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Journal of Mathematics
Volume 2013 (2013), Article ID 678946, 9 pages
http://dx.doi.org/10.1155/2013/678946
Research Article

Convergence of Common Random Fixed Point of Finite Family of Asymptotically Quasi-Nonexpansive-Type Mappings by an Implicit Random Iterative Scheme

1Department of Mathematics, J. H. Government PG College, Betul 460001, India
2Department of Mathematics, NRI Institute of Information Science & Technology, Bhopal 462021, India

Received 26 October 2012; Revised 5 February 2013; Accepted 6 February 2013

Academic Editor: Zindoga Mukandavire

Copyright © 2013 A. S. Saluja and Pankaj kumar Jhade. 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.

Abstract

We introduce a new implicit random iteration process generated by a finite family of asymptotically quasi-nonexpansive-type mappings and study necessary and sufficient conditions for the convergence of this process in a uniformly convex Banach space. The results presented in this paper extend and improve the recent ones announced by Plubtieng et al. (2007), Beg and Thakur (2009), and Saluja and Nashine (2012).

1. Introduction

Probabilistic functional analysis has come out as one of the momentous mathematical disciplines in view of its requirements in dealing with probabilistic models in applied problems. The study of random fixed points forms a central topic in this area. Random fixed point theorems for random contraction mappings on separable complete metric spaces were first proven by Špaček [1]. Subsequently, Bharucha-Reid [2] has given sufficient conditions for a stochastic analog of Schauder’s fixed point theorem for a random operator. The study of random fixed point theorems was initiated by Špaček [1] and Hanš [3, 4]. In an attempt to construct iterations for finding fixed points of random operators defined on linear spaces, random Ishikawa scheme was introduced in [5]. This iteration and also some other random iterations based on the same ideas have been applied for finding solutions of random operator equations and fixed points of random operators (see [5]).

Recently, Beg [6], Choudhury [7], Duan and Li [8], Li and Duan [9], Itoh [10], and many others have studied the fixed point of random operators. Beg and Abbas [11] studied the different random iterative algorithms for weakly contractive and asymptotically nonexpansive random operators on arbitrary Banach spaces. They also established the convergence of an implicit random iterative process to a common random fixed point for a finite family of asymptotically quasi-nonexpansive operators.

In 2007, Plubtieng et al. [12] studied the implicit random iteration process with errors, which converges strongly to a common fixed point of a finite family of asymptotically quasi-nonexpansive random operators on an unbounded set in uniformly convex Banach spaces and proves some strong convergence theorems.

An implicit process is generally desirable when no explicit scheme is available. Such a process is generally used as a “tool” to establish the convergence of an explicit scheme.

Recently, Beg and Thakur [13] introduced modified general composite implicit random iteration process and proved some strong convergence theorems for a finite family of random asymptotically nonexpansive mappings in separable Banach spaces.

Very recently, Saluja and Nashine [14] introduced a new modified general composite implicit random iteration process to give necessary and sufficient conditions for strong convergence of the iteration process to a common random fixed point of a finite family of asymptotically quasi-nonexpansive in the intermediate sense random operators in separable Banach spaces.

Motivated and inspired by the above work, in this paper, we introduced a new implicit random iteration process generated by a finite family of asymptotically quasi-nonexpansive-type random operators with mixed errors in a uniformly convex Banach spaces and give necessary and sufficient conditions for the convergence of the proposed iteration process to a common random fixed point. The results presented in this paper extend and improve the results of Plubtieng et al. [12], Beg and Thakur [13], Saluja and Nashine [14], and some others.

2. Preliminaries

Let be a measurable space with , a sigma algebra of , and let be a nonempty subset of a Banach space . A mapping is measurable if for each open subset of . The mapping is a random map if for each fixed , the mapping is measurable, and it is continuous if for each , the mapping is continuous. A measurable mapping   is the random fixed point of the random map if , for each . We denote by () and () the set of all random fixed points of a random map and the domain of the random map , respectively, and by the nth iterate of . The letter denotes the random mapping , defined by and .

