Approximating Fixed Points of Reich–Suzuki Type Nonexpansive Mappings in Hyperbolic Spaces
In this work, we prove some strong and Δ convergence results for Reich-Suzuki type nonexpansive mappings through M iterative process. A uniformly convex hyperbolic metric space is used as underlying setting for our approach. We also provide an illustrate numerical example. Our results improve and extend some recently announced results of the metric fixed-point theory.
A self-map on a subset of a metric space is called contraction if there exists some constant such that for all it follows that . If for all , then is called nonexpansive. A point is called a fixed point of whenever . Banach  theorem (1922) states that any contraction map on a complete metric space has a unique fixed point which is the limit of the sequence generated by the Picard iterates, that is, . In 1965, Kirk , Browder , and Göhde  independently proved that any self nonexpansive mapping defined on a bounded closed convex subset of a uniformly convex Banach space always has a fixed point. Now, a natural question which comes in mind is that whether the sequence of Picard iterates converges to a fixed point of a self nonexpansive mapping. The answer of this question in general is negative. Therefore, there is a need to construct some new procedures to overcome such situations and to obtain a better rate of convergence, for example, Mann , Ishikawa , Noor , S , Abbas and Nazir , and Thakur et al.  iterative processes are often used to approximate fixed points of nonexpansive mappings. In 2008, Suzuki  introduced a weaker notion of nonexpansive mappings: a self-map on a subset of a metric space is said to be Suzuki type nonexpansive if for every two elements in , holds whenever . It is easy to observe that the class of Suzuki type nonexpansive mappings properly includes the class of nonexpansive mappings.
The class of Suzuki type nonexpansive mappings was studied extensively by many authors (cf. [10, 12–21]). Very recently, Pant and Pandey  introduced Reich–Suzuki type nonexpansive mappings which in turn include the class of Suzuki type nonexpansive mappings.
Definition 1. (see ). Let be a nonempty subset of a metric space. A map is said to be Reich–Suzuki type nonexpansive if for all , there is some constant such thatApproximation of fixed points of nonexpansive and generalized nonexpansive mappings is an active area of research on its own [23–27]. Recently, Pant and Pandey  used Thakur et al.  iterative process to approximate fixed points of Reich–Suzuki type nonexpansive mappings. The purpose of this paper is to prove strong and convergence results for Reich–Suzuki type nonexpansive mappings under M iterative process , which is known to converge faster than the Thakur et al.  iterative process. In this way, we improve and extend to many results of the current literature.
Kohlenbach  suggested the concept of generalized metric spaces and so-called hyperbolic metric spaces. This type of metric spaces includes normed spaces, the Hilbert ball with the hyperbolic metric, Cartesian products of Hilbert balls, metric trees, Hadamard manifolds, and CAT (0) spaces in the sense of Gromov. The definition is given as follows:
A triplet is called a hyperbolic metric space whenever is a metric space and is a function such that for all and , the following conditions hold.()()()()The set is known as metric segment with endpoints a and b. Throughout, we will write . A subset of is called convex provided that , for every and . When there is no ambiguity, we will write for .
Definition 2. Let be a hyperbolic metric space. For any and . SetWe say that is uniformly convex whenever , for every and .
Definition 3. (see [29, 30]). A hyperbolic metric space is said to be strictly convex provided that for every and such that follows the condition . By , every uniformly convex hyperbolic metric space is strictly convex.
Recently, Ullah and Arshad  introduced -iteration process in Banach spaces. The hyperbolic space version of iteration process reads as follows:where is a sequence in . Ullah and Arshad  proved some weak and strong convergence results of iterative process for Suzuki type nonexpansive mappings in the context of Banach spaces. Ullah et al.  extended the results of Ullah and Arshad  to the setting of CAT (0) spaces. In this paper, we study M iteration process for Reich–Suzuki type nonexpansive mappings in the setting of hyperbolic spaces.
Let be a nonempty subset of hyperbolic metric space and be a bounded sequence in X. For each , define(i)Asymptotic radius of at a as(ii)Asymptotic radius of relative to B as(iii)Asymptotic center of relative to by
We know that in a complete hyperbolic space with monotone modulus of uniform convexity, every bounded sequence has a unique asymptotic center with respect to every nonempty closed convex subset of .
Definition 4. (see ). Assume that is a bounded sequence in a hyperbolic space . Then, is said to be -convergent to a point , if s is a unique asymptotic center of each subsequence of .
The following lemma gives many numbers of Reich‐Suzuki type nonexpansive mappings.
Lemma 1 (see ). Let be a nonempty subset of a hyperbolic space and . If is Suzuki nonexpansive, then is Reich–Suzuki type nonexpansive with constant .
Fixed-point set structure of Reich–Suzuki type maps is as follows.
Lemma 2 (see ). Let be a nonempty subset of a hyperbolic space and . If is Reich–Suzuki type nonexpansive, then is closed. Furthermore, if the space is strictly convex and the set is convex, then is also convex.
Lemma 3 (see ). Let be a nonempty subset of a hyperbolic metric space and is a Reich–Suzuki type nonexpansive. Then, for every and , it implies that .
