# On Common Random Fixed Points of a New Iteration with Errors for Nonself Asymptotically Quasi-Nonexpansive Type Random Mappings

Authors: R. A. Rashwan; P. K. Jhade; Dhekra Mohammed Al-Baqeri
Journal: International Journal of Analysis
 (2013)

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## Abstract

We prove some strong convergence of a new random iterative scheme with errors to common random fixed points for three and thenN 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.


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## Body

## 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(Ω,Σ) be a measurable space, C a nonempty subset of a separable Banach space E. A mapping ξ:Ω→C is called measurable if ξ-1(B⋂C)∈Σ for every Borel subset B of E.A mappingT:Ω×C→C is said to be random mapping if for each fixed x∈C, the mapping T(·,x):Ω→C is measurable.A measurable mappingξ:Ω→C is called a random fixed point of the random mapping T:Ω×C→C if T(w,ξ(w))=ξ(w) for each w∈Ω.Throughout this paper, we denote the set of all random fixed points of random mappingT by RF(T) and by Tn(w,x) for the nth iterate T(w,T(,…T(w,x))) of T.The class of asymptotically nonexpansive mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [11] proved that if C 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 subsetC of E is said to be a retract of E if there exists a continuous map P:E→C such that Px=x for all x∈C. Every closed convex subset of uniformly convex Banach space is a retract. A map P:E→E is a retraction if P2=P. It follows that if a map P is a retraction, then Py=y for all y in the range of P.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:(1)x1∈C,xn+1=P((1-αn)xn+αnT(PT)n-1xn),
where T:C→E is an asymptotically nonexpansive nonself mapping, {αn} is a real sequence in (0,1), and P is a nonexpansive retraction from E to C.Wang [29] generalized the result of Chidume et al. [28] and got some new results. He defined and studied the following iteration process:
(2)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1xn),x1∈C,n≥1,
where T1,T2:C→E are asymptotically nonexpansive nonself mappings and {αn},{βn} are real sequences in [0,1).Now, we introduce the following concepts for nonself mappingsDefinition 1 (see [28, 30, 31]).
LetC be a nonempty subset of a real separable Banach space and T:Ω×C→E a nonself random mapping. Then, T is said to be  (1)
nonexpansive random operator if for arbitraryx,y∈C, ∥T(w,x)-T(w,y)∥≤∥x-y∥, forallw∈Ω;(2)
nonself asymptotically nonexpansive random mapping if there exists a sequence of measurable functionsrn(w):Ω→[1,∞) with limn→∞rn(w)=1 for each w∈Ω such that for arbitrary x,y∈C,
(3)∥T(PT)n-1(w,x)-T(PT)n-1(w,y)∥≤rn(w)∥x-y∥,∀w∈Ω,n≥1;(3)
nonself asymptotically quasi-nonexpansive random mapping ifRF(T)≠ϕ and there exists a sequence of measurable functions rn(w):Ω→[1,∞) with limn→∞rn(w)=1 for each w∈Ω such that
(4)∥T(PT)n-1(w,η(w))-ξ(w)∥≤rn(w)∥η(w)-ξ(w)∥,∀w∈Ω,n≥1,
where ξ(w):Ω→C is a random fixed point of T and η(w):Ω→C is any measurable mapping;(4)
nonself asymptotically nonexpansive-type random mapping if(5)limsupn→∞{supx,y∈C{∥T(PT)n-1(w,x)-T(PT)n-1(w,y)∥2-∥x-y∥2∥T(PT)n-1(w,x)-T(PT)n-1(w,y)∥2}supx,y∈C}≤0,∀w∈Ω,n≥1;(5)
Nonself asymptotically quasi-nonexpansive-type random mapping ifRF(T)≠ϕ, and
(6)limsupn→∞{supξ(w)∈F{∥T(PT)n-1(w,η(w))-ξ(w)∥2-∥η(w)-ξ(w)∥2∥T(PT)n-1(w,η(w))-ξ(w)∥2}supξ(w)∈F}≤0,∀w∈Ω,n≥1,
where ξ(w):Ω→C is a random fixed point of T and η(w):Ω→C is any measurable mapping.Remark 2.
(1) IfT:Ω×C→E is a nonself asymptotically nonexpansive random mapping, then T is a nonself asymptotically nonexpansive-type random mapping.