Definition 1. Let be a nonempty subset of a separable Banach space and let be a random map. The map is said to be a nonexpansive random operator ifforarbitrary , one has for each and ;a quasi-nonexpansive random operator if and for all and ; an asymptotically nonexpansive random operator if there exists a sequence of measurable mappings with for each , such that for arbitrary , an asymptotically quasi-nonexpansive random operator if and there exists a sequence of measurable mappings with for each such that for arbitrary and, an asymptotically nonexpansive type if for all and ; an asymptotically quasi-nonexpansive-type random operator if and for all and ; a uniformly -Lipschitzian random operator if for arbitrary , one has where and is a positive constant.

Remark 2. We know that the following implications hold from the definitions: xy(8)

Example 3. Let be the real line with the usual norm and . Define by for and . Obviously ; that is, is a fixed point of ; that is, . Now we check that is asymptotically quasi-nonexpansive-type random operator. In fact, if and , then that is, Thus, is quasi-nonexpansive. It follows that is asymptotically quasi-nonexpansive with for each , and hence it is asymptotically quasi-nonexpansive-type random operator.The map is a semi compact random operator if for a sequence of measurable mappings from to , with for every , one has a subsequence of and a measurable mapping such that converges pointwise to as . A Banach space is said to be uniformly convex if the modulus of convexity of (see [15])
 for all (i.e., is a function ).

Definition 4 4 (modified general composite iteration process, cf. [14]). Let be a family of asymptotically quasi-nonexpansive in the intermediate sense random operators from to , where is a closed, convex subset of a separable Banach space with . Let be any fixed measurable map, then the sequence of function defined by where with , , where is a positive integer, and as .

Now we introduce the two-step implicit random iterative process with mixed errors for a finite family of asymptotically quasi-nonexpansive-type random operators in a Banach space as follows.

Definition 5 5 (two-step implicit random iterative scheme). Let be random operators (), where is a nonempty convex subset of a separable Banach space . Let be a measurable mapping from to and let and be bounded sequences of measurable functions from to . Define sequences of functions and as given below: which can be written in the following compact form: where .

As a matter of fact, denote the indexing set by . Let be uniformly -Lipschitzian asymptotically quasi-nonexpansive-type random operators from . We show that the scheme (14) exists.

Let and .

Define , for all by for all . The existence of is guaranteed if has a fixed point.

For any and , we have

Now is a contraction if or . As , therefore is a contraction. By the Banach contraction principle, has a unique fixed point. Thus, the existence of is established. Similarly, we can establish the existence of . Thus, the implicit algorithm (14) is well defined.

In the sequel, we will need the following lemmas.

Lemma 6 (see [16]). Let and be two nonnegative sequences satisfying where and is some positive integer. Then exists.

Lemma 7 (see [17]). Let and be two fixed real numbers. Then a Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function with such that for all and , where .

Lemma 8 (see [18]). Let be a given real number. Then for any and , there exists a nonnegative real number between and such that .

3. Main Results

In this section, we investigate the convergence of an implicit random iterative process with mixed errors for a finite family of asymptotically quasi-nonexpansive-type random operators to obtain the random solution of the common random fixed point in a uniformly convex separable Banach space.

Before proving our main results, we first prove the following crucial lemma.

Lemma 9. Let be a uniformly convex separable Banach space and let be a nonempty closed and convex subset of . Let, for each be asymptotically quasi-nonexpansive-type random operators. Suppose that . Let be measurable mapping and let be a sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the following conditions: for some ;;and are bounded, and .
Note. It is pointed out that in condition (iii), for all is equivalent to that there exists a sequence with and as such that for all .
Then for any given , there exists for each , such that for all ; for all , where and is the positive sequence appearing in the note of Lemma 9; exists.

Proof. Let , from (14), where ; it follows that
From Definition 1, for any given , there exists a positive integer such that ; that is, . We have for all .
Since , we have for each .
Substituting (22) in (20), we obtain
Next considering the fourth term in the right side of (19), we have for all .
From the Note of Lemma 9 given previously, we have for all , where is a positive sequence with .
Substituting (23)–(26) into (19), we obtain for all , .
From condition (i), we have for all .
Thus, from (27) we know that for all and for all .
Since and is bounded in , is a real sequence in ; let Then we have for all and for all .
Conclusion (1) is proved.
Again it follows from conclusion (1) that for any , we obtain for all and for all .
Thus, conclusion (2) is proved.
Similarly from conclusion (1), it is easy to see that for all .
By conditions (i),  (ii), .
Hence, from Lemma 6 we know that exists.
This proves conclusion (3) and hence completes the proof of the lemma.

