Abstract
In this article, we obtain the demiclosed principle, fixed point theorems and convergence theorems for the class of total asymptotically nonexpansive mappings on spaces with . Our results generalize the results of Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Tang et al. (Abstr. Appl. Anal. 2012:965751, 2012), Karapınar et al. (J. Appl. Math. 2014:738150, 2014) and many others.
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1 Introduction
For a real number κ, a space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.
Fixed point theory in spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on spaces (see, e.g., [3–11]). Since any space is a space for , all results for spaces immediately apply to any space with . However, there are only a few articles that contain fixed point results in the setting of spaces with .
The concept of total asymptotically nonexpansive mappings was first introduced in Banach spaces by Alber et al. [12]. It generalizes the concept of asymptotically nonexpansive mappings introduced by Goebel and Kirk [13] as well as the concept of nearly asymptotically nonexpansive mappings introduced by Sahu [14]. In 2012, Chang et al. [15] studied the demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in the setting of spaces. Since then the convergence of several iteration procedures for this type of mappings has been rapidly developed and many of articles have appeared (see, e.g., [16–24]). Among other things, under some suitable assumptions, Karapınar et al. [24] obtained the demiclosed principle, fixed point theorems, and convergence theorems for the following iteration.
Let K be a nonempty closed convex subset of a space X and be a total asymptotically nonexpansive mapping. Given , and let be defined by
where and are sequences in .
In this article, we extend Karapınar et al.’s results to the general setting of space with .
2 Preliminaries
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 of c is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by . This means that if and only if there exists such that
In this case, we write . The space is said to be a geodesic space (D-geodesic space) if every two points of X (every two points of distance smaller than D) are joined by a geodesic, and X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic joining x and y for each (for with ). A subset K of X is said to be convex if K includes every geodesic segment joining any two of its points. The set K is said to be bounded if
Now we introduce the model spaces , for more details on these spaces the reader is referred to [25]. Let . We denote by the metric space endowed with the usual Euclidean distance. We denote by the Euclidean scalar product in , that is,
Let denote the n-dimensional sphere defined by
with metric , .
Let denote the vector space endowed with the symmetric bilinear form which associates to vectors and the real number defined by
Let denote the hyperbolic n-space defined by
with metric such that
Definition 2.1 Given , we denote by the following metric spaces:
-
(i)
if , then is the Euclidean space ;
-
(ii)
if , then is obtained from the spherical space by multiplying the distance function by the constant ;
-
(iii)
if , then is obtained from the hyperbolic space by multiplying the distance function by the constant .
A geodesic triangle in a geodesic space consists of three points x, y, z in X (the vertices of △) and three geodesic segments between each pair of vertices (the edges of △). A comparison triangle for a geodesic triangle in is a triangle in such that
If , then such a comparison triangle always exists in . If , then such a triangle exists whenever , where . A point is called a comparison point for if .
A geodesic triangle in X is said to satisfy the inequality if for any and for their comparison points , one has
Definition 2.2 If , then X is called a space if and only if X is a geodesic space such that all of its geodesic triangles satisfy the inequality.
If , then X is called a space if and only if X is -geodesic and any geodesic triangle in X with satisfies the inequality.
Notice that in a space if , then the inequality implies
This is the (CN) inequality of Bruhat and Tits [26]. This inequality is extended by Dhompongsa and Panyanak [27] as
for all and . In fact, if X is a geodesic space, then the following statements are equivalent:
-
(i)
X is a space;
-
(ii)
X satisfies (CN);
-
(iii)
X satisfies (CN∗).
Let . Recall that a geodesic space is said to be R-convex for R (see [28]) if for any three points , we have
It follows from (CN∗) that a geodesic space is a space if and only if is R-convex for . The following lemma is a consequence of Proposition 3.1 in [28].
Lemma 2.3 Let and be a space with for some . Then is R-convex for .
The following lemma is also needed.
Lemma 2.4 ([[25], p.176])
Let and be a complete space with for some . Then
for all and .
We now collect some elementary facts about spaces. Most of them are proved in the setting of spaces. For completeness, we state the results in with .
Let be a bounded sequence in a space . For , we set
The asymptotic radius of is given by
and the asymptotic center of is the set
It is known from Proposition 4.1 of [8] that in a space X with , consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.
A sequence in X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of . In this case we write and call x the Δ-limit of .
Lemma 2.6 Let and be a complete space with for some . Then the following statements hold:
-
(i)
[[8], Corollary 4.4] Every sequence in X has a Δ-convergent subsequence;
-
(ii)
[[8], Proposition 4.5] If and , then , where .
By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[27], Lemma 2.8]).
Lemma 2.7 Let and be a complete space with for some . If is a sequence in X with and is a subsequence of with and the sequence converges, then .
Definition 2.8 Let K be a nonempty subset of a space . A mapping is called total asymptotically nonexpansive if there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
A point is called a fixed point of T if . We denote with the set of fixed points of T. A sequence in K is called approximate fixed point sequence for T (AFPS in short) if
Algorithm 1 The sequence defined by and
is called an Ishikawa iterative sequence (see [30]).
If for all , then Algorithm 1 reduces to the following.
Algorithm 2 The sequence defined by and
is called a Mann iterative sequence (see [31]).
The following lemma is also needed.
Lemma 2.9 ([[32], Lemma 1])
Let and be sequences of nonnegative real numbers satisfying
If , then exists.
