Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

Chugh, Renu; Kumar, Sanjay

doi:https://doi.org/10.1155/2014/781704

Abstract and Applied Analysis

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Iterative Fixed-Point Methods for Solving Nonlinear Problems: Dynamics and Applications

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Volume 2014 | Article ID 781704 | https://doi.org/10.1155/2014/781704

Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

Renu Chugh¹and Sanjay Kumar¹

Academic Editor: Alicia Cordero

Received05 Aug 2014

Accepted27 Sept 2014

Published24 Dec 2014

Abstract

We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families to -number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

1. Introduction

Let be real Banach space. The convex subset is called proximal set if for each there exists at least one such that . Every closed convex subset of uniformly convex Banach spaces is proximal. We use the following notations for multivalued mappings: : collection of all nonempty compact subsets of ; : collection of all nonempty proximal bounded subsets of ; : collection of all nonempty bounded closed subsets of .Let be Hausdorff metric induced by of defined as , for every . Let ; a multivalued mapping is said to be nonexpansive if , for all . An element is called fixed point of if , where the set of all fixed points of is denoted by . The mapping is said to be quasi nonexpansive if and for all and . It is known that every nonexpansive multivalued mapping with is quasi nonexpansive, but there exist quasi-nonexpansive mappings which are not nonexpansive. It is well known that if is quasi-nonexpansive mapping, then is closed.

Definition 1. A map is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . It is clear that if is compact, then every multivalued mapping is hemicompact.

Definition 2. A Banach space is said to satisfy Opial’s condition if imply that The study of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings. In 1969, Nadler Jr. [1] proved the convergence theorem for multivalued contraction mappings. Then in 1973, Markin [2] studied the multivalued contraction and nonexpansive mappings in Hausdorff metric space. Later in 1997, Hu et al. [3] proved the convergence theorems for finding common fixed point of two multivalued nonexpansive mappings that satisfies certain contractive conditions. Sastry and Babu [4] proved the convergence of Mann and Ishikawa iterates to a fixed point of the multivalued mapping with fixed point under certain conditions. They proved with the help of example that limit of the sequence is different from the point of initial choice. Then Abbas et al. [5] introduced the new one-step iterative processes to compute the common fixed point of two multivalued nonexpansive mappings in a real uniformly convex Banach space. Let be two multivalued nonexpansive mappings. They introduced iteration as follows: where , , such that and for and satisfying . Then they obtained strong convergence theorems for the proposed process under some basic boundary conditions.

In 2012 Bunyawat and Suantai [6] introduced the one-step iterative process as follows: where the sequence satisfying and such that for . They proved the convergence of iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach space.

Then Zhang et al. in 2013 introduced the two-step iterative process as follows: where the sequences satisfying , , such that and such that for . Zhang et al. extended the results of Bunyawat and Suantai from one countable family to two countable families and also gave a new proof for the iteration used in the paper of Abbas et al. [5].

In the same year Ahmed and Altwqi introduced the three-step iterative process as follows: where , , and and the sequences satisfying , , and proved the strong and weak convergence results for three finite families of multivalued nonexpansive mappings.

Different iterative processes have been used to approximate fixed points of multivalued mappings. Many authors have intensively studied the fixed point theorems and got some results. At the same time, they extended these results to many discipline branches, such as control theory, convex optimization, variational inequalities, differential inclusion, and economics (see [7–19]).

Motivated by [6, 20–22], in this paper, we extended the result of Zhang et al. [21] from two countable families to -number of countable families and proved weak and strong convergence results of two new multistep iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space. Also with the help of numerical example we compare the convergence step of two different multistep iterative processes. We use the following iteration processes: where , such that , and such that , , and , are sequences in which satisfies and , , and , , and and , .

Remark 3. If and 2, then multistep iteration (6) reduces to one-step and two-step iterations (3) and (4) defined by Bunyawat and Suantai and Zhang et al. whereas for , multistep iteration (6) reduces to finite three-step iteration (5) defined by Ahmed and Altwqi.

