Demiclosedness Principle for Total Asymptotically Nonexpansive Mappings in 𝐶𝐴𝑇(0) Spaces

Abstract

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings. Moreover, we establish the existence theorem and convergence theorems for modified Ishikawa iterative process in the framework of spaces. Our results generalize, extend, and unify the corresponding results on the topic in the literature.

1. Introduction

A self-mapping , on a metric space , is called nonexpansive, if and asymptotically nonexpansive, introduced by Goebel and Kirk [1], if there exists a nonnegative sequence with such that A self-mapping , on a metric space , is called asymptotic point-wise nonexpansive, introduced by Hussain and Khamsi [2], if there exists a sequence of mappings with such that It is quite natural to extend (2) and (3) in the following way: a self-mapping , on a metric space , is called asymptotically -nonexpansive, if there exists a nonnegative sequence with such that where is a strictly increasing and continuous mapping with . Notice that an asymptotically -nonexpansive is a generalization of an asymptotically nonexpansive. Indeed, if we take , we get inequality (2). Analogously, we consider the extension of (3) as follows. A self-mapping , on a metric space , is called asymptotic point-wise -nonexpansive if there exists a sequence of mappings with such that It is evident that if we replace in (5), then we derive (3).

A self-mapping , on a metric space , is called nearly Lipschitzian with respect to a fix sequence , introduced by Sahu [3], if, for each , there exists a constant such that for all , where for each and . The infimum of constants satisfying (6) is called the nearly Lipschitz constant of and is denoted by . Furthermore, is called nearly nonexpansive if for all and nearly asymptotically nonexpansive if for all and .

A self-mapping , on a metric space , is said to be asymptotically nonexpansive in the intermediate sense, introduced by Bruck et al. [4], if it is continuous and the following inequality holds: We note that if we set then as . It follows that (7) is reduced to for and .

For the examples which are asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive, see, for example, [5]. In fact, the class of nearly asymptotically nonexpansive mappings is intermediate classes between the class of asymptotically nonexpansive mappings and that of asymptotically nonexpansive in the intermediate sense mappings.

In 2006, Alber et al. [6] introduced the notion of total asymptotically nonexpansive mappings. The class of such mappings includes the asymptotically nonexpansive mappings; for more details, see, for example, [7]. This new notion unifies various definitions mentioned above.

On the context of uniformly convex Banach spaces, several papers appeared on the topic of the approximation of fixed points of mappings in the classes of nonexpansive and asymptotically nonexpansive mappings. Motivated by these results, we investigate the existence of fixed points of total asymptotically nonexpansive mappings in the context of spaces that attracted attention of several authors; see, for example, [8–18].

More precisely, we prove the convergence of modified Ishikawa iterative process, introduced by Schu [19], where lies in a nonempty closed convex subset of a space , , are real sequences in for each , and is a total asymptotically nonexpansive mapping. The notation “” is introduced in the next section.

Notice that it is not possible to get the following modified Mann iterative process: from the modified Ishikawa iterative process, since we can take , in (10). Also, as a special case the results remain true for modified Mann iteration. Our results generalize, extend, and unify the corresponding results of [13, 20–22] and the references contained therein.

2. Preliminary Remarks

Throughout the paper, the set of real numbers will be denoted by . Suppose that is a metric space, , and . A map is said to be a geodesic path joining the point to if and , with for all . In short, we use a geodesic from to instead of a geodesic path joining to . Notice that if   is an isometry, then . The image of is called a geodesic segment (or metric segment) joining and . If it is unique, this geodesic is denoted by . A metric space is called a geodesic space if every two points of are joined by a geodesic. Furthermore, is called uniquely geodesic if there is exactly one geodesic joining to for each . A subset is called convex if includes every geodesic segment joining any two of its points.

In a geodesic metric space , geodesic triangle consists of three points in and a geodesic segment between each pair of vertices. Here, the points are also called vertices of and a geodesic segment is said to be the edge of . A triangle , in the Euclidean plane , is called a comparison triangle for geodesic triangle in is where for .

Comparison Axiom. Let be a geodesic metric space and let   be a comparison triangle for a geodesic triangle in . We say that satisfies the inequality if for all and all comparison points .

A geodesic metric space is called a space [23] if all geodesic triangles of appropriate size satisfy the comparison axiom. A complete space is called “Hadamard space.”

Lemma 1 (see [20]). Let be a space. Then,(1) is uniquely geodesic;(2)let with . If such that , then ;(3)let . For each , there exists a unique point such that

In the sequel, we use the notation for the unique point satisfying (13).

Assume that is a Hadamard space. Suppose that is a bounded sequence in . Define for . The asymptotic radius of is given by The asymptotic center of is defined as follows: We denote by the collection of all bounded closed convex subsets of a Hadamard space .

Asymptotic center is exactly one point in a space (see, e.g., [24]). Furthermore, the distance function is convex in complete spaces (see, e.g., [23]).

