Abstract
We consider the class of mappings endowed with the condition in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and -convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT() spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.
1. Introduction
Choose a metric space and . Then, a self-map is called nonexpansive if
As many know, Browder [1] (see also Kirk [2] and Gohde [3]) achieved an elementary fixed point theorem for nonexpansive map in the case when is closed bounded convex in any given uniformly convex Banach space (UCBS). This remarkable theorem began the research of different results for nonexpansive maps. Among other things, Kirk [4] obtained this theorem in the nonlinear setting of CAT(0) metric spaces. The class of nonexpansive mappings is very important because of its useful applications in many fields of applied sciences. Thus, it is very natural and essential to study the extension of these maps. In 2008, Suzuki [5] considered an extension of nonexpansive maps by weakening the inequality of the nonexpansive maps in the following way: a self-map is said to hold condition if any given ,
Obviously, a nonexpansive map has condition (), but an example in [5] shows the inverse is not valid in general. In [6], Nanjaras et al. improved the main outcome of Suzuki [5] to the nonlinear setting of CAT(0) spaces. In [7], Phuengrattana used Ishikawa iteration [8] for reckoning of fixed points for the class of Suzuki mappings in a Banach space. Also, Basarir and Sahin [9] used Agarwal et al.’s [8] iteration for reckoning fixed points of Suzuki mappings in the nonlinear setting of CAT(0) spaces.
In [10], Garcia-Falset et al. suggested a new type of maps which is properly more general than Suzuki maps. A self-map of is called Garcia-Falset nonexpansive or said to be endowed with the condition if any given , there is a such that
Remark 1. If is nonexpansive, then it is easy to show for each . Hence, every nonexpansive map has condition with . Similarly, if is Suzuki map, then, it satisfies for every . Hence, every Suzuki map has condition with . Interestingly, the example constructed in this paper shows that there exists maps satisfying condition but not the converse.
After establishing the existence of a fixed point result for an operator, a natural question arises on how to compute it by any appropriate numerical method. To work on such type of problems is not easy. The well-known Banach Contraction Theorem (BCT) essentially suggests the Picard iteration to compute the unique fixed point of a given contraction in a metric space. Bagherboum [11] used Ishikawa iteration [12] for computing fixed points of Garcia-Falset mappings in the nonlinear setting of Busemann spaces. We know that the Picard iteration fails for solving nonexpansive problems. Now, we discuss some other methods which are different from the Picard iteration process.
One of the elementary iterative processes is due to Mann [13] stated as follows: where .
The Mann iteration (4) was extended to the setting of two steps by Ishikawa [12] as follows: where .
In 2007, Agarwal et al. [8] studied the following iteration for contractions and observed that it is better than the above iterative processes:
where
In this research, we consider the Agarwal iterative process (6) for the larger class of maps due to Garcia-Falset et al. [10]. We use a very general ground space called 2-uniformly convex hyperbolic metric space for establishing the main outcome. To support our results, we provide an example of maps with condition and prove that its Agarwal iterative process (6) is more effective than Mann (4) and Ishikawa iterative (5) processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT() spaces. In this way, we extend the corresponding results of Bagherboum [11], Phuengrattana [7], Basarir and Sahin [9], and Nanjaras et al. [6] as we consider the general setting of domain, larger class of maps, and faster iteration process.
Now, we present some well-known definitions and results which will be either used in the main results or to understand the given concept herein.
Definition 2 (see [14]). A given metric space is called Kolenbach–hyperbolic space or simply hyperbolic space when there is a map such that for all and , we have
(C1) ;
(C2) ;
(C3) ;
(C4) .
Notice that, for given and , we shall use for . Also, the property (C1) gives
Keep in mind that is said to be convex in if for every , .
Definition 3. A given hyperbolic space is said to be a uniformly convex hyperbolic space (UCHS), if there exist , and such that for each satisfying , and follows that:
Moreover, if is given such that for all and then, is called a modulus of uniform convexity (MUC). Also, is called monotone modulus of uniform convexity (MMUC) iff nonincreasing in the variable for every chosen fixed .