(2) IfRF(T)≠ϕ and T:Ω×C→E is a nonself asymptotically quasi-nonexpansive random mapping, then T is a nonself asymptotically quasi-nonexpansive-type random mapping.
(3) IfRF(T)≠ϕ and T:Ω×C→E is a nonself asymptotically nonexpansive-type random mapping, then T is a nonself asymptotically quasi-nonexpansive-type random mapping.Remark 3.
Observe that for any measurable mappingη(w):Ω→C and ξ(w)∈F, we have
(7)limsupn→∞{supξ(w)∈F{∥T(PT)n-1(w,η(w))-ξ(w)∥2-∥η(w)-ξ(w)∥2∥T(PT)n-1(w,η(w))-ξ(w)∥2}supξ(w)∈F}≤0,
which implies
(8)limsupn→∞{supξ(w)∈F{(∥T(PT)n-1(w,η(w))-ξ(w)∥-∥η(w)-ξ(w)∥∥T(PT)n-1(w,η(w))-ξ(w)∥)×(∥T(PT)n-1(w,η(w))-ξ(w)∥+∥η(w)-ξ(w)∥∥T(PT)n-1(w,η(w))-ξ(w)∥)(∥T(PT)n-1(w,η(w))-ξ(w)∥}supξ(w)∈F}≤0.
Therefore,
(9)limsupn→∞{supξ(w)∈F{(∥T(PT)n-1(w,η(w))-ξ(w)∥-∥η(w)-ξ(w)∥∥T(PT)n-1(w,η(w))-ξ(w)∥)(∥T(PT)n-1(w,η(w))-ξ(w)∥}supξ(w)∈F}≤0.In [25], Shahzad studied the following iterative sequences:
(10)xn+1=P((1-αn)xn+αnTP[(1-βn)xn+βnTxn]),x1∈C,n≥1,
where T:C→E is a nonexpansive nonself mapping, C is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P being a nonexpansive retraction from E to C, and {αn},{βn} are real sequences in [0,1).Recently, Thianwan [32] generalized the iteration process (10) as follows:
(11)xn+1=P((1-αn-γn)xn+αnTP[(1-βn)yn+βnTyn]+γnun),yn=P((1-αn′-γn′)xn+αn′TP×[(1-βn′)xn+βn′Txn]+γn′vn),x1∈C,n≥1,
where {αn},{βn},{γn},{αn′},{βn′},{γn′} are appropriate sequences in [0,1) and {un},{vn} are bounded sequences in C. 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:
(12)xn+1=P((1-an)xn+anS(PS)n-1×((1-αn)yn+αnS(PS)n-1yn)),yn=P((1-bn)xn+bnT(PT)n-1×((1-βn)xn+βnT(PT)n-1xn)),x1∈C,n≥1,
where {an},{bn},{αn},{βn} are appropriate sequences in [0,1).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:(13)xn+1=αn1xn+βn1(PT1)nyn+γn1(PT2)nyn,yn=αn2xn+βn2(PT1)nzn+γn2(PT2)nzn,zn=αn3xn+βn3(PT1)nxn+γn3(PT2)nxn,
where {αni},{βni},{γni}(i=1,2,3) are appropriate sequences in [a,1-a] for some a∈(0,1) satisfying αni+βni+γni=1(i=1,2,3).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:
(14)ξn+1(w)=αnT1n(w,ηn(w))+βnξn(w)+γnfn(w),ηn(w)=αn′T2n(w,ζn(w))+βn′ξn(w)+γn′fn′(w),ζn(w)=αn′′T3n(w,ξn(w))+βn′′ξn(w)+γn′′fn′′(w),(w)=αn′′T3n(w,ηn(w))+βn′′ξn(w)n≥1,w∈Ω,
where T1,T2,T3:Ω×C→C are three asymptotically nonexpansive random self-mappings, ξ1:Ω→C is an arbitrary given measurable mapping from Ω to C, {fn(w)},{fn′(w)},{fn′′(w)} are bounded sequence of measurable functions from Ω to C, and {αn}, {αn′}, {αn′′}, {βn}, {βn′}, {βn′′}, {γn}, {γn′}, {γn′′} are sequences of real numbers in [0,1] with αn+βn+γn=αn′+βn′+γn′=αn′′+βn′′+γn′′=1.Remark 4.
