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Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces
Fixed Point Theory and Applications volume 2012, Article number: 155 (2012)
Abstract
In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subset C of a complete CAT(0) space X and prove that the proposed sequence converges to a common fixed point for these semigroups of nonexpansive mappings. Note that our results generalize Cho et al. result (Nonlinear Anal. 74:6050-6059, 2011) and related results.
1 Introduction
Fixed point theory in CAT(0) spaces was first studied by Kirk [1, 2]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared; for example, one can see [3–6] and related references.
Let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map c from a closed interval to X such that , , and for all . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by . The space is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each . A subset is said to be convex if Y includes every geodesic segment joining any two of its points.
A geodesic triangle in a geodesic space consists of three points , , and in X (the vertices of Δ and a geodesic segment between each pair of vertices (the edge of Δ)). A comparison triangle for geodesic triangle in is a triangle in the Euclidean plane such that for .
A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT(0): Let Δ be a geodesic triangle in X, and let be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0) inequality if for all and all comparison points , , . It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include pre-Hilbert spaces [7], R-trees [8], the complex Hilbert ball with a hyperbolic metric [9], and many others.
If x, , are points in a CAT(0) space, and if is the midpoint of the segment , then the CAT(0) inequality implies
This is the (CN) inequality of Bruhat and Tits [10]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [[7], p. 163].
In 2008, Dhompongsa and Panyanak [11] gave the following result, and the proof is similar to the proof of the remark in [[12], p. 374].
Lemma 1.1 [11]
Let X be a CAT(0) space. Then
for all and .
By the above lemma, we know that CAT(0) space is a convex metric space. Indeed, it is a metric space X with a convex structure if there exists a mapping such that
for all and and this space X is called a convex metric space [13]. Furthermore, Takahashi [13] has proved that
for all and when X is a convex metric space with a convex structure. So, we also get the following result, and it is proved in [11].
Lemma 1.2 [11]
Let X be a CAT space and . For each , there exists a unique point such that and .
For convenience, from now on we will use the notation . Therefore, we have
Let C be a nonempty closed convex subset of a CAT(0) space X, and let T be a nonexpansive mapping on C, i.e., such that for all . We use to denote the set of fixed points of T, i.e., . Let be the set of positive integers, be the set of real numbers, and let be the set of nonnegative real numbers.
A family of mappings is called a one-parameter continuous semigroup of nonexpansive mappings on a nonempty closed convex subset C of a CAT(0) space X if the following conditions hold:
(SG)1 for each , is a nonexpansive mapping on C;
(SG)2 for all ;
(SG)3 for each , the mapping from into C is continuous.
A family of mappings is called a one-parameter strongly continuous semigroup of nonexpansive mappings on a nonempty closed convex subset C of a CAT(0) space X if conditions (SG) i , , and the following condition are satisfied:
(SG)4 for all .
Note that if C is a nonempty compact subset of a Banach space and is a semigroup of nonexpansive mappings, then [14]; see also [15–17] and others. For the example of a one-parameter continuous semigroup of nonexpansive mappings, one can see [18].
Construction of common fixed points of a nonexpansive semigroup is an important subject in the theory of nonexpansive semigroup mappings and its applications. Fox example, one can refer to [19, 20]. In [21], Shioji and Takahashi introduced the implicit iteration
-
(A)
, ,
where C is a nonempty closed convex subset of a real Hilbert space H, , is a sequence in , is a sequence of positive real numbers divergent to ∞. Under suitable conditions, Shioji and Takahashi [21] proved strong convergence of to a member of . Note that their iterate at step n is constructed through the average of a semigroup over the interval .
In 2003, Suzuki [22] introduced the following implicit iteration process in a Hilbert space:
-
(B)
,
for a nonexpansive semigroup, where C is a nonempty closed convex subset of a real Hilbert space H, , is a sequence in , is a sequence of positive real numbers. Note that is constructed directly from the . So, Zegeye and Shahzad [23] viewed Suzuki’s iteration process (B) as an extension of the implicit process (A) to nonexpansive semigroups.
