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Nonexpansive semigroups in spaces
Fixed Point Theory and Applications volume 2014, Article number: 44 (2014)
The main purpose of this paper is to study Browder type convergence theorems for a nonexpansive semigroup with geometric approaches in a space. Besides, we determine a necessary and sufficient condition for convergence of a Browder type iteration associated to a uniformly asymptotically regular nonexpansive semigroup on the unit sphere in an infinite-dimensional Hilbert space.
Let be a metric space, C a closed convex subset of X and a mapping. Recall that T is nonexpansive on C if , for all . We denote by the fixed point set of the mapping T. A one-parameter family of self-mappings of C is called a strongly continuous nonexpansive semigroup on C if the following conditions are satisfied:
for each , is a nonexpansive mapping on C;
, for all ;
, for all ;
for each , the mapping from into C is continuous.
Let denote the common fixed point set of all mappings in .
There have been considerably many interesting results of iterative methods for approximating fixed points of nonexpansive mappings, nonexpansive semigroups, and their generalizations which solve some variational inequalities problems due to their various applications in several physical problems, such as in operations research, economics, and engineering design; see, e.g., [1–4] and the references therein.
Suppose that X is a real Hilbert space and u is an arbitrary point of X. If T is nonexpansive on C, then for each there exists a unique such that because the mapping is a contraction. In 1967, Browder  was the pioneer to consider an implicit scheme and prove the following strong convergence theorem of this algorithm in a Hilbert space.
Theorem 1.1 Let C be a bounded closed convex subset of a Hilbert space and T a nonexpansive mapping on C. Let u be an arbitrary point of C and define by
Then as , converges strongly to a point of nearest to u.
The extension work of Browder’s type convergence theorems has been tremendously studied for not only one single nonexpansive mapping, but, most significantly, a semigroup of nonexpansive mappings; see, e.g., [6, 7] and the references therein.
This paper is devoted to studying Browder’s type iterations for a nonexpansive semigroup in a space, where , which is a specific type of metric space. Intuitively, triangles in a space are ‘slimmer’ than corresponding ‘model triangles’ in a standard space of constant curvature κ (see Section 2). Complete spaces are often called Hadamard spaces. A few recent new convergence results of classical iterations on spaces with are obtained; see, e.g., [8–10] and the references therein.
Theorem 1.2 (, Theorem 4.4])
Let C be a bounded closed convex subset of a space, a strongly continuous nonexpansive semigroup on C, and two sequences , such that . Choose arbitrarily a point and for each let be the fixed point of the mapping . Then and converges strongly to a point of nearest to .
In 2010, Acedo and Suzuki  proved a Browder type convergence theorem for uniformly asymptotically regular ( for short) nonexpansive semigroups (see Section 2) in Hilbert spaces under a weaker condition on and than that in Theorem 1.2.
Theorem 1.3 (, Theorem 2.3])
Let C be a closed convex subset of a Hilbert space, a and strongly continuous nonexpansive semigroup on C such that , and two sequences , such that . Fix and define a sequence in C by
Then converges strongly to a point of nearest to .
The preceding two theorems lead naturally to the question of whether or not they can be extended to a space with . The purpose of this article is to investigate this question with the geometric approaches in spaces.
This paper is organized as follows. In Section 2 we recall the definition of geodesic metric spaces and summarize some useful lemmas and the main properties of spaces. In Section 3 we present some technical results about Δ-convergence of a sequence in a complete space. In Section 4 we establish our main results (Theorems 4.2, 4.3, 4.4) of Browder’s iterations for nonexpansive semigroups in spaces and conclude that Theorems 1.2 and 1.3 can be generalized to spaces under the same respective conditions on the coefficients and . It is noteworthy that, however, without the assumption, the sequence established in Theorem 1.2 is not necessarily convergent to the nearest point of to if even when (and therefore ); see Example 4.5. Furthermore, we determine a necessary and sufficient condition for a Browder type convergence theorem associated to a nonexpansive semigroup on the unit sphere in an infinite-dimensional Hilbert space. We also propose an open problem in Section 5.