IfT1=T2=T3=T and γn=γn′=γn′′=0, then (14) becomes as follows:
(15)ξn+1(w)=αnTn(w,ηn(w))+βnξn(w),ηn(w)=αn′Tn(w,ζn(w))+βn′ξn(w),ζn(w)=αn′′T3n(w,ξn(w))+βn′′ξn(w),n≥1,w∈Ω,
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:
(16)ξn+1(w)=P((PT)n-1(w,ηn(w))(1-αn)ξn(w)+αnT(PT)n-1(w,ηn(w))),ηn(w)=P((1-βn)ξn(w)+βnT(PT)n-1(w,ξn(w))),n≥1,w∈Ω,
where T:Ω×C→E is an asymptotically nonexpansive nonself random mapping, ξ1:Ω→C is an arbitrary given measurable mapping from Ω to C, {αn},{βn} are sequences in [0,1], and P is a nonexpansive retraction from E to C.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 N-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.
LetT1,T2,T3:Ω×C→C be three nonself random mappings, where C is a nonempty closed convex subset of a separable Banach space E, and P:E→C is a nonexpansive retraction of E onto C. Let ξ1(w):Ω→C be a measurable mapping. Suppose that {ξn(w)} is generated iteratively by ξ1(w)∈C, having
(17)ξn+1(w)=P[(1-an-σn)ξn(w)+anT1(PT1)n-1×((PT1)n-1(w,ηn(w))(w,ηn(w))w,(1-αn)ηn(w)+αnT1(PT1)n-1(w,ηn(w)))+σnfn(w)],ηn(w)=P[(1-bn-δn)ξn(w)+bnT2(PT2)n-1×((PT1)n-1(w,ηn(w))w,(1-βn)ζn(w)+βnT2(PT2)n-1(w,ζn(w)))+δngn(w)],ζn(w)=P[+γnT3(PT3)n-1(w,ξn(w)))(1-cn-λn)ξn(w)+cnT3(PT3)n-1×((w,ξn(w))(PT1)n-1(w,ηn(w))w,(1-γn)ξn(w)+γnT3(PT3)n-1(w,ξn(w)))+λnhn(w)(1-cn-λn)(1-cn-λn)],
for all n≥1, w∈Ω, where {an},{bn},{cn},{αn},{βn},{γn},{σn},{δn}, and {λn} are sequences in [0,1] such that an+σn≤1,bn+δn≤1,cn+λn≤1, and {fn(w)},{gn(w)},{hn(w)} are bounded sequences of measurable functions from Ω to C for all w∈Ω.Definition5 can be extended to N nonself random mappings as follows.Definition 6.
LetT1,T2,…,TN:Ω×C→C be N nonself random mappings, where C is a nonempty closed convex subset of a separable Banach space E, and P:E→C is a nonexpansive retraction of E onto C. Let ξ1(w):Ω→C be a measurable mapping. Define sequences function {ξn(N)(w)}, {ξn(N-1)(w)},…,{ξn(1)(w)} in C as follows:
(18)ξn+1(w)=ξn(N)(w)=P[(1-an(N)-σn(N))ξn(w)+an(N)TN(PTN)n-1×(w,(1-αn(N))ξn(N-1)(w)+αn(N)TN×(PTN)n-1(w,ξn(N-1)(w)))+σn(N)fn(N)],ξn(N-1)(w)=P[(1-an(N-1)-σn(N-1))ξn(w)+an(N-1)TN-1(PTN-1)n-1×(w,(1-αn(N-1))ξn(N-2)(w)+αn(N-1)TN-1(PTN-1)n-1(w,ξn(N-2)(w)))+σn(N-1)fn(N-1)],⋯=⋯⋯=⋯ξn(1)(w)=P[(1-an(1)-σn(1))ξn(w)+an(1)T1(PT1)n-1×(w,(1-αn(1))ξn(w)+αn(1)T1(PT1)n-1×(w,ξn(w))αn(1)T1(PT1)n-1)+σn(1)fn(1)],n≥1,w∈Ω,
where {an(i)},{αn(i)}, and {σn(i)}(i=1,2,…,N) are sequences in [0,1] such that an(i)+σn(i)≤1, for all (i=1,2,…,N), and {fn(i)}(i=1,2,…,N) are bounded sequences of measurable functions from Ω to C for all w∈Ω.The following lemma is useful for proving our results.Lemma 7 (see [39]).