In 2005, Suzuki [24] considered an iterative process for a one-parameter continuous semigroup of nonexpansive mappings on C, where C is a nonempty compact convex subset of a Banach space E, defined by
where and . Then Suzuki [24] proved that converges strongly to a common fixed point of if
In 2009, Dhompongsa et al. [25] gave the following important result for a strongly continuous semigroup of nonexpansive mappings.
Theorem 1.1 [25]
Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X, and let be a strongly continuous semigroup of nonexpansive mappings on C. Let and be sequences of real numbers satisfying , , and . Let , and let be a sequence in C with
Then and converges to the element of nearest to .
In 2011, Cho et al. [26] gave the following result for a continuous semigroup of nonexpansive mappings on a nonempty compact convex subset C of a complete CAT(0) space X.
Theorem 1.2 [26]
Let C be a nonempty compact convex subset of a complete CAT(0) space X, and be a one-parameter continuous semigroup of nonexpansive mappings on C. Let be a sequence in satisfying
For any , define a sequence in C by
Then converges to a common fixed point of the semigroup .
Remark 1.1 By Theorems 1.1 and 1.2, we know that
-
(a)
if C is a nonempty bounded closed convex subset of a complete CAT(0) space X, and is a strongly continuous semigroup of nonexpansive mappings on C, then ;
-
(b)
if C is a compact convex subset of a complete CAT(0) space X, and is a one-parameter continuous semigroup of nonexpansive mappings on C, then .
Motivated by the above works and related results, we study the following iteration processes for some families of one-parameter continuous semigroups of nonexpansive mappings on a nonempty compact convex subset C of a complete CAT(0) space X.
Let and be one-parameter continuous semigroups of nonexpansive mappings on a nonempty compact convex subset C of a complete CAT(0) space X. We consider the following iteration processes (1.3), (1.4), and (1.5):
where and are sequences in , and are sequences in . We prove that the proposed sequences converge to a common fixed point of these families of mappings. Note that our results generalize Theorem 1.1 (i.e., Theorem 3.5 in [26]).
Besides, Thong [27] considered an implicit iteration for nonexpansive semigroups on a nonempty compact convex subset C of a real Banach space E as follows:
where is a sequence in , and is a sequence in .
In this paper, motivated by [27], we also consider the following implicit iteration process for some families of one-parameter continuous semigroups of nonexpansive mappings on a nonempty compact convex subset C of a complete CAT(0) space X:
where and are sequences in , and are sequences in .
For a special case of the iteration process (1.6), we have the following types:
We prove the proposed sequences converge to a common fixed point of three families of mappings. Our result for the iteration process (1.7) is similar to (1.5) on complete CAT(0) spaces. Our results for the iteration processes (1.6) and (1.8) generalize Theorem 1.2. Note that the iteration process (1.8) is also a special case of the iteration processes (1.3) and (1.4).
2 Preliminaries
In 2005, Suzuki [24] gave the following result, and it is an important tool in this paper.
Lemma 2.1 [24]
Let be a real sequence and Ï„ be a real number satisfying . Suppose that either of the following holds:
-
(i)
or
-
(ii)
.
Then Ï„ is a cluster point of . Moreover, for any and , there exists such that for all with .
Lemma 2.2 [11]
Let X be a CAT(0) space. Then
for all and .
Definition 2.1 Let be a bounded sequence in a CAT(0) space X, and let C be a subset of X. Now, we use the following notations:
-
(i)
.
-
(ii)
.
-
(iii)
.
-
(iv)
.
-
(v)
.
Note that is called an asymptotic center of if . It is known that in a CAT(0) space, consists of exactly one point [28].
Definition 2.2 [6]
Let be a CAT(0) space. A sequence in X is said to be Δ-convergent to if x is the unique asymptotic center of for every subsequence of . That is, for every subsequence of . In this case, we write Δ- and call x the Δ-limit of .
