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Implicit Mann approximation with perturbations for nonexpansive semigroups in CAT(0) spaces
Fixed Point Theory and Applications volume 2012, Article number: 145 (2012)
Abstract
In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the △-convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Mann iteration schemes with bounded perturbations for nonexpansive semigroups in CAT(0) spaces. The results presented in this paper extend and enrich the existing literature.
MSC:47H05, 47H10, 47J25.
1 Introduction
Let be a metric space and K be a subset of X. A mapping is said to be nonexpansive if for all . We denote the set of all nonnegative real numbers by and the set of all fixed points of T by , i.e.,
For each , let be nonexpansive mappings and denote the common fixed points set of the family by . A family of mappings is said to be uniformly asymptotically regular if, for any bounded subset B of K,
for all .
A nonexpansive semigroup is a family
of mappings on K such that
-
(1)
for all and ;
-
(2)
is nonexpansive for each ;
-
(3)
for each , the mapping from to K is continuous.
We denote by the common fixed points set of nonexpansive semigroup Γ, i.e.,
Note that, if K is a nonempty, compact and convex subset of a Banach space, then is nonempty (see [1–3]).
A geodesic from x to y in X is a mapping Ψ from a closed interval to X such that , and for all . In particular, Ψ is an isometry and . The image Θ of Ψ is called a geodesic (or metric) segment joining x and y. The space is said to be a geodesic space if any two points of X are joined by a geodesic segment, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for any , which is denoted by and is called the segment joining x and y. A subset K of a geodesic space X is said to be convex if for any , .
A geodesic triangle in a geodesic metric space consists of three points , , in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle in is a triangle in such that for all . It is known that such a triangle always exists (see [4]). A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom (CA):
(CA) Let △ be a geodesic triangle in and let be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if, for all and all comparison points ,
The complete CAT(0) spaces are often called Hadamard spaces (see [5]). For any , we denote by the unique point which satisfies
It is known that if is a CAT(0) space and , then for any , there exists a unique point . For any , the following inequality holds:
where (for metric spaces of hyperbolic type, see [6]).
Recently, Cho et al. [7] studied the strong convergence of an explicit Mann iteration sequence for approximating a common fixed point of Γ in a CAT(0) space, where is generated by the following iterative scheme for a nonexpansive semigroup :
where and . The existence of fixed points, an invariant approximation and convergence theorems for several mappings in CAT(0) spaces have been studied by many authors (see [8–19]).
On the other hand, Thong [20] considered an implicit Mann iteration process for a nonexpansive semigroup on a closed convex subset C of a Banach space, as follows:
Under different conditions, Thong [20] proved the weak convergence and strong convergence results of implicit Mann iteration scheme (1.1) for nonexpansive semigroups in certain Banach spaces. In the last twenty years, many authors have studied the convergence of implicit iteration sequences for nonexpansive mappings, nonexpansive semigroups and pseudocontractive semigroups in Banach spaces (see [21–24] and the references therein). Readers may consult [11, 25, 26] for the convergence of Ishikawa iteration sequences for nonexpansive mappings and nonexpansive semigroups in certain Banach spaces. In the literature of approximating convergence for nonexpansive mappings and nonexpansive semigroups in CAT(0) spaces, explicit iteration schemes are very abundant but implicit iteration process remains unaddressed. Therefore, it is of interest to investigate the convergence of implicit Mann type process with perturbations for nonexpansive semigroups in CAT(0) spaces.
Motivated and inspired by the work mentioned above, we consider the following implicit Mann iteration scheme with perturbations for a family of nonexpansive mappings in a CAT(0) space:
where and are given sequences of real numbers, is a bounded sequence in K. We prove that generated by (1.2) is △-convergent to some point in under appropriate conditions. We also consider the following implicit Mann iteration process with perturbations for a nonexpansive semigroup in a CAT(0) space:
where and are given sequences of real numbers, is a bounded sequence in K. Under various and appropriate conditions, we obtain that generated by (1.3) converges strongly to a common fixed point of Γ. We extend the strong convergence result in [20] and establish the △-convergence results of implicit Mann type approximation for nonexpansive semigroups in CAT(0) spaces.
2 Definitions and lemmas
Let be a bounded sequence in a CAT(0) space . For any , denote
-
(i)
is called the asymptotic radius of ;
-
(ii)
is called the asymptotic radius of with respect to K;
-
(iii)
the set is called the asymptotic center of ;
-
(iv)
the set is called the asymptotic center of with respect to K.
