Fixed Point Approximation of Generalized Nonexpansive Mappings in Hyperbolic Spaces

Abstract

We prove strong and Δ-convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces using S-iteration process due to Agarwal et al. As uniformly convex hyperbolic spaces contain Banach spaces as well as CAT(0) spaces, our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces.

1. Introduction

Let be a metric space and let be a nonempty subset of . A mapping is said to be as follows:(i)nonexpansive if for all ;(ii)quasi-nonexpansive if for all and , where denotes the set of fixed points of .

We know that there exist many generalizations of nonexpansive and quasi-nonexpansive mappings. Garcia-Falset et al. [1] introduced two generalizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings (see [2]).

Definition 1 (see [1]). Let be a mapping defined on a subset of metric space and . Then is said to satisfy the condition , if, for all , is said to satisfy the condition whenever satisfies the condition for some .

Definition 2 (see [1]). Let be a mapping defined on a subset of a metric space and . Then is said to satisfy the condition if, for all ,

In the case , then the condition implies the condition . Suzuki (see [2]) said that satisfies the condition , when .

The following example shows that the class of mappings satisfying the conditions and , for some , is larger than the class of mappings satisfying the condition .

Example 3 (see [1]). For a given , define a mapping on by Then the mapping satisfies the condition but it fails the condition , whenever . Moreover, satisfies the condition for .

The basic properties and details of CAT(0) spaces can be found in the literature [3–5]. In [6], Lim introduced a concept of convergence in a general metric space which is called “-convergence.” In 2008, Kirk and Panyanak [7] specialized Lim’s concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. Since then, the existence problem and the -convergence problem of iterative sequences to a fixed point for various classes of nonexpansive mappings in the frame work of CAT(0) spaces have been rapidly developed (see [1, 8–12]).

In [13], Leustean proved that CAT(0) spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity quadratic in . Thus, the class of uniformly convex hyperbolic spaces are a natural generalization of both uniformly convex Banach spaces and CAT(0) spaces.

Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [14]. It is noted that they are different from Gromov hyperbolic spaces [15] or from other notions of hyperbolic spaces that can be found in literature (see [16–19]).

A hyperbolic space is a metric space together with a convexity mapping satisfying ; ; ; ,for all and .

A metric space is said to be a convex metric space in the sense of Takahashi [20], where a triple satisfy only   (see [21–23]). We get the notion of the space of hyperbolic type in the sense of Goebel and Kirk [16], where a triple satisfies . The was already considered by Itoh [24] under the name of “condition III” and it is used by Reich and Shafrir [18] and Kirk [17] to define their notions of hyperbolic spaces.

The class of hyperbolic spaces include normed spaces and convex subsets thereof, the Hilbert space ball equipped with the hyperbolic metric [25], Hadrmard manifold, and the CAT(0) spaces in the sense of Gromov (see [15]).

If and , then we use the notation   for  . The following holds even for the more general setting of convex metric space [20]: for all and ,

A hyperbolic space is uniformly convex [13] if, for any and , there exists such that, for all , provided , , and .

A mapping , providing such a for given and , is called a modulus of uniform convexity. We called that is monotone if it decreases with for fix .

Recently, Agarwal et al. [26] introduced S-iteration process as follows (see [27]).

Let be a convex subset of a linear space and let be a mapping from into itself. Then the iterative sequence generated from and defined by where and are sequences in satisfying the certain condition. It is observed that rate of convergence of S-iteration process is similar to the Picard iteration process but faster than the Mann iteration process for contraction mapping (see [26, 27]).

The purpose of this paper is to prove the strong and -convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces by using S-iteration process. Our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces [10–12, 28, 29].

2. Preliminaries

Let be a nonempty subset of metric space and let be any bounded sequence in . Consider a continuous functional defined by Then, the infimum of over is said to be the asymptotic radius of with respect to and is denoted by .

A point is said to be an asymptotic center of the sequence with respect to if the set of all asymptotic centers of with respect to is denoted by . This set may be empty or a singleton or contain infinitely many points.

If the asymptotic radius and the asymptotic center are taken with respect to , then these are simply denoted by and , respectively. We know that, for , if and only if .

It is known that every bounded sequence has a unique asymptotic center with respect to each closed convex subset in uniformly convex Banach spaces and even CAT(0) spaces.

The following lemma is due to Leuştean [30] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Lemma 4 (see [30]). Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity . Then every bounded sequence in has a unique asymptotic center with respect to any nonempty closed convex subset of .

Recall that a sequence in is said to be -convergent to , if is the unique asymptotic center of for every subsequence of . In this case, we write - and call the -limit of .

Lemma 5 (see [31]). Let be a uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let and be a sequence in for some . If and are sequences in such that for some , then we have .

