Abstract
In this article, we introduce a novel iterative algorithm to approximate fixed point of mappings with condition. We establish some strong and weak convergence results in a uniformly convex Banach space. Using a numerical example, we compare the speed of the proposed algorithm with some leading iterative algorithms.
1. Introduction and Preliminaries
Throughout the article, stands for the set of all positive numbers and for the set of all real numbers. Let be a Banach space, a subset of and . An element is called a fixed point of if . We denote the set by . is called quasi-nonexpansive (resp., nonexpansive) if and for all and (resp., , for all ). It is well known that if is uniformly convex and is convex closed bounded then is nonempty (see [1–3]). In [4], Suzuki generalized the concept of nonexpansive mappings as follows.
Definition 1. Let be a subset of a Banach space . A mapping is said to satisfy the condition (or Suzuki mappings) if for each .
Remark 2. It is easy to see that every nonexpansive mapping satisfies the condition.
Using the following example, Suzuki [4] showed that condition is more general than the nonexpansiveness.
Example 3 (see [5]). Define a mapping by Here, satisfies condition, but not nonexpansive.
Recently in 2018, Patir et al. [5] generalized the notion of condition as follows.
Definition 4. Let be a subset of a Banach space . A mapping is said to satisfy the condition if there exists and satisfying such that for each , implies .
Remark 5. It is easy to see that a mapping with condition satisfies the condition.
Using the following example, Patir et al. [5] showed that condition is more general than the condition.
Example 6 (see [5]). Define a mapping by Here, satisfies condition, but not .
The famous theorem of Banach suggests a Picard iterative algorithm, that is, to approximate the unique fixed point of , where is a contraction mapping. It is now well-known that for a nonexpansive mapping the Picard iterative algorithm may fail to converge to the fixed point of . Therefore, it is natural to investigate new algorithms to overcome such problem and to get better speed of convergence. Nowadays, the well-known algorithms are the Mann [6], Ishikawa [7], Agarwal et al. [8], Noor [9], Abbas and Nazir [10], Thakur et al. [11], and so on. For some recent literature on iterative algorithms, we refer the reader to [12–18].
Mann [6] iterative algorithm is defined as where .
Ishikawa [7] iterative algorithm reads as follows: where .
Agarwal et al. [8] generalized iterative algorithm (6) is as follows: where .
In 2016, Thakur et al. [11] used a new iterative algorithm. With the help of a numerical example, they showed that this algorithm is better than the Picard, Mann, Ishikawa, Agarwal, Noor, and Abbas iterative algorithms for the class of Suzuki mappings. The algorithm reads as follows: where .
Motivated by the above, in this article, we propose a new iterative algorithm and prove some weak and strong convergence results for mapping satisfying condition in the context of uniformly convex Banach spaces. We also present a new numerical example of a mapping satisfying condition and numerically compare speed of the proposed iterative algorithm with the speed of leading iterative algorithms. Our results are new and never investigated before.
Following are some basic definitions and results needed in the sequel.
Definition 7 (see [19]). Recall that a Banach space is said to be uniformly convex if for each there is a such that for every two elements in ,
Definition 8. Let be a subset of a Banach space and a bounded sequence in . For each define (i)asymptotic radius of at by (ii)asymptotic radius of relative to by (iii)asymptotic center of relative to by The set is singleton provided that is uniformly convex. Also, is nonempty convex provided that is convex and weakly compact, see, e.g., [20, 21].
Definition 9 (see [22]). A Banach space is said to have Opial’s property if for each sequence in which weakly converges to and for every - it follows that
Definition 10 (see [23]). Let be a subset of a Banach space . A mapping is said to satisfy condition if there is a function satisfying and for all such that for all , where represents distance of from .
The following lemma gives many examples of mappings with condition.
Lemma 11. Let be a subset of a Banach space . If satisfies condition, then satisfies the condition.
Lemma 12 (see [5]). Let be a subset of a Banach space and satisfies condition. If is a fixed point of , then for each
Theorem 13 (see [5]). Let be a subset of a Banach space having Opial property. Let satisfy condition. If be such that (i) converges weakly to (ii)then .
Proposition 14 (see [5]). Let be a subset of a Banach space . If satisfies the condition on , then for all and , (i)(ii)at least one of the following ((a) and (b)) holds: (a)(b) Condition (a) implies and condition (b) implies .(iii)We need the following useful lemma from [24].
