Abstract

We prove some strong convergence of a new random iterative scheme with errors to common random fixed points for three and then nonself asymptotically quasi-nonexpansive-type random mappings in a real separable Banach space. Our results extend and improve the recent results in Kiziltunc, 2011, Thianwan, 2008, Deng et al., 2012, and Zhou and Wang, 2007 as well as many others.

1. Introduction and Preliminaries

The theory of random operators is an important branch of probabilistic analysis which plays a key role in many applied areas. The study of random fixed points forms a central topic in this area. Research of this direction was initiated by Prague School of Probabilistic in connection with random operator theory [1–3]. Random fixed point theory has attracted much attention in recent times since the publication of the survey article by Bharucha-Reid [4] in 1976, in which the stochastic versions of some well-known fixed point theorems were proved.

A lot of efforts have been devoted to random fixed point theory and applications (e.g. see [5–10] and many others).

Let be a measurable space, a nonempty subset of a separable Banach space . A mapping is called measurable if for every Borel subset of .

A mapping is said to be random mapping if for each fixed , the mapping is measurable.

A measurable mapping is called a random fixed point of the random mapping if for each .

Throughout this paper, we denote the set of all random fixed points of random mapping by and by for the th iterate of .

The class of asymptotically nonexpansive mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [11] proved that if is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

Iterative techniques for asymptotically nonexpansive self-mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors (e.g. see [12–15]).

The strong and weak convergences of the sequence of Mann iterates to a fixed point of quasi-nonexpansive mappings were studied by Petryshyn and Williamson [16]. Subsequently, the convergence of Ishikawa iterates of quasi-nonexpansive mappings in Banach spaces were discussed by Ghosh and Debnath [17]. The previous results and some obtained necessary and sufficient conditions for Ishikawa iterative sequence to converge a fixed point for asymptotically quasi-nonexpansive mappings were extended by Liu [18, 19].

In 2000, Noor [20] introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces. Xu and Noor [21] introduced and studied a three-step iterative scheme for asymptotically nonexpansive mappings, and they proved weak and strong convergences theorems for asymptotically nonexpansive mappings in Banach spaces. In 2005, Suantai [22] defined a new three-step iteration, which is an extension of Noor iterations, and gave some weak and strong convergences theorems of such iterations for asymptotically nonexpansive mappings in uniformly convex Banach spaces.

For nonself nonexpansive mappings, some authors (e.g., see [23–27]) have studied the strong and weak convergences theorems in Hilbert space or uniformly convex Banach spaces.

A subset of is said to be a retract of if there exists a continuous map such that for all . Every closed convex subset of uniformly convex Banach space is a retract. A map is a retraction if . It follows that if a map is a retraction, then for all in the range of .

The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [28] in 2003 as the generalization of asymptotically nonexpansive self-mappings.

They studied the following iteration process: where is an asymptotically nonexpansive nonself mapping, is a real sequence in , and is a nonexpansive retraction from to .

Wang [29] generalized the result of Chidume et al. [28] and got some new results. He defined and studied the following iteration process: where are asymptotically nonexpansive nonself mappings and are real sequences in .

Now, we introduce the following concepts for nonself mappings

Definition 1 (see [28, 30, 31]). Let be a nonempty subset of a real separable Banach space and a nonself random mapping. Then, is said to be (1)nonexpansive random operator if for arbitrary , , ;(2)nonself asymptotically nonexpansive random mapping if there exists a sequence of measurable functions with for each such that for arbitrary , (3)nonself asymptotically quasi-nonexpansive random mapping if and there exists a sequence of measurable functions with for each such that where is a random fixed point of and is any measurable mapping;(4)nonself asymptotically nonexpansive-type random mapping if (5)Nonself asymptotically quasi-nonexpansive-type random mapping if , and where is a random fixed point of and is any measurable mapping.

Remark 2. (1) If is a nonself asymptotically nonexpansive random mapping, then is a nonself asymptotically nonexpansive-type random mapping.
(2) If and is a nonself asymptotically quasi-nonexpansive random mapping, then is a nonself asymptotically quasi-nonexpansive-type random mapping.
(3) If and is a nonself asymptotically nonexpansive-type random mapping, then is a nonself asymptotically quasi-nonexpansive-type random mapping.

Remark 3. Observe that for any measurable mapping and , we have which implies Therefore,

In [25], Shahzad studied the following iterative sequences: where is a nonexpansive nonself mapping, is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space with being a nonexpansive retraction from to , and are real sequences in .

Recently, Thianwan [32] generalized the iteration process (10) as follows: where are appropriate sequences in and are bounded sequences in . He proved weak and strong convergences theorems for nonexpansive nonself mappings in uniformly convex Banach spaces.