Lemma 4 (see [6]). Let be a uniformly convex Banach space, a positive number, and a closed ball of . Then, for any given sequence and for any given sequence of positive number with , there exists a continuous, strictly increasing, and convex function with such that, for any positive integer , with

Lemma 5 (see [23]). Suppose that is a uniformly convex Banach space and for all positive integers . Also suppose that and are two sequences of such that , , and hold for some ; then .

Lemma 6 (see [24, 25]). Let be a sequence of nonnegative real numbers satisfying the following property: , where , , and satisfy the following restrictions:(i); (ii); (iii).Then, converges to zero as .

2. Main Results

2.1. Weak and Strong Convergence Results for New Multistep Iterative Scheme (6)

Theorem 7. Let be a nonempty closed convex subset of a uniformly convex Banach space with Opial’s condition. For , let be sequences of multivalued quasi-nonexpansive mappings from into with and . Let be the sequence defined by (6) and then it converges weakly to a point .

Proof. Let ; first we prove that is bounded and exists. Now from Lemma 5 and (6), we have From (6), we get Similarly, we have Now continuing like this at last, we get By putting (13) in (14), we get So from (15), we say that is nondecreasing and bounded and hence, is bounded and exists.
Now we prove that and , where .
From (15), we can write as Since we assume that , , , , , , and exist, we have and from the continuity of , we have Now we will prove that , for each .
From (9) and (10), we have Now, by putting (10) in (12), we have Now continuing like this at last, we have we can write it as Since we assume that , , , , , , and exists, we have and from the continuity of , we have So by repeating these steps for different values of , we have Next, we prove that , for each , where .
From (6), we have From (23), we have By using triangle inequality, we have Together with (23) and (25), we have Now, we prove that converges weakly to a point . Since we have proved that is bounded, there exists a subsequence of such that converges weakly to ; using (25), we can say that converges weakly to , for . Now suppose that there exist , such that , for ; then by Opial’s condition we have As are -multivalued quasi-nonexpansive mappings, we have where .
Taking of both sides of (30) and from (17) and (27), we have Now combining (28) with (31) and (29) with (32), we have which gives contradiction, so we have , for and ; this implies . Now we prove that converges weakly to . Let be another subsequence of that converges weakly to some . Again as above we conclude that . We show that . Let , since exists for every . From (1), we have It implies that , a contradiction. So we have . It means that converges weakly to as .
For , , , and , Theorem 7 reduces to the following corollary.

Corollary 8. Let be a nonempty closed convex subset of a uniformly convex Banach space with Opial’s condition. Let , , and be three multivalued quasi-nonexpansive mappings from into with and . Let be the iteration defined as where such that , such that , and such that , , , and are sequences in which satisfies and , , , and ; then converges weakly to a point .

Theorem 9. Let be a uniformly convex real Banach space and let be a bounded and closed convex subset of . For , let be a sequence of multivalued quasi-nonexpansive and continuous mappings from into with , and . Let be a sequence defined by (2) with and , are sequences in which satisfies and , , and , , and and , , for all . Assume that one of is hemicompact. Then converges strongly to common fixed point of .

Proof. Let is hemicompact for some , then from (17) and (27), we have , for all , and , for all , . So there exists a subsequence of such that ; using (25), we can say that . From continuity of , , we have and . This implies that , , and . Since exists, it follows that converges strongly to .

Theorem 10. Let be a uniformly convex real Banach space and let be a compact convex subset of . For , let be a sequence of multivalued quasi-nonexpansive mappings from into with , , and . Let be a sequence defined by (2) with and , are sequences in which satisfies and , , and , , and and , , for all . Then converges strongly to common fixed point of .

Proof. Since is compact, there exists a subsequence of such that for some ; also from (25), we can say that Now, we have Hence this implies that and converges strongly to common fixed point of .

Remark 11. If in iterative process defined by (6) we use and 2, Theorems 7, 9, and 10 reduce into convergence results proved by Bunyawat and Suantai [6, 22] and Zhang et al. [21].
For and (any finite number), Theorems 7, 9, and 10 reduce into result proved by Ahmed and Altwqi [26].

