Approximating Fixed Points of Enriched Nonexpansive Mappings in Geodesic Spaces

Shukla, Rahul; Panicker, Rekha

doi:https://doi.org/10.1155/2022/6161839

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Volume 2022 | Article ID 6161839 | https://doi.org/10.1155/2022/6161839

Approximating Fixed Points of Enriched Nonexpansive Mappings in Geodesic Spaces

Rahul Shukla¹and Rekha Panicker¹

Academic Editor: Santosh Kumar

Received22 Feb 2022

Accepted12 Apr 2022

Published04 May 2022

Abstract

In this paper, we consider the class of enriched nonexpansive mappings in the setting of geodesic spaces. We obtain a number of fixed point theorems for these mappings in geodesic spaces. Further, we employ the SP iterative method and present some new convergence theorems for the class of enriched nonexpansive mappings under different assumptions. We present some results concerning and strong convergence.

1. Introduction

Nonexpansive mappings are those class of nonlinear mappings which have Lipschitz constant equal to one. A nonexpansive mapping needs not to admit a fixed point in a complete space. However, Browder [1], Göhde [2], and Kirk [3] independently ensured the existence of fixed points of nonexpansive mappings in Banach spaces under certain geometric assumptions. Many mathematicians have generalized and extended these results and considered a number of nonlinear mappings, see [4–9] (see also the references therein).

In 2019, Berinde [10] considered a new class of nonlinear mappings by enriching nonexpansive mappings, known as enriched nonexpansive mappings. He obtained some fixed point theorems for these classes of mappings in Hilbert spaces. It was observed in [10, 11] that class of enriched nonexpansive mappings has strong relations with averaged and nonexpansive mappings.

On the other hand, in 1970, Takahashi [12] considered the structure of convexity outside linear spaces. These spaces are fruitful in the context of fixed point theory. Goebel and Kirk [13] employed Krasnosel’skiĭ-Mann iterative method to find fixed points of nonexpansive mappings in hyperbolic type spaces. In the recent years, a number of papers have appeared in the literature dealing with the fixed point theorems in nonlinear spaces, see [14–24].

The class of enriched nonexpansive mappings has been studied only in linear spaces. Now it is natural to extend this class of mappings outside of linear spaces (or in nonlinear spaces) and ensure the existence of fixed points. The aim of this paper is to study the class of enriched nonexpansive mappings in geodesic spaces. We observe that for every -enriched nonexpansive mapping, one can define a nonexpansive mapping, and the set of fixed points of both the mappings remains the same. Therefore, the existence of fixed points for -enriched nonexpansive mappings is equivalent to existence of fixed points for nonexpansive mappings. However, the convergence of fixed points for -enriched nonexpansive mappings is slightly different than the convergence of fixed points for nonexpansive mappings. We prove that Krasnosel’skiĭ method converges to fixed point of mapping. Further, we use SP iterative method to reckon fixed points of -enriched nonexpansive mappings under certain assumptions. These results are new even in Hilbert spaces. Our results extend, complement, and generalize some results from [10, 11, 16, 19, 25–27].

2. Preliminaries

Let be a metric space and . A mapping is called as geodesic path from to if for all The image of forms a geodesic joining and . It is noted that the geodesic segment joining and is not unique, in general. For more details of geodesic spaces, see [14, 21].

Definition 1 (see [28]). A triplet is called as a hyperbolic metric space if is a metric space, and function satisfies the following assumptions for all and
(W1)
(W2)
(W3)
(W4)

Remark 2. If for all then it can be seen that all normed linear spaces are hyperbolic metric space.

Remark 3. If conditions (W1)–(W3) are satisfied, then is hyperbolic type space considered by Goebel and Kirk [13]. Reich and Shafrir [22] also obtained some important results in hyperbolic metric spaces.
We shall write to denote a point of space. For indicates geodesic segments. A subset of hyperbolic metric space (or hyperbolic space) is called convex if whenever

Remark 4. Leutean [20] proved that the class of spaces is the class of complete uniformly convex hyperbolic spaces (in short, complete UC-hyperbolic space), see the definition of UC-hyperbolic space in [19].
If is a Busemann space, then there is a unique convexity mapping in such a way that is -hyperbolic space with unique geodesics. In other words, for all and any , there is an element which is unique (say ) in such a way Let be three points in metric space ; the point is said to lie between and if and these points are distinct pairwise. Thus, if lies between and , then lies between and