Let be a metric space. For any subset E of X and , the diameter of E and the distance from x to E are defined, respectively, by
We always denote the open ball and the closed ball centered at x with radius by and , respectively.
Let C be a closed convex subset of X and let be a family of self-mappings of C. Then is called
asymptotically regular on C if for any and any ,
uniformly asymptotically regular (in short ) on C if for any and any bounded subset D of C,
For , a geodesic path joining x to y (or a geodesic from x to y) is an isometric mapping such that , , i.e., , for all . Therefore . The image of γ is called a geodesic (segment) from x to y and we shall denote a definite choice of this geodesic segment by . We remark that composing γ with a translation (this is still an isometry), one can always choose the interval to be , where . A point in the geodesic will be written as , where , and so and .
Let . The metric space is said to be
a geodesic (metric) space if any two points in X are joined by a geodesic;
uniquely geodesic if there is exactly one geodesic joining x to y for all ;
r-geodesic space if any two points with are joined by a geodesic;
r-uniquely geodesic if any two points with are joined by a unique geodesic in X.
A subset C of X is convex if every pair of points can be joined by a geodesic in X and the image of every such geodesic is contained in C. If this condition holds for all points with , then C is said to be r-convex.
The n-dimensional sphere is the set , where denote the Euclidean scalar product. It is endowed with the following metric: Let be the function that assigns to each pair the unique number such that
Then is a metric , I.2.1].
Definition 2.1 Given a real number κ, we denote by the following metric spaces:
if , then is the Euclidean space ;
if , then is obtained from the sphere by multiplying the distance function by ;
if , then is obtained from the sphere by multiplying the distance function by , where is the hyperbolic n-space.
It is well known that is a geodesic metric space. If , then is uniquely geodesic. If , then is -uniquely geodesic, and any open ball (respectively, closed) ball of radius (respectively, ) in X is convex , I.2.11]. The diameter of will be denoted and thus is if , and ∞ otherwise.
Given two distinct points with there is a natural way to parameterize a unique geodesic joining A to B: consider the path , , where the initial vector is the unit vector in the direction of . We shall refer to the image of c as a minimal great arc joining A to B.
The spherical angle between two minimal great arcs issuing from a point of , with the initial vectors u and v, say, is the unique number such that . A spherical triangle △ in consists of a choice of three distinct points (its vertices) , and three minimal great arcs (its sides), one joining each other of vertices. The vertex angle at C is defined to be the spherical angle between the sides of △ joining C to A and C to B.
Proposition 2.2 (The Spherical Law of Cosines , I.2.13])
Let a geodesic triangle in () have sides a, b, c and angles α, β, γ at the vertices opposite to the sides of length a, b, c, respectively. Then
In particular, fixing a, b and κ, c is a strictly increasing function of γ, varying from to as γ varies from 0 to π.
A geodesic triangle △ in a metric space X consists of three points , its vertices, and a choice of three geodesic segments , , joining them, its sides. Such a geodesic triangle will be denoted or (less accurately if X is not uniquely geodesic) . If a point lies in the union of , and , then we write .
A triangle in is called a comparison triangle for in X if , and . Such a triangle always exists if the perimeter of △ is less than ; it is unique up to an isometry of , I.2.14]. A point is called a comparison point for if .
A geodesic triangle △ in X is said to satisfy the inequality if, given a comparison triangle for △, for all ,
where are respective comparison points of x, y.
Definition 2.3 If , then X is called a space if X is a geodesic space all of whose geodesic triangles satisfy the inequality.
If , then X is called a space if X is -geodesic and all geodesic triangles in X of perimeter less than satisfy the inequality.
In particular, Hilbert spaces are . A space is a space for every . A space X is uniquely geodesic (if ) and any open (respectively, closed) ball of radius (respectively, ) in X is convex , II.1.4].