A sequence in a CAT(0) space X is said to be △-convergent to a point x in X, if x is the unique asymptotic center of for all subsequences . In this case, we write △- and x is called the △-limit of .
For the sake of convenience, we restate the following lemmas that shall be used.
Lemma 2.1 [11]
Let be a CAT(0) space. Then,
for all and .
Lemma 2.2 [11]
Let be a CAT(0) space. Then,
for all and .
Lemma 2.3 [11]
Let K be a closed convex subset of a complete CAT(0) space and be a nonexpansive mapping. Suppose that is a bounded sequence in K such that and converges for all . Then, , where the union is taken over all subsequences of . Moreover, consists of exactly one point.
Lemma 2.4 [7]
Let and be bounded sequences in a CAT(0) space X. Let be a sequence in such that . Define for all and suppose that
Then .
Lemma 2.5 [26]
Let , and be sequences of nonnegative real numbers such that
where is some nonnegative integer. If and , then exists.
3 Main results
To focus on the convergence results of this present paper, it is necessary to show that the sequences generated by implicit Mann iteration processes (1.2) and (1.3) are well defined.
Lemma 3.1 Let K be a nonempty, closed and convex subset of a complete CAT(0) space X and be nonexpansive mappings. Suppose that is a bounded sequence in K, and are given parameter sequences. Then the sequence generated by implicit Mann iteration process (1.2) is well defined.
Proof For each and any given , define a mapping by
It can be verified that for each fixed , is a contractive mapping. Indeed, if setting and , then we have and . It follows from Lemmas 2.2 and 2.1 that
Consequently, we have that and . Thus,
which shows that for each , is a contractive mapping. By induction, Banach’s fixed theorem yields that the sequence generated by (1.2) is well defined. This completes the proof. □
We need the following lemma for our main results. The analogs of [7], Lemma 3.1] and [28], Lemma 2.2] are given below. We sketch the proof here for the convenience of the reader.
Lemma 3.2 Let K be a nonempty, closed and convex subset of a complete CAT(0) space X, be a bounded sequence in K and be nonexpansive mappings. Let and be given sequences such that . Suppose that generated by (1.2) is bounded (or, equivalently, is bounded) and either
holds. If , then .
Proof First, we show the equivalence between the boundedness of and the boundedness of . If is bounded, then set
for some given point and , . Setting , we know that there exists such that for all ,
It follows from Lemma 2.1 that
Hence, for all , from (3.1) we have
which means that is bounded.
Conversely, if is bounded, then set
for some given point . Denote . From Lemma 2.1, we have
By induction, we know that , which shows that is bounded.
Then, we prove Lemma 3.2. If , then we have
Similarly, if , then from inequality (3.2) we have
It follows from Lemma 2.4 that . Note that
We have that . This completes the proof. □
As a direct consequence of Lemma 3.2, the following lemma is immediate.
Lemma 3.3 Let K be a nonempty, closed and convex subset of a complete CAT(0) space X, be a bounded sequence in K and be nonexpansive mappings. Let and be given sequences such that . Suppose that generated by (1.2) is bounded (or, equivalently, is bounded) and
holds. If , then .
We now present our main results. The following theorem discusses the △-convergence of implicit Mann iteration sequence (1.2) with perturbations for a family of nonexpansive mappings in CAT(0) spaces.
Theorem 3.1 Let K be a nonempty, closed and convex subset of a complete CAT(0) space X, and be uniformly asymptotically regular and nonexpansive mappings such that . Let be a bounded sequence in K, and be given sequences such that . Then the sequence generated by (1.2) is well defined. Suppose that and either
holds. Then △-converges to some point in .
Proof By Lemma 3.1, we know that the sequence generated by (1.2) is well defined. For any , from (1.2) and Lemma 2.1, we have
Since , it follows from (3.1) that for all ,
Since and is bounded, Lemma 2.5 yields that converges and thus is bounded.
Applying Lemma 3.2, we have . We prove that for each ,
Because the family of nonexpansive mappings is uniformly asymptotically regular, we have
Since converges for any , an application of Lemma 2.3 yields that consists of exactly one point and is contained in , for all . This shows that △-converges to some point in . This completes the proof. □
By Lemma 3.3 and Theorem 3.1, the following theorem holds trivially.
Theorem 3.2 Let K be a nonempty, closed and convex subset of a complete CAT(0) space X, and be uniformly asymptotically regular and nonexpansive mappings such that . Let be a bounded sequence in K, and be given sequences such that . Then the sequence generated by (1.2) is well defined. Suppose that and
holds. Then △-converges to some point in .