Notice that if are points of a space and if is the midpoint of the segment , which we will denote by , then the inequality implies that because equality holds in the Euclidean metric. Here, (17) is known as the inequality; see Bruhat and Tits [25].

Finally, we note that a geodesic metric space is a space if and only if it satisfies inequality (17) (see, e.g., [23]).

Lemma 2   (see [8]). Let be a space. Then, the following inequality, is satisfied for all and .

Definition 3 (see [14]). A sequence in is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case, one writes and call the of .

Lemma 4 (see [8]). Assume that is a Hadamard space and . If is a bounded sequence in , then the asymptotic center of is in .

Lemma 5   (see [8]). Assume that is a Hadamard space. Each bounded sequence in has a -convergent subsequence.

space has the Opial property; that is, for a given , we have where   -converges to and given with . Moreover, these metric spaces offer a nice example of uniformly convex metric spaces.

3. Convergent Theorems

In this section, we first recollect some elementary definitions and basic results on the topic in the framework of spaces.

Definition 6   (see [6]). Let be a space and be a subset of . A self-mapping on a subset is called total asymptotically nonexpansive if there are nonnegative real sequences and ,  , with ,   as , and strictly increasing and continuous function with such that

Remark 7. If , then inequality (20) turns into which is nearly asymptotically nonexpansive (6).
In addition, if for all , then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. In the case , a self-mapping is uniformly continuous. Notice that a self-mapping can be uniformly continuous even if . If and for all , then we obtain the class of nonexpansive mappings from (20).

Definition 8   (see [18, 26]). Assume that is a Hadamard space and . A self-mapping on is called demiclosed at zero, if, for each sequence that -converges to a point in , , implies that or let one formally say that is demiclosed at zero if the conditions , -converges to and , imply .

Lemma 9. Assume that is a Hadamard space and . For a bounded sequence in , one sets . Then there is a point such that

Proof. It is immediate consequence of existence of the asymptotic center and Lemma 4.

Lemma 10. Assume that is a Hadamard space and . Suppose that a self-mapping on is a total asymptotically nonexpansive mapping. For a point in , let . Then , where is such that for the same in Lemma 9.

Proof. Since is total asymptotically nonexpansive, we have for any . If we let go to infinity, we get Let go to infinity, which implies that .

Theorem 11. Assume that is a Hadamard space and . If is a uniformly continuous total asymptotically nonexpansive mapping, then has a fixed point. Moreover, the fixed point set is closed and convex.

Proof. Define , for each . Let such that . We have seen that as . The inequality implies the following: If we let go to infinity, we get which implies that Therefore, is a Cauchy sequence in and hence converges to some ; that is, . Since is continuous, then and this proves that . Again, since is continuous, is closed. In order to prove that is convex, it is enough to prove that , whenever . Indeed, set . The inequality implies that for any . Since when go to infinity, , which implies .

It is known that the demiclosed principle plays an important role in studying the asymptotic behavior for nonexpansive mappings (see [12, 27–30]). In [29], Xu proved the demiclosed principle for asymptotically nonexpansive mappings in the setting of a uniformly convex Banach space. Nanjaras and Panyanak [12] extended Xu’s result to spaces. A demiclosed principle for asymptotically nonexpansive mappings in the intermediate sense on a real uniformly convex Banach space was proved by Yanga et al. [30]. Motivated by them we will establish demiclosed principle and existence theorem for total asymptotically nonexpansive mappings in the context of spaces. Also the next theorem shows that the result of Theorem 11 holds without the boundness condition imposed on , provided that there exists a bounded approximate fixed point sequence ; that is, .

Theorem 12. Assume that is a Hadamard space and . Suppose that is a uniformly continuous total asymptotically nonexpansive mapping. Let be a bounded approximate fixed point sequence. If , then we have .

Proof. Since is an approximate fixed point sequence, then we have for any . Hence, , for each . In particular, we have . The inequality implies that for any . If we let , we will get for any . The definition of implies that for any , or Letting , we will get . By the continuity of ,

Consequently, we derive the following corollaries which can be found in [20].

Corollary 13. Assume that is a Hadamard space and . Suppose that is a uniformly continuous nearly asymptotically nonexpansive mapping. If is a bounded sequence in such that , then has a fixed point.

Proof. Every bounded sequence in has a -convergent subsequence, by Lemma 5, which can be showed again by . Now apply Theorem 12.

Corollary 14. Assume that is a Hadamard space and . Suppose that is a uniformly continuous nearly asymptotically nonexpansive mapping. If is a bounded sequence in which -converges to and , then and .

4. Approximation

Assume that is a Hadamard space and . Suppose that a self-mapping is total asymptotically nonexpansive. Consider the following iteration process, namely, modified Ishikawa iteration scheme: where and are real sequences in for each .