Definition 4. Suppose a pair is a given hyperbolic space. For any and , define: and keeping in mind that infimum is taken over every satisfying , , and . Then, is called -uniformly convex hyperbolic space (2-UCHS) provided that
The following remark suggests that 2-UCHS is a more general domain than many other linear and nonlinear spaces.
Remark 5. Any given UCBS, CAT(0) space, and CAT() spaces having and , can be considered as a -UCHS (see [15–17]).
Definition 6. Choose a complete 2-UCHS such that is bounded and is closed as well as convex. The asymptotic radius of corresponding to is . Moreover, the asymptotic center of corresponding to is the set . In addition, if the underlying space has MMUC, then the set is singleton (see, e.g., [18]).
Definition 7. Choose a complete 2-UCHS such that is closed as well as convex. Fix an element . Assume that is a bounded sequence in . Then, is called -convergent to iff the asymptotic center for all subsequence of . When is -limit of , then we shall write .
Lemma 8 (see [19]). Any bounded sequence in a given 2-UCHS having MMUC has a -convergent subsequence.
Lemma 9 (see [20]). Choose a complete 2-UCHS endowed with MMUC such that and is bounded. Then, the asymptotic center of is always in .
Mappings endowed with condition enjoy the following useful properties which we have combined in a proposition form. For details, we refer the reader to [10].
Proposition 10. Choose a complete 2-UCHS such that . Suppose . (i)When the map has condition with nonempty fixed point set, then for each and , we have (ii)When the map has condition then fixed point set of is always closed in
We now state an important characterization of a 2-UCHS, which was proved in [15].
Lemma 11. Choose a 2-UCHS . Then, the following inequality holds for each and .
2. Approximation Results
We now want to show our desirable strong and -convergence results by considering the Agarwal iteration (6). It should be noted that will be considered complete and endowed with the MMUC. The notation will simply denote the set of all fixed points of .
We suggest the following useful lemma which will provide a key role in main outcome.
Lemma 12. Choose a 2-UCHS such that is closed as well as convex. Furthermore, let be endowed with condition having . If is the Agarwal iterates (6), then, exists for all .
Proof. Assume that . By using Proposition 10 (i), one has Hence, From the above observations, we have . Hence, the real sequence is bounded below by and nonincreasing as well, so exists for every .
The existence of a fixed point for a self-map having condition in the nonlinear domain 2-UCBS can be established by the following way. This result provides a fundamental key for establishing the main results in the sequel.
Theorem 13. Choose a 2-UCHS such that is closed as well as convex. Furthermore, let be endowed with condition . If is the Agarwal iterates (6) with restriction . Then, iff is bounded and .
Proof. Suppose that the iterative sequence is bounded having . We may fix an element, namely, . By using condition of , we have It follows that . Since is singleton set, one has . Thus, . Conversely, we suppose that and . Then, by the proof of Lemma 12, is bounded. By Lemma 11, we have Thus, Since , it follows that Thus, and hence
In some sense, the notion of -convergence in nonlinear domains provides the analog of the notion of weak convergence in linear setting. To obtain the -convergence for operators having the condition in a 2-UCHS by using the Agarwal iterative sequence (6), we shall propose the following techniques.
Theorem 14. Choose a 2-UCHS such that is closed as well as convex. Furthermore, let be endowed with condition and . If is the Agarwal iterates (6) with restriction , then, -converges to a fixed point of .
Proof. It has been observed in Theorem 13 that the iterative sequence is bounded having . We may consider the and notice that the union has been imposed on all possible subsequences of the iterative sequence . We want to show that . Suppose . Then, one can find a subsequence of in such a way that . By Lemmas 8 and 9, one can find a subsequence of in such a way that . Keeping in mind and applying condition of , one has
If we apply on both of the sides of above, then, we have . By Lemma 12, exists. Next, we show that . Assume on contrary that . Then, using the property of uniqueness of asymptotic centers, we get
The above strict inequality suggests a contradiction. So we must have and .