Lemma 2.4 Let be a space and let . Then
, II.2. Exercise 2.3(1)] for and , we have
for , we have
, Lemma 3.3] if , for with , we have
Let p, q, r be three distinct points of X with . The κ-comparison angle between q and r at p, denoted , is the angle at in a comparison triangle for .
Let and let be two geodesic paths with . Given and , we consider the comparison triangle and the κ-comparison angle . The (Alexandrov) angle or the upper angle between the geodesic paths γ and is the number defined by
If X is uniquely geodesic, and , the angle of in X at p is the (Alexandrov) angle between the geodesic segments and issuing from p and is denoted .
Proposition 2.5 (, I.1.13, 1.14 and II.1.8])
Let X be a metric space and let γ, , be three geodesics issuing from a common point. Then
if is a geodesic with , and if are defined by and , then .
3 Basic properties and Δ-convergence
This section contains a number of primary results in  which are crucial to the study of our problem. The following proposition states very useful properties of the metric projection in a complete space.
Proposition 3.1 (, Proposition 3.5])
Let X be a complete space and let be nonempty closed and π-convex. Suppose that such that . Then the following are satisfied:
There exists a unique point such that .
If and with , then .
If , then for any ,
The mapping of X onto C in Proposition 3.1 is called the metric projection.
The next result shows the existence property of fixed points for a nonexpansive mapping.
Proposition 3.2 (, Theorem 3.9])
Let X be a complete space such that . Then every nonexpansive mapping has at least one fixed point.
The rest of this section is devoted to presenting several closely related characterizations of Δ-convergence. For this purpose, we start with some basic definitions of an asymptotic radius and an asymptotic center. Let be a bounded sequence in a complete space X. For and , let . The asymptotic radius of is given by
the asymptotic radius with respective to C of is given by
the asymptotic center of is given by the set
the asymptotic center with respective to C of is given by the set
Proposition 3.3 (, Proposition 4.1])
Let X be a complete space and nonempty closed and π-convex. If be a sequence in X such that , then consists of exactly one point.
Definition 3.4 A sequence in X is said to Δ-converge to if x is the unique asymptotic center of every subsequence of . In this case, we write and x is called the Δ-limit of .
The next result is an immediate consequence of the preceding proposition.
Proposition 3.5 (, Corollary 4.4])
Let X be a complete space and a sequence in X. If , then has a Δ-convergent subsequence.
4 Main results
Since the validity of all our results, including the proofs as well, on spaces can be restored on any space with by rescaling without major changes, we will pay our attention to spaces. In addition, when we deal with a space, the hypothesis for each theorem in this section is replaced by and so can be dropped if (in this case, ; refer to Section 2 for the definition).
To verify our result, the following basic property of asymptotical regularity is required; also cf. , Proposition 2.1] in a topological vector space.
Lemma 4.1 Let C be a subset of a metric space and let be a family of self-mappings of C such that , for all . If is asymptotically regular, then
Proof Fix . Then . To prove , let x be a fixed point of . For , we obtain
and therefore . □
Let be a complete space, C a closed π-convex subset of X and a nonexpansive mapping. Then is closed. Also it is seen that is π-convex. Indeed, for with , we have
and similarly . Thus both equalities must hold and hence and belong to the unique geodesic . This implies that .
Now consider a family of nonexpansive self-mappings of C such that . From the previous remark, each fixed point set is closed and convex, hence so is . Fix with . Let (the uniqueness follows from Proposition 3.1(i)) and . Then for each , maps the closed π-convex set into itself. For any and , define the mapping by . Observe that is a contraction. In fact, for , we apply Lemma 2.4(iii) to get
Then has a unique fixed point in .
Let denote the maximum integer no greater than t. We now extend Theorem 1.3 in Hilbert spaces to spaces as the following result.