Finally, we study the strong convergence of implicit Mann iteration sequence (1.3) with perturbations for a nonexpansive semigroup in CAT(0) spaces, under various and appropriate conditions.
Theorem 3.3 Let K be a compact and convex subset of a complete CAT(0) space X, and be a nonexpansive semigroup on K. Let be a bounded sequence in K, and be given sequences such that . Then the sequence generated by implicit scheme (1.3) is well defined. Suppose that is a sequence in such that
If , then converges strongly to some point in .
Proof Following the proof details of the main result of [2], we know that in complete CAT(0) spaces is nonempty (see [2, 7]). From Lemma 3.1, it is easy to see that the sequence generated by (1.3) is well defined. We show that
Assume for the contrary that (3.4) does not hold. There exist a subsequence , a sequence and an such that for all ,
Since K is compact, there exists a convergent subsequence of . Without loss of generality, we assume that as . Obviously, and so
which is a contradiction. Formula (3.4) follows readily. By Lemma 3.3, we have
Following the proof of [7], Theorem 3.5], we can show that there exists a subsequence convergent to , where is a common fixed point of . Since is a cluster of , we have . It follows from (1.3) and (3.3) that exists. Hence, , which completes the proof. □
Remark 3.1 The proofs of Theorems 3.1 and 3.3 are respectively similar to [7], Theorems 3.4 and 3.5]. As we know, the existing literature of approximating convergence for several mappings in CAT(0) spaces restricts to explicit iteration schemes. Theorems 3.1-3.3 extend and develop some existing results such as [7], Theorems 3.4 and 3.5] and [11], Theorems 3.2-3.4] from explicit Mann iteration schemes to implicit Mann iteration processes with perturbations.
We prove another strong convergence theorem which is different from Theorem 3.3.
Theorem 3.4 Let K be a compact and convex subset of a complete CAT(0) space X, and be a nonexpansive semigroup on K. Let be a bounded sequence in K, and be given sequences. Then the sequence generated by scheme (1.3) is well defined. If
and , then converges strongly to a common fixed point of Γ.
Proof Following the proof details of the main result of [2], we know that in complete CAT(0) spaces is nonempty (see [2, 7]). From Lemma 3.1, we know that generated by (1.3) is well defined.
Claim 1: If is a sequence of nonnegative real numbers such that , then
Assume for the contrary that (3.5) does not hold. There exist a subsequence , a sequence and an such that for all ,
Since K is compact, there exists a convergent subsequence of . Without loss of generality, we assume that as . Obviously, and so
which is a contradiction. Formula (3.5) follows readily.
Claim 2: . Since K is a compact and convex subset of X, there exists a subsequence such that as . It follows from (1.3) and Lemma 2.1 that
Thus, we have
For any given , let and . Since K is compact and is bounded, it follows that and . Also, we know that and for sufficiently large , because and . Consequently,
Hence, it follows from (3.5) that
For any given , from (3.6) we know that
where is the integer part of . Since , it follows from (3.5) and (3.7) that
Therefore, from the above equality, we know that . From (3.3), Lemma 2.5 yields that exists and thus converges strongly to as . This completes the proof. □
Remark 3.2 The main results presented in this paper can be immediately applied to any CAT(k) space with , because any CAT(k) space is a CAT() space for any (see [4, 7]).
If X is a Banach space, from Theorem 3.4, we have the following corollary.
Corollary 3.1 Let K be a compact and convex subset of a Banach space X, and be a nonexpansive semigroup on K. Let be a bounded sequence in K, and be given sequences. Then the sequence generated by
is well defined. If
and , then converges strongly to a common fixed point of Γ.
Remark 3.3 When , Corollary 3.1 reduces to [20], Theorem 2.3]. Therefore, Theorem 3.4 and Corollary 3.1 extend [20], Theorem 2.3] from implicit Mann iteration processes to implicit Mann iteration processes with bounded perturbations.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11026063, 11171237, 11101069). This work was also supported by a Hong Kong Polytechnic University Postdoctoral Fellowship (Number G-YX4N) and an Internal Competitive Research Grant (Number A-PL05).
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Li, Xs., Yip, Tl. Implicit Mann approximation with perturbations for nonexpansive semigroups in CAT(0) spaces. Fixed Point Theory Appl 2012, 145 (2012). https://doi.org/10.1186/1687-1812-2012-145
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DOI: https://doi.org/10.1186/1687-1812-2012-145