Finally, we show that is -convergent in , that is, we want to show is singleton. Assume that is a subsequence of the sequence . In the view of Lemmas 8 and 9, one can find a subsequence of in such a way that -. Suppose and . We have already established that and . Now, we have to show that . Suppose not, then since exists and also the asymptotic centers are consist of exactly one point, so
which is obviously a contradiction and so . Hence, .☐
Normally, we are interested in a strong convergence. To obtain the strong convergence theorems, we essentially impose some other assumptions on the domain or the operator. We first take a compact domain and establish a strong convergence theorem as follows.
Theorem 15. Choose a 2-UCHS such that is compact as well as convex. Furthermore, let be endowed with condition and . If is the Agarwal iterates (6) with restriction . Then, converges strongly to a fixed point of .
Proof. Since is compact, we may select a subsequence of such that . By using condition of , we have In the view of Theorem 13, . Hence, if we apply on both of the sides of the above inequality, we get by uniqueness of limits in metric spaces, . Hence, is the fixed point of . Moreover, by Lemma 12, exists. So is strongly convergent to . This completes the proof.☐
The compactness of the domain in the above theorem provided us a significant help. Now, one thinks about how to replace the compactness of the domain by any other assumption. We replace it by the following condition, which was essentially defined by Senter and Dotson [21].
Definition 16. A self-map on a subset of a 2-UCHS is said to satisfy condition if there exists some nondecreasing function satisfying , for every and for each .
Theorem 17. Choose a 2-UCHS such that is closed as well as convex. Furthermore, let be endowed with conditions and and . If is the Agarwal iterates (6) with restriction . Then, converges strongly to a fixed point of .
Proof. Theorem 13 gives . By condition one has Thus, one can choose a subsequence of and in such that for every natural number . Proof of Lemma 12 suggests that is nonicreasing, and so Now, From the above, one can easily conclude that the sequence form a Cauchy sequence in the closed subset of . Hence, and so is a fixed point of . But Lemma 12 suggests that exists. Thus, the element is the strong limit point of the sequence .☐
3. Numerical Observations
This section is devoted to some numerical computations. We first suggest a nontrivial example of endowed with the condition , but not with . This example shows that the class of Garcia-Falset is properly more general than the class of Suzuki mappings.
Example 1. Suppose and is a self-map defined as follows:
We fix the value of . We shall prove that for each .
Case I. Assume that . Then, . Thus,
Case II. Here, we choose . Then and . Thus,
Case III. If and . Then and . So,
Case IV. If and . Then and and so
Next, we provide the values for which does not satisfy condition . Choose and . It is easy to show but . For any choice of , let and , then, the observations are given in Table 1.
Now, we study the behavior of Agarwal, Ishikawa, and Mann iterations under different cases. Notice that is the stopping criteria, , and we stop the iterates for . In this case, Figures 1, 2, and 3 show that the Agarwal iteration process converges faster to the fixed point as compared to the other iterations.
4. Conclusions
In this paper, we have initiated the finding of fixed points for the generalized class of mappings due to Garcia-Falset et al. [10] in the general setting of 2-UCHS. We established several strong and -convergence theorems under the Agarwal iterative process [8]. We have presented a new example of mapping having condition but not and proved that its Agarwal iterative process [8] is more effective than the corresponding Mann [13] and Ishikawa [12] iterative processes. In view of the above discussion, our results simultaneously hold in UCBS, CAT(0), and some CAT() spaces. The present results can be extended and applied to many fields of applied sciences and differential equations. Thus, our results extend the corresponding results of Bagherboum [11], Phuengrattana [7], Basarir and Sahin [9], and Nanjaras et al. [6] from the general setting of domains and faster iteration process.
Data Availability
The data used to support the findings of this study are included in the references within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
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.