Note that the modified Ishikawa iterative process coincides with the following modified Mann iterative process if for each then In this section we want to show that is an approximate fixed point sequence. Due to this, we use the following lemma which can be found in [31].

Lemma 15. Let , , and be sequences of nonnegative real numbers such that, for all , Let and . Then exists.

Lemma 16. Assume that is a Hadamard space and . Suppose that a self-mapping is a uniformly continuous total asymptotically nonexpansive with . Suppose also that there exist constants , such that for all . Let . Starting from arbitrary define the sequence by (37). Suppose that and . Then exists and .

Proof. Let ; then
Since is increasing function, it results that if and if . In either case we obtain for each . Therefore, Thus, so one can write Thus, we get the following inequality: For some , since and , are bounded, due to Lemma 15 the sequence has a limit and so it is bounded. By Lemma 2, we have Since exists, is bounded and it follows from (42) that is also bounded. Then, there exist constants such that for some . Thus, which implies that Since and exists, therefore .

Note that if the domain of is bounded, we can omit the conditions of existence of constants such that for all and .

Theorem 17. Assume that is a Hadamard space and . Assume that is a uniformly continuous total asymptotically nonexpansive mapping with and there exist constants such that for all . Let . Starting from arbitrary define the sequence by (37), where are sequences in such that . Suppose that and . Then is -convergent to a fixed point of .

Proof. Since , by Lemma 16, , so by uniform continuity of , . Therefore, one can write
Also and hence
Set , where the union is taken over by all subsequences of . We assert that . Let ; then there is a subsequence of such that . By Lemmas 4 and 5 there exists a subsequence of such that . We have seen that , so by Theorem 12 and exists by Lemma 16. We will show that . Suppose, on the contrary, that . By the uniqueness of asymptotic centers, which is a contradiction. Hence, we get that . To show that   -converges to a fixed point of , it suffices to show that consists of exactly one point. Let be a subsequence of . By Lemmas 4 and 5 there exists a subsequence of such that . Let and . We have seen that and . It is sufficient to show that to finalize the proof. Suppose, on the contrary, is not equal to . Since is convergent, then by the uniqueness of asymptotic centers, which is a contradiction, and hence the conclusion follows.

Corollary 18. Assume that is a Hadamard space and . Suppose that is a uniformly continuous total asymptotically nonexpansive mapping with and there exist constants such that for all . Let . Starting from arbitrary define the sequence by (37), where are sequences in . Suppose that and . Then the condition as implies that

Note that, in the case , we can state Theorem 17 in the following manner.

Lemma 19. Assume that is a Hadamard space and . Assume that is a uniformly continuous total asymptotically nonexpansive mapping with . Suppose also that there exist constants such that for all . Let . Starting from arbitrary define the sequence by (38), where is a sequence in . Suppose that and . Then the condition as implies that

Proof. We have from (38) Therefore, , and also Since is uniformly continuous, the hypotheses as implies that

Proposition 20   (see [32, Lemma 2.9]). Let be a complete space and let . Suppose that is a sequence in for some and are sequences in such that , , and for some . Then

Theorem 21. Assume that is a Hadamard space and . Suppose that a self-mapping is uniformly continuous total asymptotically nonexpansive mapping with and suppose that there exist constants such that for all . Let , and is a sequence in for all . Starting from arbitrary define the sequence by (38). Suppose that and . Then .

Proof. First we show that for is bounded and it has a limit Since is increasing function, it results that if and if . In either case we obtain for each . Thus, we get the following inequality: However, and ; therefore, due to Lemma 15 the sequence has a limit and it is bounded. Assume that . Since for all , then Additionally, since then Hence, By Proposition 20, we have . By Lemma 19, . This completes the proof.

Recall that a mapping is said to be semicompact if is closed and for any bounded sequence with , there exist and satisfying .

The next theorem extends corresponding results of Beg [33], Chang [34], and Osilike and Aniagbosor [22] for a more general class of non-Lipschitzian mappings in the framework of spaces. It also extends corresponding results of Dhompongsa and Panyanak [8] from the class of nonexpansive mappings to a more general class of non-Lipschitzian mappings in the same space setting. Moreover, it extends corresponding results of Abbas et al. [20].

Theorem 22. Assume that is a Hadamard space and . Suppose that a self-mapping is uniformly continuous total asymptotically nonexpansive mapping with ; suppose that there exist constants such that for all . Let . Starting from arbitrary define the sequence by (37), where are sequences in for all , such that . Suppose that and , and also suppose that is semicompact for some . Then the sequence converges strongly to some fixed point of .

Proof. By Theorem 17, we have . Since is uniform continuous, it follows the estimation that . Since is semicompact, there exist a subsequence of and with . Again since is uniformly continuous and it follows from the estimation, that ; that is, . By Lemma 16, the limit of exists as . Since , therefore . This accomplishes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that help in improving this paper.