
.
From Theorem 1, it follows that the sequence is bounded. Subsequently, there exists a subsequence of , such that —converges to . Because E satisfies the (S) property, we deduce that —converges to q.
By Lemma 7, for any
, we have
, that is
From (
9), it follows that
By adding
to both sides of the above inequality, we can conclude that
which yields
Noticing that
is closed and convex, we can take
, such that
From (
10), it follows that
From (
10), (
11) and (
12), we get
. Similarly, we can deduce that
which yields
. □
By a similar method, we can obtain the following result on account of Theorem 2.
Corollary 1. Let E be a complete CAT(0) space and K be a nonempty closed convex subset of E. Let be two -generalized hybrid multi-valued mappings satisfying , either or . Suppose that . If is a sequence generated by (2) such that —converges to q, and in which , then . Here, we omit the proof of Corollary 1, since it is essentially similar to the proof of Theorem 2.
4. Application
In 2000, Moudafi [
26] gave a viscosity approximation method for finding fixed points of nonexpansive mappings. Exactly, suppose that
X is a Hilbert space and
C is a nonempty closed convex subset of
X. Let
be a nonexpansive mapping with a nonempty fixed point set
. Staring with an arbitrary initial point
, define a sequence
, by
where
is a contraction and
is a sequence in (0,1). In [
26], under certain appropriate conditions imposed on
, the author proved that
converges strongly to a fixed point
of
T, which satisfies the following variational inequality:
In 2012, Shi and Chen [
27] used the property
to generalize the result of Moudafi to CAT(0) space and Wangkeeree and Preechasilp [
22] omitted the property
from Shi and Chen’s result by the concept of quasi-linearization introduced by Berg and Nikolaev [
19]. Immediately after, Panyanak and Suantai [
28] extended the results of [
22] to multivalued nonexpansive mappings with the endpoint condition. Next, we prove strong convergence of a new viscosity approximation method for a finite family of
-generalized hybrid multi-valued mappings in CAT(0) spaces.
Proposition 4. Let K be a nonempty convex subset of a CAT(0) space E, and f be a ()-generalized hybrid multi-valued mapping defined on K with , which satisfies and, either or . Subsequently, f is quasi-nonexpansive.
Proof. Because of Definition 3, for any
, we get
Let
be a fixed point of
f, then we have
and, hence,
Since
and
, we deduce that
which implies that
f is quasi-nonexpansive. Similarly, we get the desired result in the case of
. □
Theorem 3. Let K be a closed convex subset of a complete CAT(0) space E, which satisfies the (S) property, and let be -generalized hybrid multi-valued mappings satisfying and, either or . Let and be a contraction on K with coefficient . Let be a sequence that is generated by where satisfy and , , and . If as , then there is a subsequence of converges strongly to , such that , which is equivalent to the following variational inequality: Proof. Because
, from Proposition 1, then
. Moreover, from Proposition 4, it follows that
f and
g are quasi-nonexpansive. It follows from Lemma 2 that
and
. Now, we show that
is bounded. Indeed, for any
, we have
among which
Subsequently,
which implies
We can obtain that is bounded, which implies that , , and are bounded.
Next, we show that
and
. Indeed, for any
, we have
among which
Afterwards,
which implies
Noticing that
,
and
is bounded, we get
Let
, then
yields
. On the other hand, we notice that
which implies
as
.
Now, we show that
contains a subsequence converging strongly to
, such that
, which is equivalent to the following variational inequality:
Because is bounded, there exists a subsequence of such that —converges to some . Since E satisfies the (S) property, we deduce that —converges to . Because and , similar to the proof of Theorem 2, we can get .
It follows from (1) of Lemma 6 that
which implies
. Because
converges weakly to
, it follows from Lemma 7 that
as
.
Finally, we show that
which solves the variational inequality (
13). For any
, we deduce
which implies that
Noticing that
and
, taking
, we obtain
It means that
solves the variational inequality (
13) and
by Lemma 5. □