1. Introduction
In [
1], Lennard et al. introduced a class of nonlinear operators in Banach spaces called
cascading non-expansive mappings which generalizes the non-expansive mappings. These mappings arise naturally in the setting of Banach spaces which contain an isomorphic copy of
or
and some important concepts like reflexivity [
1] and weak compactness [
2] have been characterized in terms of the fixed point property for this family of operators.
Although these mappings were introduced in the framework of Banach spaces, its definition depends fundamentally on three properties of the space, the metric, the completeness, and the concept of convexity, so it makes sense to define cascading operators in metric spaces where there is a notion of convexity.
A nonlinear setting which is natural to extend the concept of cascading operator is that of uniquely geodesic metric spaces, since in these spaces the notion of geodesic allows a definition of convex sets. In relation to fixed point problems, it has been specially fruitful to consider the subclass of spaces, which possesses a metric structure that is similar to the one in Hilbert spaces.
In
Section 3, we introduce the cascading operators in the setting of
spaces and, following the reasoning by Lennard et al. in [
1], we distinguish this family from other collections of operators, which encompass the most common generalizations of
asymptotically non-expansive mappings, studied in metric spaces.
In
Section 4, mainly inspired by the results by Dhompongsa et al. in [
3] about asymptotically non-expansive mappings and Khamsi et al. in [
4] concerning
asymptotically pointwise non-expansive mappings, we establish a
demiclosedness principle for cascading non-expansive mappings in
spaces and derive some fixed point results for this family of operators. We also prove a Δ-
convergence theorem for a Mann iteration process ([
5]) using a cascading operator.
2. Preliminaries
A
geodesic joining two points
in a metric space
is a mapping
such that
,
and, for any
, we have that:
A metric space is geodesic if every two points in X are joined by a geodesic. is said to be uniquely geodesic, if, for every , there is exactly one geodesic joining x and y for each , which we denote by . The point in is also denoted by .
As a subclass of the uniquely geodesic spaces, we have the
spaces which usually are considered as the nonlinear analogue of Hilbert spaces. These spaces were introduced by Aleksandrov in [
6].
Definition 1. Let be a uniquely geodesic metric space, . We say that is if The inequality from above is known as the CN inequality of Bruhat and Tits ([
7]).
The following result is very useful in order to perform calculations in spaces.
Proposition 1. ([7] Proposition 2.2) Let X be a space. Then, for all and , Definition 2. A subset C of a uniquely geodesic space is convex if, for any , we have that . If , we define A class of mappings widely studied (see [
8,
9,
10] among others) in the setting of metric fixed point theory is the class of asymptotically non-expansive mappings.
Definition 3. Let be a metric space. A mapping is said to be asymptotically non-expansive if there exists a sequence of positive numbers , with , such that, for all y , These functions were defined first in the context of normed spaces by Kirk in [
8] and properly extend the collection of non-expansive mappings, that is, those functions
, such that
.
Finally, given a metric space
X, a nonempty set
and a mapping
,
3. Cascading Non-Expansive Mappings
In this section, we introduce the notion of cascading non-expansive mappings in the setting of complete spaces and compare it to other types of functions that include the most common generalizations of asymptotically non-expansive mappings studied both in metric spaces and Banach spaces.
Definition 4. Let be a complete space and a closed convex set.
Define , ,..., . If there exists with as such that, for all we say that T is a cascading non-expansive mapping. Next, we recall the notions of totally asymptotically non-expansive mapping, asymptotically pointwise non-expansive mapping and mapping of an asymptotically non-expansive type.
Definition 5. [11] Let be a metric space. A mapping is called totally asymptotically non-expansive if there are nonnegative real sequences and with , as and a strictly increasing and continuous function with such that: Remark 1. This definition unifies several generalizations of the asymptotically non-expansive mappings.
If , we get the nearly asymptotically non-expansive mappings ([12]). If and for all , and
, we recover the asymptotically non-expansive mappings in the intermediate sense ([13]). If and for all , , we have the asymptotically non-expansive mappings.