3 Main results
3.1 Existence theorems
Theorem 3.1 Let and be a complete space with for some . Let K be a nonempty closed convex subset of X, and let be a continuous total asymptotically nonexpansive mapping. Then T has a fixed point in K.
Proof Fix . We can consider the sequence as a bounded sequence in K. Let be a function defined by
Then there exists such that . Since T is total asymptotically nonexpansive, for each , we have
Let . Taking in (2), we get that
This implies that
In view of (1), we have
Taking , we get that
yielding
By (3) and (4), we have . Therefore, is a Cauchy sequence in K and hence converges to some point . Since T is continuous,
□
From Theorem 3.1 we shall now derive a result for spaces which can also be found in [24].
Corollary 3.2 Let be a complete space and K be a nonempty bounded closed convex subset of X. If is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.
Proof It is well known that every convex subset of a space, equipped with the induced metric, is a space (cf. [25]). Then is a space and hence it is a space for all . Notice also that K is R-convex for . Since K is bounded, we can choose and so that . The conclusion follows from Theorem 3.1. □
3.2 Demiclosed principle
Theorem 3.3 Let and be a complete space with for some . Let K be a nonempty closed convex subset of X, and let be a uniformly continuous total asymptotically nonexpansive mapping. If is an AFPS for T such that , then and .
Proof By Lemma 2.6, . As in Theorem 3.1, we define for each . Since , by induction we can show that for all (cf. [16]). This implies that
In (5), taking , we have
Hence
In view of (1), we have
where . Since , letting , we get that
yielding
By (6) and (7), we have . Since T is continuous,
□
As we have observed in Corollary 3.2, we can derive the following result from Theorem 3.3.
Corollary 3.4 ([[24], Theorem 12])
Let be a complete space, K be a nonempty bounded closed convex subset of X, and be a uniformly continuous total asymptotically nonexpansive mapping. If is an AFPS for T such that , then and .
3.3 Convergence theorems
We begin this section by proving a crucial lemma.
Lemma 3.5 Let and be a complete space with for some . Let K be a nonempty closed convex subset of X, and be a uniformly continuous total asymptotically nonexpansive mapping with and . Let and be a sequence in K defined by
where and are sequences in such that . Then is an AFPS for T and exists for all .
Proof It follows from Theorem 3.1 that . Let and . Since T is total asymptotically nonexpansive, by Lemma 2.4 we have
This implies that
Since and , by Lemma 2.9 exists. Next, we show that is an AFPS for T. In view of (1), we have
This implies that
Again by (1), we have
Substituting this into (8), we get that
yielding
Since and , we have
This implies by that
By the uniform continuity of T, we have
It follows from (9) and the definitions of and that
By (9), (10), and (11), we have
□
Now, we are ready to prove our Δ-convergence theorem.
Theorem 3.6 Let and be a complete space with for some . Let K be a nonempty closed convex subset of X, and let be a uniformly continuous total asymptotically nonexpansive mapping with and . Let and be a sequence in K defined by
where and are sequences in such that . Then Δ-converges to a fixed point of T.
Proof Let where the union is taken over all subsequences of . We can complete the proof by showing that is contained in and consists of exactly one point. Let , then there exists a subsequence of such that . By Lemma 2.6, there exists a subsequence of such that . Hence by Lemma 3.5 and Theorem 3.3. Since exists, by Lemma 2.7. This shows that . Next, we show that consists of exactly one point. Let be a subsequence of with , and let . Since , by Lemma 3.5 exists. Again, by Lemma 2.7, . This completes the proof. □
As a consequence of Theorem 3.6, we obtain the following.
Corollary 3.7 ([[24], Theorem 17])
Let be a complete space, K be a nonempty bounded closed convex subset of X, and be a uniformly continuous total asymptotically nonexpansive mapping with and . Let and be a sequence in K defined by
where and are sequences in such that . Then Δ-converges to a fixed point of T.
Recall that a mapping is said to be semi-compact if K is closed and each bounded AFPS for T in K has a convergent subsequence. Now, we prove a strong convergence theorem for uniformly continuous total asymptotically nonexpansive semi-compact mappings.
Theorem 3.8 Let and be a complete space with for some . Let K be a nonempty closed convex subset of X, and let be a uniformly continuous total asymptotically nonexpansive mapping with and . Let and be a sequence in K defined by
where and are sequences in such that . Suppose that is semi-compact for some . Then converges strongly to a fixed point of T.
Proof By Lemma 3.5, . Since T is uniformly continuous, we have
as . That is, is an AFPS for . By the semi-compactness of , there exist a subsequence of and such that . Again, by the uniform continuity of T, we have
That is, . By Lemma 3.5, exists, thus p is the strong limit of the sequence itself. □
Corollary 3.9 ([[24], Theorem 22])
Let be a complete space, K be a nonempty bounded closed convex subset of X, and be a uniformly continuous total asymptotically nonexpansive mapping with and . Let and be a sequence in K defined by
where and are sequences in such that . Suppose that is semi-compact for some . Then converges strongly to a fixed point of T.
Remark 3.10 The results in this article also hold for the class of weakly total asymptotically nonexpansive mappings in the following sense. A mapping is called weakly total asymptotically nonexpansive if there exist nonnegative real sequences , with , as and a nondecreasing function such that
Author’s contributions
The author completed the paper himself. The author read and approved the final manuscript.
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Panyanak, B. On total asymptotically nonexpansive mappings in spaces. J Inequal Appl 2014, 336 (2014). https://doi.org/10.1186/1029-242X-2014-336
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DOI: https://doi.org/10.1186/1029-242X-2014-336