Lemma 2.3 [6]
Let be a CAT(0) space. Then every bounded sequence in X has a Δ-convergent subsequence.
Lemma 2.4 [29]
Let C be a nonempty closed convex subset of a CAT(0) space X. If is a bounded sequence in C, then the asymptotic center of is in C.
Lemma 2.5 [6]
Let C be a nonempty closed convex subset of a complete CAT(0) space X, and let be a nonexpansive mapping. Let be a bounded sequence in C with Δ- and . Then and .
Lemma 2.6 [30]
Let X be a CAT(0) space. Let and be two bounded sequences in X with . If Δ-, then Δ-.
Next, we give the following results and these results show that the intersection of the fixed point sets for a continuous semigroup of nonexpansive mappings is nonempty. Note that Theorem 2.1 is different from Theorem 1.1.
A: Common fixed point for a strongly nonexpansive semigroup
Lemma 2.7 Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. Let be a continuous semigroup of nonexpansive mappings on C. Let be a sequence in . Let be a sequence in , and let be defined as
Assume that . Then if and only if .
Proof Suppose that . Since C is bounded, there exist a subsequence of and such that Δ-. By Lemma 2.5, .
Take any and let t be fixed. For with , we have
for some (note that C is bounded). Hence, we know that
for each . Since Δ-, for each . Therefore, .
Conversely, suppose that . Now, take any , and let w be fixed. Then we have
And this implies that and exists. Hence,
By assumption, . Next, we get
This implies that and . Furthermore, it follows that
Therefore, the proof is completed. □
By Lemma 2.7, we get the following theorem, and it is different from Theorem 1.1.
Theorem 2.1 Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. Let be a strongly continuous semigroup of nonexpansive mappings on C. Let be a sequence in , and be a sequence in with . Let be defined as
Then . Furthermore, if C is a compact set, then for some .
Proof Since is a strongly continuous semigroup of nonexpansive mappings on C, it is easy to see that . By Lemma 2.7, , and the proof is completed. □
B: Common fixed point for a nonexpansive semigroup
Lemma 2.8 Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X, and be a one-parameter continuous semigroup of nonexpansive mappings on C. Let be a sequence in satisfying
For any , define a sequence in C by
Then if and only if .
Proof Suppose that . Following the same argument as in the proof of Theorem 4 in [24], we get . Conversely, if , it is easy to see that . □
By Lemma 2.8, we get the following theorem. Notice also that it is a consequence of Theorem 1.2.
Theorem 2.2 Let C be a nonempty compact convex subset of a complete CAT(0) space X, and be a one-parameter continuous semigroup of nonexpansive mappings on C. Let be a sequence in satisfying
For any , define a sequence in C by
Then .
Proof Following the same argument as in the proof of Theorem 3.5 in [26], we can prove that . By Lemma 2.8, . □
3 Main results
Theorem 3.1 Let C be a nonempty compact convex subset of a complete CAT(0) space X. Let and be continuous semigroups of nonexpansive mappings on C. Let and be sequences in . Let and be sequences in . Suppose that
Let be defined as
Assume that
-
(i)
, ;
-
(ii)
;
-
(iii)
either or holds;
-
(iv)
either or holds.
Then for some .