2.2. Strong and Weak Convergence Results for New Multistep Iterative Scheme (7)

Theorem 12. Let be a nonempty closed convex subset of a uniformly convex Banach space with Opial’s condition. For , let be sequences of multivalued quasi-nonexpansive mappings from into with and . Let be the sequence defined by (7); then it converges weakly to a point .

Proof. Let , first we prove that is bounded and exists. Now, from Lemma 5 and (7), we have Similarly, we get Now continuing like this at last, we have By putting (41) in (42), we get So from (43), we say that is nondecreasing and bounded and hence, is bounded and exists.
Now we prove that , and , where .
From (43), we can write as Now from (7), as we assume that the following conditions hold: , , , , , , and exist. So from these conditions, we have and then using the continuity of , we have From (38) and (39), we have Now by putting (39) in (40), we have Now continuing like this at last, we have we can write it as Since we assume that , and exists, we have and from the continuity of , we have So by repeating these steps for different values of , we have Now, we prove that , from (7), we have Taking limit of both sides of (53) and using (45), we have Again using (7), we have From (51) and (54), we have . Now by repeating these steps for all values of , we get Now to prove weak convergence of to a point , we will use (43); that is, is bounded, so there exists a subsequence of such that converges weakly to ; then using (56), we claim that converges weakly to , for . Now assume that there exist , such that , for ; then by Opial’s condition we have As are -multivalued quasi nonexpansive mappings, we have where .
Taking of both sides of (59) and from (45) and (51), we have Now combining (57) with (60) and (58) with (61), we have which gives contradiction, so we have , for and ; this implies that . Now we prove that converges weakly to . Let be another subsequence of that converges weakly to some . Again as above we conclude that . We show that . Let , since exists for every . From (1), we have It implies that , a contradiction. So we have . It means that converges weakly to as .

Theorem 13. Let be a uniformly convex real Banach space and let be a bounded and closed convex subset of . For , let be a sequence of multivalued quasi-nonexpansive and continuous mappings from into with , and . Let be a sequence defined by (7) with and , are sequences in which satisfies and , , , , for all . Assume that one of is hemicompact. Then converges strongly to common fixed point of .

Proof. Let is hemicompact for some ; then from (45) and (50), we have , for all , and , for all , . So there exists a subsequence of such that using (56); we can say that . From continuity of , , we have and . This implies that , and . Since exists, it follows that converges strongly to .

Theorem 14. Let be a uniformly convex real Banach space and let be a compact convex subset of . For , let be a sequence of multivalued quasi-nonexpansive mappings from into with , and . Let be a sequence defined by (7) with and , are sequences in which satisfies and , , , , for all . Then converges strongly to common fixed point of .

Proof. Since is compact, there exists a subsequence of , such that for some ; also from (56), we can say that Now, we have Hence this implies that and converges strongly to common fixed point of .

Remark 15. Since iterations used in [4–6, 15, 21] are special case of iterative scheme (6), motivated from them, we generalize in the following sense. (1)Since we prove our result for quasi-nonexpansive mappings so generalizes from nonexpansive to quasi nonexpansive.(2)We generalize from single-valued to multivalued mappings.(3)Our results extended from one and two countable families to -number of countable families of multivalued quasi-nonexpansive mappings.(4)We prove weak and strong convergence results for new multistep iterative scheme (7). With the help of numerical example of multivalued quasi-nonexpansive mappings and computational program in C++ we prove fast rate of convergence of new multistep iterative scheme (7).

3. Numerical Computation

We use the following numerical example of finite family of multivalued quasi-nonexpansive mappings to compare the converging steps of one-step, two-step, and two new multistep iterative procedures. Let be -countable family of multivalued quasi-nonexpansive mappings defined as Now using the initial value and different initial conditions used in result proved above in C++ program, we get the following observation for different iterations.

4. Conclusion

After analyzing the comparison shown in Table 1 we conclude that the iterative scheme (7) converges faster than other existing iterative schemes and rate of convergence increases as the number of step of iterations increases for multivalued quasi-nonexpansive mappings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright

Copyright © 2014 Renu Chugh and Sanjay Kumar. 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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