Lemma 5 (see [14]). Let be a uniquely geodesic space. Let be pairwise distinct points. A point lies between and if and only if

Proposition 6 (see [14]). Let be a metric space and be pairwise distinct points. The following are equivalent: (a) lies between and , and lies between and (b) lies between and , and lies between and

Let be a bounded sequence in a hyperbolic space and with A functional can be defined as follows:

The asymptotic radius of with respect to (in short, wrt) is defined as

A point in is called an asymptotic center of wrt if

is denoted as set of all asymptotic centers of wrt A bounded sequence in a hyperbolic space is said to -converge to if is the unique asymptotic center for every subsequence of . A sequence is called Fejér monotone wrt if for all for all

Definition 7 (see [29]). A mapping is called quasi-nonexpansive if where

Definition 8 (see [30]). The mapping with is said to have Condition (I) if the following assumptions are satisfied: (a) a function which is nondecreasing(b)For , and (c)For all , where

Definition 9. Let be a metric space and with A mapping is called as compact if has a compact closure.

Proposition 10 (see [20]). Let be a complete UC-hyperbolic space, with . Suppose that is convex and closed, and is bounded sequence in Then, has a unique asymptotic center with respect to

Lemma 11 (see [17]). Let be same as in Proposition 10. Let and be a sequence with For some if and are sequences in with , , and Then,

Lemma 12. Let and be same as in Proposition 10. Let be a mapping. For , consider as follows: for all Then,

Lemma 13 (see [20]). Let be a bounded sequence in and . Let and be two sequences in with for all and . Suppose that and there exists such that Then, .

Lemma 14 (see [14]). Let be a metric space, such that . If is Fejér monotone wrt , and for every subsequence of Then, the sequence -converges to

Lemma 15 (see [16]). Let be a complete UC-hyperbolic space and such that and is closed convex. Let be a bounded sequence in and a function defined as follows: for any . is called as type function, and it is unique.
Then, there is a minimum point (unique) and

Proposition 16 (see [13]). Let and be same as in Lemma 15 with is bounded. Let be a nonexpansive mapping. Let and . Define a sequence in by Krasnosel’skiĭ iterative method [31]. Then,

The proof of the following theorem is motivated from [16].

Theorem 17. Let , , and be same as in Proposition 16. Then,

Proof. For a given and for any , a sequence can be defined: From Proposition 16, it implies that From Lemma 15, there is a minimum point (unique) in such a way that From the definition of mapping , Then,

3. Main Results

In 2019, Berinde [10] considered a new class of mappings which is defined below.

Definition 18. Let be a Banach space and a mapping. The mapping is called -enriched nonexpansive if in such a way that for all

It can be noted that 0-enriched mapping is nonexpansive mapping. Even both the class of mappings, that is, quasi-nonexpansive and -enriched nonexpansive, are independent in nature, cf. [27].

Remark 19. Take , and it is straight forward from (19) that Take, then From the above inequality, we can take convex combination of and the identity mappings.

In view of Remark 19, we consider Definition 18 in -hyperbolic spaces.

Definition 20. Let be a -hyperbolic space, a subset of such that , and a mapping. The mapping is called -enriched nonexpansive if in such a way that for all , where

We prove the following important lemma which will be utilized throughout this paper.

Lemma 21. Let be a uniquely geodesic space. For some , let be pairwise distinct points with and Then where

Proof. From Lemma 5, lies between and . And lies between and . From Proposition 6, lies between and . Thus, and for some Since is uniquely geodesic space, we have Since , we have Again, since , we have From (25), (27), and (28), one can conclude Therefore, , and

Theorem 22. Let be a complete UC-hyperbolic space and such that Assume that is closed, bounded, and convex. Let be a -enriched nonexpansive mapping. Then, Moreover, for given , any , there exists such that the sequence generated by (Krasnosel’skiĭ method) -converges to an element of