Theorem 10. Let be a uniformly convex separable Banach space and let be a nonempty closed and convex subset of . Let, for each , be asymptotically quasi-nonexpansive-type random operators satisfying the following condition: there exists constant and such that for all , and . Suppose that . Let be measurable mapping and let be sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the conditions (i)–(iii) of Lemma 9.
Then converges strongly to a common random fixed point of if and only if

Proof. The necessity is obvious. We will only prove the sufficiency. Suppose that condition (35) is satisfied. Then from Lemma 9(3), we have Now first we prove that is a Cauchy sequence. In fact, by conditions (ii), (iii) and (36), for given , there exists a positive integer such that for any (where is the positive integer appeared in Lemma 9), we have By the definition of infimum, it follows from (37) that for any given , there exists a such that On the other hand, it follows from Lemma 9(2) that for the given and for any , we have where .
Therefore, from (38)–(40), for any , we have This implies that is a Cauchy sequence. Since the space is complete, the sequence is convergent; let .
Now, we prove that is a common fixed point of .
Since and , for given , there exists a positive integer such that for , we have It follows from the second inequality in (42) that there exists such that Moreover, it follows from (21) that for any , we have
Therefore, from (42)–(44), for any , we have
This implies that (as ).
Again since for any , by condition (34) and (42)–(44), we have
This shows that (as ).
From the uniqueness of limit, we have ; that is, is a common fixed point of for all .
This completes the proof of the theorem.

If , in Theorem 10, we obtain the following result.

Corollary 11. Let be a uniformly convex separable Banach space and let be a nonempty closed and convex subset of . Let, for each , be asymptotically quasi-nonexpansive-type random operators satisfying the following condition: there exists constant and such that for all , , and . Suppose that . Let be measurable and defined by the implicit random iterative scheme with mixed errors where satisfying the following conditions:, for some ;; is bounded, such that .Then converges strongly to a common random fixed point of if and only if .

Theorem 12. Let be a uniformly convex separable Banach space and let be a nonempty closed and convex subset of with . Let be asymptotically quasi-nonexpansive-type random operator with and there exists one member in to be semicompact. Let be measurable mapping and let be the sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the following conditions:, for some ;, ;, are bounded, () such that , and .Then the sequence converges strongly to a common random fixed point of the random operators .

Proof. From Lemma 9(1), for all , .
From Lemma 6 and conditions (ii), (iii), we have that exists. From Lemmas 7 and 8 for all (where is the positive integer appeared in Lemma 9),we have where Substituting (24) into (51), using Lemma 8, we obtain where .
Substituting (23) and (26) into (53), we have And so, we obtain Therefore, from condition (i), we have Consequently, Since is strictly increasing and continuous at 0 with , it follows that Hence, from (14), we have as well as as   for all .
For convenience, let .
Since for , we have .
Notice that Thus, by , we have Therefore, for , we have Notice that . Thus, and the previous inequality becomes which yields that From it follows that Furthermore, for all , we obtain which implies that Thus, By the hypothesis that there exists in to be semicompact, we may assume that is semicompact without loss of generality. By the definition of semicompactness, there exists a subsequence of such that . By (69) again, we have It shows that . Replace by in (50); from Lemma 6, we easily know that exists so that converges strongly to a common random fixed point .
This completes the proof of the theorem.

Remark 13. Our results extend and improve the corresponding results of Plubtieng et al. [12] and Beg and Thakur [13] to the case of a more general class of asymptotically quasi-nonexpansive random operators considered in this paper. Our results also generalize the results of Saluja and Nashine [14] to the class of more general class of asymptotically quasi-nonexpansive-type random operators and two-step implicit random iteration process considered in this paper.

Acknowledgment

The authors are thankful to the anonymous referees for their critical remarks and valuable comments, which helped them to improve the presentation of this paper.

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