Lemma 15. Let be a uniformly convex Banach space and for every . If and are two sequences in such that , , and for some , then .
2. Main Results
In this section, we first introduce a new iterative algorithm, namely, JA iteration, to approximate fixed point of mappings satisfying condition as follows: where .
Now we construct a new example of mapping which satisfies condition but not . We then use this example to compare the rate of convergence of JA iterative algorithm (12) with the leading Thakur (8) and leading two-step Agarwal iterative algorithm (7). This example also shows that the converse of Lemma 11 does not hold in general.
Example 16. Let be endowed with absolute valued norm. Define a mapping by Choose and . We see that but . Thus, is not Suzuki mapping. Choose and , we prove that satisfies the condition.
Case 1. For , we have
Case 2. For and , we have
Case 3. For , we have
Hence, satisfies the condition. Now we show that algorithm (12) converges fast to as compared to the leading Thakur (8) and Agarwal (7). Choose , . Table 1 and Figure 1 show the efficiency of the proposed iterative algorithm.
Now using Example 16, we obtained the influence of initial points and parameters for the Agarwal (7), Thakur (8), and JA (12) iterations. In Table 2, items in bold show that the JA iteration (12) converges faster than others.
Now we establish some convergence theorems for mappings with condition using algorithm (12). Similar results for the other algorithms can be established on the same line of proofs.
Lemma 17. Let be a closed convex subset of a Banach space and satisfies the condition with . If is a sequence generated by (12), then exists for each .
Proof. Let . By Lemma 12, we have They imply that
Thus, is bounded and nonincreasing, and hence, exists for each .
Theorem 18. Let be a closed convex subset of a uniformly convex Banach space and a mapping satisfying the condition. If is a sequence generated by (12), then if and only if is bounded and .
Proof. Suppose and . Then, by Lemma 17, exists and is bounded. Put
By the proof of Lemma 17 together with (19), we have
By Lemma 12, we have
Again by the proof of Lemma 17, we have
It follows that
So, we can get . Therefore,
From (25), we have
Hence,
Now from (19), (21), and (27) together with Lemma 15, we obtain
Conversely, let . By Proposition 14 (iii), for , , (by Proposition 14 (i))
So . Since is uniformly convex Banach space, so .
Theorem 19. Let be a closed convex subset of a uniformly convex Banach space having Opial property. If satisfies the condition with , then generated by (12) converges weakly to an element of .
Proof. By Theorem 18, the sequence is bounded and . Since is uniformly convex, we have that is reflexive. Thus, a subsequence of exists such that converges weakly to some . By Theorem 13, . It is suffice to show that converges weakly to . Indeed, if does not converge weakly to , then there exists a subsequence of and such that converges weakly to and . Again by Theorem 13, . By Lemma 17 together with Opial property, we have
This is a contradiction. So, . Thus, converges weakly to .
Theorem 20. Let be a closed and convex subset of a uniformly convex Banach space and satisfies the condition with . If is a sequence generated by (12), then converges to an element of if and only if or .
Proof. The necessity is obvious.
Conversely, suppose that and . From Lemma 17, exists for each . Therefore, , by assumption. We prove that is a Cauchy sequence in . As , for a given , there exists such that for each ,
In particular, . Therefore, there exists such that
Now for ,
This shows that is a Cauchy sequence in . As is a closed subset of a Banach space , so there exists a point such that . Now gives that . The closeness of follows from Lemma 12. Hence, .
We now prove the following theorem using condition .
Theorem 21. Let be a closed and convex subset of a uniformly convex Banach space and satisfies the condition with . If is a sequence generated by (12), then converges strongly to an element of provided that satisfies the condition .
Proof. From Theorem 18, we have Thus, by condition , we obtain
Now all the requirements of Theorem 20 are satisfied; therefore, by its conclusion converges strongly to a fixed point of .
Remark 22. In the view of Lemma 11, our results contain the results for mappings with condition.
3. Conclusions
We have introduced a new faster iterative algorithm, namely, JA iteration, for the approximation of fixed points of the class of mapping with condition. Moreover, the class of mappings with condition is essentially more general than the class of Suzuki mappings. Thus, our results are better in both the senses (faster iteration and general mapping). Hence, our results are the good extension of the corresponding results of Abbas and Nazir [10], Thakur et al. [11], and references therein.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
The first author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.