Proof. By the definition of mapping , we get for all and Set the mapping as follows: Thus, from (31), we get, for all , and is a nonexpansive mapping. For any and a given , we can define a sequence From Proposition 16, it follows that From Theorem 17, ; thus, from Lemma 12, Further, for any Thus, from (W1) Hence, the sequence is monotone nonincreasing. It implies that is Fejér monotone sequence wrt . In view of Proposition 10, the sequence has unique asymptotic center wrt . Suppose is a subsequence of and is unique asymptotic center of wrt . Now, From (35) and Lemma 13, it follows that From Lemma 14, the sequence -converges to an element of From Lemma 21 with and , we have for all since and It follows that Thus, for any , the sequence defined by (30) -converges to a point in

Remark 23. It can be seen that Theorem 22 generalizes the results in [10] (Theorem 3.3) from Hilbert spaces to hyperbolic spaces.

Theorem 24. Let and be same as in Theorem 22. Suppose such that , and is closed and convex. Assume satisfies Condition (I) with For fixed and any , there exists such that the sequence generated by (Krasnosel’skiĭ method) strongly converges to an element of

Proof. By the similar technique in proof of Theorem 22, one can set a mapping as in (32), and is nonexpansive. Let and define For all From (41), we have Thus, and are monotone nonincreasing sequences and and exist. Let From (42) By (44), we have In view of (44), (45), (46), and Lemma 11, it implies: Since we have Since satisfies Condition (I) and (49), we obtain By (47), and One can easily show that is a Cauchy sequence. For the sake of completeness, we prove this claim. From (51), for given , a in such a way that for all Hence, so there is in such a way Therefore, for all , and is a Cauchy sequence. By the closedness of in , converges to a point . Now From (47), . Therefore, the sequence strongly converges to a point in Further, for all with

Remark 25. Theorem 24 generalizes the results in [11] (Theorem 3.2) from Hilbert spaces to hyperbolic spaces.

Theorem 26. Let and be same as in Theorem 22. Let such that and be a closed and convex. Suppose that is compact mapping with For fixed , is a sequence generated as follows: strongly converges to an element of

Proof. We set the nonexpansive mapping as in the proof of Theorem 22. For given and for any , define a sequence Following largely as in Theorem 24 and from Lemma 11Since we get From the above equation and (60) Since the range of under is subset of a compact set, there is a subsequence of strongly converges to By (63), the subsequence strongly converges to Since is nonexpansive mapping and by the triangle inequality, we obtain Thus, subsequence strongly converges to and Since exists, it follows that the sequence strongly converges to an element of

4. SP Iterative Method

In this section, we present some convergence results for SP iterative process. For a fix and the mapping , the SP iterative method in the setting of hyperbolic metric spaces can be defined as follows [26]: where , , and are sequences in .

Similar to [25] (Lemma 4), we model the following lemma.

Lemma 27. Let and be same as in Theorem 24. Let be a nonexpansive mapping with For fixed and for all , with , the sequence is defined by Then, the following holds: (1) exists (2)

Proof. From (W1), we get Further, from (67) and (68), we get Thus, is a monotone nonincreasing sequence. Hence, exists. Let From (69) and (70), we have Thus Since the mapping is nonexpansive and from (72) From (70), (73), (74), and Lemma 11, it follows:

Theorem 28. Let and be same as in Theorem 24. Let be a -enriched nonexpansive mapping with For fixed for all , with , the sequence generated by (65) -converges to an element of

Proof. For given and for all , with , we can consider a sequence : where is a mapping defined as in (32). Using Lemma 21, we have where , , and . By Lemma 27 and repeating the technique of proof of Theorem 22, one can complete the proof.

Theorem 29. Let , , and be same as in Theorem 24. For fixed , for all , with , the sequence generated by (65) strongly converges to an element of

Proof. Using proof of Theorem 28, Lemma 27, and Theorem 24, one can complete the proof.

Theorem 30. Let and be same as in Theorem 24. Let be a compact -enriched nonexpansive mapping with For fixed , for all , with , the sequence generated by (65) strongly converges to a point in

Proof. Using proof of Theorem 28, Lemma 27, and Theorem 26, one can complete the proof.

Remark 31. Theorems 28–30 are new even in Hilbert spaces.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors contributed equally to this work.

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Copyright © 2022 Rahul Shukla and Rekha Panicker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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