Let{an},{bn} and {mn} be nonnegative real sequences satisfying
(19)an+1≤(1+mn)an+bn,n≥1.
If ∑n=1∞‍mn<∞ and ∑n=1∞‍bn<∞, then(1)
limn→∞an exists;(2)
limn→∞an=0 whenever liminfn→∞an=0.


## 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 N 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.
LetE be a real separable Banach space and C a nonempty closed convex subset of E with P being a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,3, be three asymptotically quasi-nonexpansive-type nonself random mappings with F=⋂i=13‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)},{ηn(w)} and {ζn(w)} are the sequences defined as in (17) where {an},  {bn}, {cn}, {αn}, {βn}, {γn}, {σn}, {δn}, and {λn} are sequences in [0,1] such that an+σn≤1,bn+δn≤1,cn+λn≤1 and {fn(w)}, {gn(w)}, {hn(w)} are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn<∞, ∑n=1∞‍δn<∞, and ∑n=1∞‍λn<∞. Then, {ξn(w)} converge to a common random fixed point of T1, T2, and T3 if and only if
(20)liminfn→∞d(ξn(w),F)=0,w∈Ω.Proof.
The necessity of (20) is obvious.
Next, we prove the sufficiency of (20). Let ξ(w)∈F=⋂i=13‍RF(Ti); by the boundedness of the sequences of measurable functions {fn(w)},{gn(w)},{hn(w)}, we put for each w∈Ω,
(21)M(w)=max{supn≥1,ξ∈F∥fn(w)-ξ(w)∥∨×supn≥1,ξ∈F∥gn(w)-ξ(w)∥∨×supn≥1,ξ∈F∥hn(w)-ξ(w)∥supn≥1,ξ∈F}.
Then, M(w)<∞ for each w∈Ω.
Sinceξ(w)∈F and η(w):Ω→C is any measurable mapping, we have
(22)limsupn→∞{supξ(w)∈F{∥Ti(PTi)n-1(w,η(w))-ξ(w)∥-∥η(w)-ξ(w)∥}supξ(w)∈F}≤0,i=1,2,3.
It follows that for any given ϵ>0, there exists a positive integer n0 such that for n≥n0 and ξ(w)∈F, we have
(23)supξ(w)∈F{∥Ti(PTi)n-1(w,η(w))-ξ(w)∥-∥η(w)-ξ(w)∥Ti(PTi)n-1}≤ϵ,i=1,2,3.
Since {ξn(w)},{ηn(w)}, and {ζn(w)}⊂E, then we have for w∈Ω,
(24){∥T1(PT1)n-1(w,ηn(w))-ξ(w)∥-∥ηn(w)-ξ(w)∥}≤ϵ,∀n≥n0,∀ξ∈F,(25){∥T2(PT2)n-1(w,ζn(w))-ξ(w)∥-∥ζn(w)-ξ(w)∥}≤ϵ,∀n≥n0,∀ξ∈F,(26){∥T3(PT3)n-1(w,ξn(w))-ξ(w)∥-∥ξn(w)-ξ(w)∥}≤ϵ,∀n≥n0,∀ξ∈F.
Setting for w∈Ω,
(27)μn(w)=(1-αn)ηn(w)+αnT1(PT1)n-1(w,ηn(w)),vn(w)=(1-βn)ζn(w)+βnT2(PT2)n-1(w,ζn(w)),τn(w)=(1-γn)ξn(w)+γnT3(PT3)n-1(w,ξn(w)).