3. Main Results

We begin with the definition of Fejér monotone sequences.

Definition 6. Let be a nonempty subset of hyperbolic space and let be a sequence in . Then is said to be Fejér monotone with respect to if for all and

Example 7. Let be a nonempty subset of hyperbolic space and let be a quasi-nonexpansive (in particular, nonexpansive) mapping such that . Then the sequence of Picard iteration is Fejér monotone with respect to .

We can easily prove the following proposition.

Proposition 8. Let be a sequence in and let be a nonempty subset of . Suppose that is Fejér monotone with respect to . Then we have the following: (1) is bounded;(2)the sequence is decreasing and convergent for all .

We now define S-iteration process in hyperbolic spaces:

Let be a nonempty closed convex subset of a hyperbolic space and let be a mapping of into itself. For any the sequence of S-iteration process is defined as where and are real sequences with , .

Lemma 9. Let be a nonempty closed convex subset of a hyperbolic space and let be a mapping which satisfies the condition for some . If is a sequence defined by (11), then is Fejér monotone with respect to .

Proof. Let . Then we have for all . Since satisfies the condition , for some , we have Using (11), we have Again, using (11) and (14), we have that is, for all . Thus, is Fejér monotone with respect to .

Lemma 10. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be a mapping which satisfies the conditions and on . If is a sequence defined by (11), then is nonempty if and only if is bounded and .

Proof. Suppose that the fixed point set is nonempty and . From Lemma 9, we know that is Fejér monotone with respect to . Hence, by Proposition 8, is bounded and exists. Let . Since for all , from the condition , we have Therefore, for all . Taking the limit supremum on both sides, we get Similarly, we have Taking the limit supremum on both sides of (14), we have Since by using (19), (20), and Lemma 5, we get Next, we know that Hence, from (23) and (24), we have Now we observe that which yields that From the estimates of (21) and (27), we have that Thus, from (11), we have which gives
Conversely, suppose that is bounded and . Let . Then, by Lemma 4, . As satisfies the condition on , there exists such that which implies that By using the uniqueness of asymptotic center, , so is a fixed point of.

Theorem 11. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be a mapping which satisfies conditions and , for some on with . Then the sequence which is defined by (11), is -convergent to a fixed point of .

Proof. From Lemma 10, we know that is a bounded sequence; therefore, has a -convergent subsequence. We now prove that every -convergent subsequence of has unique -limt in . For this, let and be -limits of the subsequences and of , respectively. By Lemma 4, and . Since is bounded sequence, by Lemma 10, . We claim that is a fixed point of . Since satisfies the condition , there exists a such that Taking the limit supremum on both sides, we have Hence, we obtain By uniqueness of the asymptotic center, .
Similarly, we can prove that . Thus, and are fixed points of  . Now we show that . If not, then by the uniqueness of asymptotic center, which is a contradiction. Hence .

Theorem 12. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be a mapping which satisfies conditions and , for some on with . Then the sequence which is defined by (11) converges strongly to a fixed point of if and only if where .

Proof. Necessity is obvious; we have to prove only sufficient part. First, we show that the fixed point set is closed; let be a sequence in which converges to some point . Since from the condition , we have By taking the limit on both sides, we obtain In view of the uniqueness of the limit, we have , so that is closed. Suppose that Then, from (15) it follows from Lemma 9 and Proposition 8 that exists. Hence we know that .
Consider a subsequence of such that where is in . By Lemma 9, we have which implies that This shows that is a Cauchy sequence. Since is closed, is a convergent sequence. Let . Then, we know that converges to . In fact, since we have Since exists, the sequence is convergent to .

We recall the definition of condition due to Senter and Doston [32].

Let be a nonempty subset of a metric space . A mapping is said to satisfy condition , if there is a nondecreasing function with , for all such that for all , where .

Theorem 13. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be a mapping which satisfies conditions and , for some on . Moreover, satisfies condition with . Then the sequence which is defined by (11) converges strongly to some fixed point of.

Proof. As in the proof of Theorem 12, it can be shown that is closed. Observe that, by Lemma 9, we have . It follows from condition that Therefore, we have Since is a nondecreasing mapping satisfying , for all , we have Rest of the proof follows in lines of Theorem 12.

Remark 14. Our Theorems 11, 12, and 13 improve and extend the previous well-known results from Banach spaces and CAT(0) spaces to more general class of uniformly convex hyperbolic spaces (see [10, 28, 29], in particular, Theorems 3.4 and 3.6 of [12]). In our results, we considered the faster iteration process to approximate the fixed point of underlying mapping in the framework of uniformly convex hyperbolic spaces.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

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

Acknowledgments

This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the Republic of Korea (2014046293).