In 2011, Kiziltunc [33] studied the strong convergence to a common fixed point of a new iterative scheme for two nonself asymptotically quasi-nonexpansive-type mappings in Banach spaces defined as follows: where are appropriate sequences in .

More recently, Deng et al. [34] obtained the strong and weak convergences theorems for common fixed points of two nonself asymptotically nonexpansive mappings in Banach spaces.

The iterative scheme is defined as follows: where are appropriate sequences in for some satisfying .

For random operators, Beg and Abbas [30] studied the different random iterative algorithms for weakly contractive and asymptotically nonexpansive random operators on arbitrary Banach space. They also established convergence of an implicit random iterative process to a common fixed point for a finite family of asymptotically quasi-nonexpansive operators. Plubtieng et al. [35, 36] studied weak and strong convergences theorems for a modified random Noor iterative scheme with errors for three asymptotically nonexpansive self-mappings in Banach space defined as follows: where are three asymptotically nonexpansive random self-mappings, is an arbitrary given measurable mapping from to , are bounded sequence of measurable functions from to , and , , , , , , , , are sequences of real numbers in with .

Remark 4. If and , then (14) becomes as follows: which was studied by Beg and Abbas in [30].

For nonself random mappings, Zhou and Wang [37] studied the approximation of the following iteration process: where is an asymptotically nonexpansive nonself random mapping, is an arbitrary given measurable mapping from to , are sequences in , and is a nonexpansive retraction from to .

Saluja [38] and many other authors extended the results of Zhou and Wang [37] by studying multistep random iteration scheme with errors for common random fixed point of a finite family of nonself asymptotically nonexpansive random mapping in real uniformly separable Banach spaces.

Inspired and motivated by [32–34, 37] and others, we introduced a new three-step and -step random iterative scheme with errors for asymptotically quasi-nonexpansive-type nonself random mappings in a separable Banach space. Some strong convergences theorems are established for these new random iterative schemes with errors in separable Banach space. The iterative scheme for three nonself random mappings is defined as follows.

Definition 5. Let be three nonself random mappings, where is a nonempty closed convex subset of a separable Banach space , and is a nonexpansive retraction of onto . Let be a measurable mapping. Suppose that is generated iteratively by , having for all , , where , and are sequences in such that , and are bounded sequences of measurable functions from to for all .

Definition 5 can be extended to nonself random mappings as follows.

Definition 6. Let be nonself random mappings, where is a nonempty closed convex subset of a separable Banach space , and is a nonexpansive retraction of onto . Let be a measurable mapping. Define sequences function , in as follows: where , and are sequences in such that , for all , and are bounded sequences of measurable functions from to for all .

The following lemma is useful for proving our results.

Lemma 7 (see [39]). Let and be nonnegative real sequences satisfying If and , then(1) exists;(2) whenever .

2. Main Results

In this section, we will first prove the strong convergence of the iterative scheme (17) to a common random fixed point for three asymptotically quasi-nonexpansive-type nonself random mappings in a separable Banach space. Then, we extend the obtained results to asymptotically quasi-nonexpansive-type nonself random mappings by using the iterative scheme (18). Finally, we use Theorem 8 and Condition (A) [40] to obtain a convergences theorem for scheme (17).

Theorem 8. Let be a real separable Banach space and a nonempty closed convex subset of with being a nonexpansive retraction. Let , be three asymptotically quasi-nonexpansive-type nonself random mappings with , for all . Suppose that and are the sequences defined as in (17) where ,  , , , , , , , and are sequences in such that and , , are bounded sequences of measurable functions from to with the following restrictions: , , and . Then, converge to a common random fixed point of , , and if and only if

Proof. The necessity of (20) is obvious.
Next, we prove the sufficiency of (20). Let ; by the boundedness of the sequences of measurable functions , we put for each , Then, for each .
Since and is any measurable mapping, we have It follows that for any given , there exists a positive integer such that for and , we have Since , and , then we have for , Setting for , Thus, for and , using (17) and (24), we have In addition, by (24), we obtain Again using (17) and (25), we have In addition, by (25), we have Also, by (17) and (26), we have In addition, by (26), we have Substituting (29), (30), (31), (32), and (33) into (28) and simplifying, we obtain Let ; then, for all .
It follows by (34) that From (35) and for all , we have By Lemma 7 and (36), it follows that exists for all and .
Since , then we have Next, we prove that is a Cauchy sequence in for each .
For , and , we have by (35) that Therefore, by using (38), we have for each , Since and by (39), we have for each , Since and , for given , there exists a positive integer such that and . We have or this shows that is a Cauchy sequence in for each .
Since is complete and is a closed subset of and so it is complete, then there exists a such that as , for all .
Now, we show that .
By contradiction, we assume that does not belong to . Since is closed set, . By using the fact that , it follows that for all , This implies that which is a contradiction. Hence, .