Theorem 4.2 Let X be a complete space, C a closed π-convex subset of X, a and strongly continuous nonexpansive semigroup on C with , and two sequences , such that . Choose arbitrarily a point such that . Let and . Define a sequence in by the implicit iteration
Then converges strongly to the point p of nearest to .
Proof If , then reduces to the constant sequence . Suppose that . We shall prove that any subsequence of contains a subsequence converging strongly to p from which it follows that also converges strongly to the point p. Let be any subsequence of . Proposition 3.5 asserts that has a Δ-convergent subsequence. By passing to a subsequence we may assume that Δ-converges to a point y. Let and be the respective corresponding subsequences of and .
Observe that . To see this, take . There are three cases:
By passing to a subsequence if necessary it may be assumed that . We discuss each case as follows.
Suppose that . Fix and we obtain
This implies that
hence for all , that is, .
If , then
Hence from which it follows that by Lemma 4.1.
If , fix . For all sufficiently large n we get
where . It follows that and therefore . This finishes the proof that y is a common fixed point of all mappings in .
Next, we claim that converges strongly to y. We suppose on the contrary that
From , we obtain
and hence taking the limit superior as yields
since . Recall that
Recall that . We have and hence
According to (4.2) and (4.3), by passing to a subsequence again we may assume that
Let be a comparison triangle for in . Observe that
For, contrarily, if , since , then ; see Proposition 2.5. By the law of cosines in (Proposition 2.2), since , it follows that
which is a contradiction because is nonexpansive. Notice that
Hence (4.4) implies that
The previous two inequalities, together with the inequality, yield
Let be a comparison triangle for in . Since , Proposition 3.1(i) assures that there is a unique point nearest to . Let such that and . By passing to a subsequence again it may be assumed that and converge, respectively, to and . Since
this guarantees that .
On the other hand, according to (4.6), by passing to a subsequence we may suppose that
Let , , and . Then by (4.5), and . Observe that . Otherwise, since , we have which implies that
But this contradicts to (4.5). Therefore and so by Proposition 3.1(ii). The law of cosines in yields
Since , this implies that
It follows that
where . Taking the limit as yields , which is a contradiction. Thus and so converges strongly to y, as claimed.
It remains to prove that . Let and consider a comparison triangle for . From
it is seen that . Hence and so
We then take the limit as and obtain
which shows that y is the nearest point in to . Consequently, we conclude that converges strongly to , which completes the proof. □
It is worthy emphasizing that a uniformly asymptotical regularity hypothesis of Theorem 4.2 is superfluous when ; see case (i) in the proof of Theorem 4.2. Thus, we state the result as follows.
Theorem 4.3 Let X be a complete space, C a closed π-convex subset of X, a strongly continuous nonexpansive semigroup on C with , and two sequences , such that . Choose arbitrarily a point such that . Let and . Then the sequence in defined by (4.1) converges strongly to the point p.
Furthermore, if X is a space (recall that ) in Theorem 4.3, then the corresponding assumption is , which can be dropped. Moreover, the sequence defined by (4.1) is bounded. In fact, since
this shows that . Then is bounded and so is . We restate Theorem 1.2 without assuming boundedness of C as follows.
Theorem 4.4 Let X be a complete space, C a closed convex subset of X, a strongly continuous nonexpansive semigroup on C with , and two sequences , such that . Choose arbitrarily a point . Then the sequence in C defined by (4.1) converges strongly to a point of nearest to .
Nonetheless, Theorem 4.2 has a uniformly asymptotical regularity hypothesis that cannot in general be removed when as illustrated in the following example.
Example 4.5 Consider the complex plane , where is the absolute value, so that it is a complete space. For , define a self-mapping of ℂ by , . Then is a strongly continuous nonexpansive semigroup with . The family is not on ℂ because
Let , , , so that and . Choose . Define a sequence in ℂ by (4.1), that is,
Hence the sequence is divergent.
Next if we take , , , then and . Choose and define a sequence in ℂ by (4.1) to get
However, the sequence converges to .