Definition 6. [14] Let be a metric space. A mapping is called asymptotically pointwise non-expansive if there exists a sequence of mappings such that, for every , , it is verified that: Definition 7. [15] Let be a metric space. A mapping is said to be an asymptotically non-expansive type if, for every , In [
1], some examples are given that prove that the collections of cascading non-expansive mappings and asymptotically non-expansive mappings differ, in the sense that, in general, neither collection is contained in the other. Taking as reference these examples, we distinguish the collection of cascading non-expansive mappings from the respective classes of functions given in Definitions 5, 6, and 7 in the setting of
spaces. We recall that a linear space
X is
if and only if
X is pre-Hilbert ([
7]) Proposition 1.14 p. 167.
Example 1. [1] (Example 2.5) Let and . Let denote the set of rational numbers and be the set of irrational numbers. Define such that: If, for all , , where , then , but From this, we conclude that U is not a cascading non-expansive mapping. Observe that, for all and , and hence T is totally asymptotically non-expansive, asymptotic pointwise non-expansive and of an asymptotically non-expansive type.
The example shows that the family of asymptotically non-expansive maps is not contained in the class cascading operators in spaces.
Example 2. In , we consider the following norm:where is a sequence such that , is decreasing. It is straightforward to see that is a Hilbert space and hence ([7] Proposition 1.14 p. 167). Let and, for , Let be such that It is easy to check that, for every , .
Let us see that T is a cascading non-expansive mapping. If , then If , then ; therefore, T is a cascading non-expansive mapping.
However,and, as , we deduce that T is neither totally asymptotically non-expansive, pointwise asymptotically non-expansive nor of an asymptotically non-expansive type. This example shows that the family of cascading operators is not contained in the family of asymptotically non-expansive maps in spaces.
Example 3. Consider the following equivalence relation over the set : Now, we define a metric on as: It is easy to see that is a complete -tree ([7] p. 167), and, consequently, it is a complete space. Let and, for every , .
Let be such that is a strictly decreasing sequence and . Define such that, if , then Observe that T is well defined. Let us see that T is a cascading non-expansive mapping. It can easily be checked that, for every , .
Let and consider the following cases:
Case 1. , () Case 2. , with . As when , it follows that T is cascading non-expansive. However, T is not asymptotically non-expansive because, if , with , then:due to . Analogously, . Then, However, when . Consequently,and T does not belong to the families given in Definitions 5–7. Remark 2. In [16], the author studied several generalizations of the asymptotically pointwise non-expansive mappings in the context of complete spaces. However, throughout similar examples to those given above, it can be proved that the collection of cascading non-expansive mappings differs from such generalizations. 4. Fixed Point Results for Cascading Operators
Cascading non-expansive operators constitute a new object of study in the framework of
spaces and, in general, they do not contain and are not contained in the collection of asymptotically non-expansive mappings as it was illustrated in
Section 3. Thus, the theorems in this section are new and do not follow from the results related to asymptotically non-expansive maps.
Let
be a complete
space,
be a bounded sequence in
X and
. Let
. The asymptotic radius of
is given by
and the asymptotic center of
is the set
It is a well known fact that, in complete
spaces,
is a singleton ([
17] Proposition 3.2).
The following notions of convergence were introduced by Lim and Kakavandi, respectively, in the setting of metric spaces. These notions resemble the weak convergence defined in Banach spaces and in fact they coincide with the weak convergence in Hilbert spaces ([
18], p. 3452).
Definition 8. Let be a complete space and be a bounded sequence in X.
(i) ([19] p. 180) We say that Δ-converges to if for every subsequence of . (ii) ([18]) We say that weakly converges to if Remark 3. From Example 4.7 in [20], if follows that these notions of convergence are different. Let be a Banach space, a nonempty closed convex set, and be a mapping. If denotes the identity map, it is said that is demiclosed at zero, if, for any sequence such that weakly converges to x and , we have that .