Lemma 4 (see ). Let be a nonempty subset of a hyperbolic metric space and is a Reich–Suzuki type nonexpansive. Then, for all , it follows that
Lemma 5 (see ). Let be a complete hyperbolic space with a monotone modulus of uniform convexity and . If and , are sequences in such that , , and for some , then .
3. Convergence Results in Hyperbolic Spaces
Throughout this section, the letter will stand for a hyperbolic space with a monotone modulus of uniform convexity.
Lemma 6. Let be a nonempty closed convex subset of . Let be a Reich–Suzuki type nonexpansive map with . Let be the sequence generated by (3). Then, exists for .
Proof. Let . By Lemma 3, we haveHence, the sequence is bounded below and decreasing. Thus, exists for .
The following theorem will be used in the upcoming results.
Theorem 1. Let be a nonempty closed convex subset of and let be a Reich–Suzuki type nonexpansive map. Let be the sequence defined by (3). Then, if and only if is bounded and .
Proof. We assume that is bounded and . Let . We shall prove that . By Lemma 4, we haveHence, and is singleton set. We must have . Hence, .
Conversely, we assume that and . We shall prove that is bounded and . By Lemma 6, exists and is bounded. PutFrom the proof of Lemma 6, it follows thatBy Lemma 3, we haveAgain from the proof of Lemma 6, it follows thatFrom (12) and (15), we getFrom (16), we haveNow from (10), (13), and (17) together with Lemma 5, we obtainThe convergence result is as follows.
Theorem 2. Let be a nonempty closed convex subset of . If is a Reich–Suzuki type nonexpansive mapping with , then is defined by (3). converges to an element of .
Proof. By Theorem 1, the sequence is bounded. Hence, one can find a -convergent subsequence of . Next, it is our aim to prove that each -convergent subsequence of has a unique limit in the set . For this purpose, we assume that has two -convergent subsequences, namely, and , with limits and , respectively. In view of Theorem 1, the sequence is bounded and . We claim that . Now,Using Lemma 4, we haveSince the asymptotic center of has a unique element, . Similarly, . By the uniqueness of asymptotic center of a sequence, we havewhich is a contradiction. Hence, the conclusions are reached.
Now, we establish a strong convergence theorem for Reich–Suzuki type nonexpansive maps using iteration process (3).
Theorem 3. Let be a nonempty compact convex subset of and be a Reich–Suzuki type nonexpansive map with . If is generated by (3), then converges strongly to the fixed point of .
Proof. By Theorem 1, . By the compactness assumption, there exists a subsequence of such that converges strongly to some in . By Lemma 4, we haveBy the uniqueness of limits in metric spaces, we must have . By Lemma 6, exists, and hence is the strong limit of .
Example 1. Let be a subset of endowed with the usual metric, that is, . Define byHere, first we shall show that the mapping is not Suzuki type nonexpansive. To do this, we choose and . The straightforward calculations giveNext, we shall show that the mapping is Reich–Suzuki type nonexpansive. To do this, we choose . We consider different situations as given below.
Case 1. For , we have
Case 2. For , we have
Case 3. For and , we have
Case 4. For and , we have
Case 5. For and , we haveThus, is Reich–Suzuki type nonexpansive mapping with . Set . Hence, the requirements of Theorem 2 are fulfilled. Now, the conclusions of Theorems 2 and 3 are reached. However, we cannot directly apply any result in [9, 10, 14, 16, 17, 19–22, 34] and references cited therein because in this situation, is not Suzuki type nonexpansive mapping.
For the next strong convergence result, compactness assumption is not necessary; however, the following condition will be added.
Definition 5. Let be a nonempty subset of . A sequence in is called Fejer monotone with respect to , ifA mapping is said to satisfy condition  if one can construct a nondecreasing function with properties and for every such that for each .
The following facts can be found in .
Proposition 1. Let be a nonempty closed subset of . Let be a Fejer monotone sequence with respect to . Then, converges strongly to some point of if and only if .
Theorem 4. Let be a nonempty closed convex subset of and be a generalized Reich–Suzuki type nonexpansive map with . If satisfies condition (I), then generated by (3) converges strongly to the fixed point of .
Proof. From Theorem 1, we haveSince the mapping satisfies the condition , it follows thatBy Lemma 2, the set is closed. It follows from Lemma 6 that is Fejer monotone sequence with respect to the set . The conclusions are based on Proposition 1.
(1)Our results extend the corresponding results of Ullah and Arshad  in two ways: (i) from the class of Suzuki type nonexpansive maps to the class of Reich–Suzuki type nonexpansive maps and (ii) from Banach spaces to the general setting of hyperbolic spaces.(2)Our results extend the corresponding results of Ullah et al.  in two ways: (i) from the class of Suzuki type nonexpansive maps to the class of Reich–Suzuki type nonexpansive maps and (ii) from CAT (0) spaces to the general setting of hyperbolic spaces.(3)Our results also extend and improve the corresponding results proved in [9, 10, 14, 16, 17, 19–22, 34] and references cited therein.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.
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