These two examples explain that strong convergence of the sequence defined by (4.1) in Theorem 4.2 ceases to be true without the assumption if .
The next example shows that the condition in Theorem 4.2 is also necessary when X is the unit sphere in , the infinite-dimensional Hilbert space of square-summable sequences.
Example 4.6 Let be the unit sphere in endowed with the following intrinsic metric : for , such that
This space is a space.
Take one element of the canonical basis of . Let be a closed ball centered at v in , where . For , define a mapping by
and for ,
For , from Lemma 2.4(iii), we obtain
Then is nonexpansive. In fact, v is the unique fixed point of for . Furthermore, it is seen that is a and strongly continuous nonexpansive semigroup on C.
Choose with so that . For and , define by
We remark that . Let , be two sequences and define a sequence in by
We shall prove that if the sequence converges to the common fixed point v of , then .
First, assume that . Since and for , we obtain from (4.7) and (4.8) that
Therefore , which implies that
Let so that . Using (4.9), (4.10), and (4.11), we have
This shows that because and so .
To prove that , we recall the inequality for all . Since
it follows that
Consequently, and therefore as desired.
The following result is an immediate consequence of Theorem 4.2 and Example 4.6.
Theorem 4.7 Let X be the unit sphere in an infinite-dimensional Hilbert space, and let , be two sequences. Then the following conditions are equivalent:
Let C be a closed convex subset of X with , a and strongly continuous nonexpansive semigroup on C with , such that , , and is a sequence in defined by (4.1).
Then converges strongly to p.
A semitopological semigroup S is a semigroup equipped with a Hausdorff topology such that for each the mappings and from S into S are continuous. A semitopological semigroup S is left (respectively, right) reversible if any two closed right (respectively, left) ideals of S have nonvoid intersection, i.e., (respectively, ), for , where denotes the closure of a set E in a topological space. The class S of all left reversible semitopological semigroups includes all groups, all commuting semigroups, and all normal left amenable semitopological semigroups, i.e., the space of bounded continuous functions on S has a left invariant mean; see [2, 3, 13–15]. If S is left (respectively, right) reversible, is a directed system when the binary relation ⪯ on S is defined by if and only if (respectively, ), .
Let S be a semitopological semigroup and C a closed convex subset of a metric space . A family is called a representation of S on C if for each , is a mapping from C into C and , for all . The representation is called a strongly continuous nonexpansive semigroup on C (or a continuous representation of S as nonexpansive mappings on C) if the following conditions are satisfied (see ):
for each , is a nonexpansive mapping on C;
for each , the mapping from S into C is continuous.
The representation of S is called
asymptotically regular on C if for any and any ,
uniformly asymptotically regular (in short ) on C if for any and any bounded subset D of C,
Problem 5.1 Let X be a complete space, C a closed π-convex subset of X, S a commutative (or left reversible) semitopological semigroup, and a continuous representation of S as nonexpansive mappings on C. Can we obtain the result analogous to Theorem 4.2 corresponding to a continuous representation of S as nonexpansive mappings on C?
To answer this problem, the following conjecture is required and hence needs to be verified.
Conjecture 5.2 Let S be a commutative (or left reversible) semitopological semigroup, C a subset of a metric space and a representation of S on C. If is asymptotically regular, then
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This research was supported by a grant NSC 102-2115-M-259-004 from the National Science Council of Taiwan. The author, therefore, thanks the NSC financial support. The author would like to express the most sincere thanks to the referees for their careful reading of the manuscript, important comments and the citation of Ref. . The author is also very grateful to Professor Anthony Lau for giving valuable remarks and Refs.  and , and for suggesting presenting Problem 5.1 for future studies.
The author declares that she has no competing interests.
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Huang, S. Nonexpansive semigroups in spaces. Fixed Point Theory Appl 2014, 44 (2014). https://doi.org/10.1186/1687-1812-2014-44
- nonexpansive mapping
- asymptotically regular
- uniformly asymptotically regular