Proof Take any , and let w be fixed. Then for each ,
and
Hence, exists, and and are bounded sequences. By (3.2), we get
And this implies that
By (3.1) and (3.4), we have
By assumption and (3.5),
By (3.2), we also have
and this inequality and assumption imply that
By (3.6), we have
By (3.7), it is easy to see that
By (3.7), (3.8), and (3.9), it is easy to see that
Furthermore, by (3.6), it follows that
By (3.10), it follows that
Next, fix with
Following the same argument as in the proof of Theorem 4 in [24], we choose a subsequence of such that
For completeness, we give the following proof. By (3.6) and (3.10), there exists such that
for all . By Lemma 2.1, we note that is a cluster point of and . Hence, there exists such that
By (3.6) and (3.10) again, there exists such that
for all . By Lemma 2.1 again, we note that is a cluster point of and . Hence, there exists such that
Continuing this argument, we can define a subsequence of satisfying
for all . Then it is obvious that for all ,
and
We also have
Similarly,
Since is a bounded sequence, there exist a subsequence of and such that Δ-. Let , , , and . Then we get
and
Since Δ-, we know that . By (3.8), Δ-, and Lemma 2.6, we know that Δ-, and this implies that . Also, we have
Similarly, .
Take any , and let t be fixed. Then for with , we have
By (3.11) and (3.13), we get
Since Δ-, we get for each . Therefore, is a common fixed point of . Furthermore, we know that by following the same argument.
In fact, since C is a compact set, we may assume that . So, is a cluster point of and . Since exists for each , . Therefore, the proof is completed. □
Remark 3.1 Theorem 3.1 generalizes Theorem 1.2.
Theorem 3.2 Let C be a nonempty compact convex subset of a complete CAT(0) space X. Let and be continuous semigroups of nonexpansive mappings on C. Let and be sequences in . Let and be sequences in . Suppose that
Let be defined as
Assume that
-
(i)
, ;
-
(ii)
;
-
(iii)
either or holds;
-
(iv)
either or holds.
Then for some .
Proof Take any , and let w be fixed. Then for each ,
and
Hence, exists, , and and are bounded sequences. By (3.14) and , we have
By assumption and (3.16),
By (3.15) and exists,
By (3.17), we have
Next, following the same argument as in the proof of Theorem 3.1, we get the proof of Theorem 3.2. □
Remark 3.2 Theorem 1.2 is also a special case of Theorem 3.2.
Theorem 3.3 Let C be a nonempty compact convex subset of a complete CAT(0) space X. Let and be continuous semigroups of nonexpansive mappings on C. Let and be sequences in . Let and be sequences in with , . Suppose that . Let be defined as
Assume that
-
(i)
;
-
(ii)
either or holds;
-
(iii)
.
Then for some .
Proof Take any and let w be fixed. Then for each ,
and
By (3.20) and (3.21),
Hence, exists, , and are bounded sequences. By (3.20) and assumptions,
By (3.21) and assumptions,
By (3.23) and (3.24), we have
Furthermore, by (3.23) and (3.25), it follows that
Following the same argument as in the proof of Theorem 3.1, there exist a subsequence of and such that , and is a common fixed point of . Hence, .
For , by (3.26), we get , and it is easy to see that .
Next, for each ,
for some .
By assumptions and (3.26), we know that
for each , and this implies that is also a common fixed point of . Now, and this implies that exists. Since , we know that , and the proof is completed. □
Remark 3.3 Theorem 3.3 is also a generalization of Theorem 1.2.
The following result is similar to Theorem 2.3 in [27].
Corollary 3.1 Let C be a nonempty compact convex subset of a complete CAT(0) space X. Let be a continuous semigroup of nonexpansive mappings on C. Let be a sequence in . Let be a sequence in with . Let be defined as
Assume that . Then for some .
Proof For each , let be defined by for each . Clearly, is a continuous semigroup of nonexpansive mappings on C. Since C is a compact set, . By Theorem 3.3, we get the conclusion. □
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This research was supported by the National Science Council of Republic of China.
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LJL: Problem resign, coordinator, discussion, revise the important part, and submit. CSC: Responsible for the important results of this paper, discuss, and draft. ZTY: Responsible for the results of this paper, discuss, and draft.
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Lin, LJ., Chuang, CS. & Yu, ZT. Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces. Fixed Point Theory Appl 2012, 155 (2012). https://doi.org/10.1186/1687-1812-2012-155
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DOI: https://doi.org/10.1186/1687-1812-2012-155