One of the fundamental results in metric fixed point theory for non-expansive mappings is the demiclosedness principle of Browder [
21], which establishes that, if
X is an uniformly convex Banach space,
is a closed convex set and
is a non-expansive mapping, then
is demiclosed.
Several works ([
4,
12,
17,
22,
23] among others) have been devoted to prove demiclosedness principles both in Banach and metric spaces for mappings which generalize the non-expansive ones. The following theorem could be interpreted as a demiclosedness principle for cascading non-expansive mappings with respect to the convergence given in (
i) in Definition 8.
Theorem 1. Let be a complete space and C a closed convex subset of X. Let be a cascading non-expansive mapping and be given as in Definition 4 with . If Δ-converges to w and , then .
Proof. Since , there exists a subsequence such that
whenever
. Let us see that, if
, then
-converges to
w. Indeed, as
-converges to
w, for any
:
From this, we conclude that
, but, since Proposition 3.2 in [
17] implies that
is a singleton, it follows that
and therefore
-converges to
w. By Proposition 3.2 in [
24],
. Observe that
when
.
Hence, for , .
In particular,
but, since
-converges to
w,
.
Consequently, and, since is a singleton, we get that . □
Theorem 1 also holds when we consider the notion of convergence given in ii) in Definition 8.
Corollary 1. Let be a complete space and C a closed convex subset of X. Let be a cascading non-expansive mapping and be given as in Definition 4 with . If is such that and weakly converges to w, then .
Proof. If
weakly converges to
w, Proposition 2.5 in [
20] implies that
-converges to
w, and the conclusion follows from Theorem 1. □
By considering the hypothesis of boundedness over C, we have that:
Corollary 2. Let be a complete space and C a closed convex subset of X. Let be a cascading non-expansive mapping and be given as in Definition 4 with . If C is bounded, the set of fixed points of T, denoted by , is a nonempty closed convex set.
Proof. Let
and
. From [
24] (p. 3690),
has a subsequence
which
-converges to
w and by Proposition 3.2 in [
24]
Consequently,
is a nonempty set, and, since
is non-expansive, the conclusions follows from Theorem 5.1 in [
4]. □
Lemma 1. Let be a complete space and C a closed convex bounded subset of X. Let be a cascading non-expansive mapping and be as in Definition 4 with . Consider the following variant of the Mann iteration process ([5]):where is any element in C and there exist , such that, for all , . It holds that: - 1 .
If , then exists.
- 2 .
.
Proof. Remember that, by Corollary 2, is a nonempty set. Let .
□
The following example shows a simple application of Theorem 1 and generalizes Example 3.
Example 4. Let X be the space described in Example 3. For simplicity, we write to represent the class if and define . Let be a cascading operator for which the sequence is given as in Definition 4, , and there exists , such that for all , (Example 3 shows that such T exists) Then, .
Let be the point for T satisfying (3). If and , Lemma 1 implies that . Define and asand . From condition (3), by passing to a subsequence if necessary, we may assume that, for all , . Thus, and . Since convergence in metric implies Δ-convergence, Δ-converges to and, from Theorem 1, it follows that . Lemma 2. Let be a complete space and be a cascading non-expansive mapping with as in Definition 4 and . Let be a sequence in C such that and converges for all . Then, Δ-converges to a fixed point of T.
Proof. It is similar to the proof of Lemma 2.10 in [
3]. □
Finally, from Lemmas 1 and 2, we conclude that the Mann iteration process defined in Equation (
2),
-converges to a fixed point of
T.
Theorem 2. Suppose that C is a closed convex bounded subset of and let be a cascading non-expansive mapping with as in Definition 4 and . Let be any initial point in C and the sequence defined in Equation (2). Then, Δ-converges to a fixed point of T. Proof. By Lemma 1, when and, for any , exists. Therefore, Lemma 2 implies that -converges to a fixed point of T. □
The theorems introduced in
Section 4 are inspired by some well known results previously studied in
spaces for asymptotically non-expansive maps. It would be interesting to find general fixed point theorems which include both families of maps and to determine conditions under which the two families coincide.