Thus, for ξ(w)∈F and w∈Ω, using (17) and (24), we have
(28)∥ξn+1(w)-ξ(w)∥=∥P[(1-an-σn)ξn(w)+anT1(PT1)n-1×(w,μn(w))+σnfn(w)(PT1)n-1]-ξ(w)anT1(PT1)n-1∥≤∥(1-an-σn)ξn(w)+anT1(PT1)n-1(w,(1-αn)ηn(w)+αnT1(PT1)n-1(w,ηn(w)))+σnfn(w)-ξ(w)anT1(PT1)n-1∥=∥(1-an-σn)ξn(w)+anξ(w)+σnξ(w)-ξ(w)+an(T1(PT1)n-1(w,μn(w))-ξ(w))+σn(fn(w)-ξ(w))∥≤(1-an-σn)∥ξn(w)-ξ(w)∥+an∥T1(PT1)n-1(w,μn(w))-ξ(w)∥+σn∥fn(w)-ξ(w)∥≤(1-an-σn)∥ξn(w)-ξ(w)∥+an[∥T1(PT1)n-1(w,μn(w))-ξ(w)∥-∥μn(w)-ξ(w)∥(PT1)n-1]+an∥μn(w)-ξ(w)∥+σnM(w)≤(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an∥μn(w)-ξ(w)∥+σnM(w).
In addition, by (24), we obtain
(29)∥μn(w)-ξ(w)∥=∥(1-αn)ηn(w)+αnT1(PT1)n-1(w,ηn(w))-ξ(w)∥≤(1-αn)∥ηn(w)-ξ(w)∥+αn∥T1(PT1)n-1(w,ηn(w))-ξ(w)∥≤(1-αn)∥ηn(w)-ξ(w)∥+αnϵ+αn∥ηn(w)-ξ(w)∥=∥ηn(w)-ξ(w)∥+αnϵ.
Again using (17) and (25), we have
(30)∥ηn(w)-ξ(w)∥=∥P[(1-bn-δn)ξn(w)+bnT2(PT2)n-1(w,vn(w))+δngn(w)(PT3)n-1]-ξ(w)anT1(PT1)n-1∥≤∥(1-bn-δn)ξn(w)+bnT2(PT2)n-1×(w,(1-βn)ζn(w)+βnT2(PT2)n-1(w,ζn(w)))+δngn(w)-ξ(w)anT1(PT1)n-1∥=∥(1-bn-δn)ξn(w)+bnξ(w)+δnξ(w)-ξ(w)+bn(T2(PT2)n-1(w,vn(w))-ξ(w))+δn(gn(w)-ξ(w))∥≤(1-bn-δn)∥ξn(w)-ξ(w)∥+bn∥T2(PT2)n-1(w,vn(w))-ξ(w)∥+δn∥gn(w)-ξ(w)∥≤(1-bn-δn)∥ξn(w)-ξ(w)∥+bnϵ+bn∥vn(w)-ξ(w)∥+δnM(w).
In addition, by (25), we have
(31)∥vn(w)-ξ(w)∥=∥(1-βn)ζn(w)+βnT2(PT2)n-1(w,ζn(w))-ξ(w)∥≤(1-βn)∥ζn(w)-ξ(w)∥+βn∥T2(PT2)n-1(w,ζn(w))-ξ(w)∥≤(1-βn)∥ζn(w)-ξ(w)∥+βnϵ+βn∥ζn(w)-ξ(w)∥=∥ζn(w)-ξ(w)∥+βnϵ.
Also, by (17) and (26), we have
(32)∥ζn(w)-ξ(w)∥=∥P[(1-cn-λn)ξn(w)+cnT3(PT3)n-1×(w,τn(w))+λnhn(w)(PT3)n-1]-ξ(w)∥≤∥(1-cn-λn)ξn(w)+cnT3(PT3)n-1×(w,(1-γn)ξn(w)+γnT3(PT3)n-1(w,ξn(w)))+λnhn(w)-ξ(w)(PT3)n-1∥=∥(1-cn-λn)ξn(w)+cnξ(w)+λnξ(w)-ξ(w)+cn(T3(PT3)n-1(w,τn(w))-ξ(w))+λn(hn(w)-ξ(w))∥≤(1-cn-λn)∥ξn(w)-ξ(w)∥+cn∥T3(PT3)n-1(w,τn(w))-ξ(w)∥+λn∥hn(w)-ξ(w)∥≤(1-cn-λn)∥ξn(w)-ξ(w)∥+cnϵ+cn∥τn(w)-ξ(w)∥+λnM(w).
In addition, by (26), we have
(33)∥τn(w)-ξ(w)∥=∥(1-γn)ξn(w)+γnT3(PT3)n-1(w,ξn(w))-ξ(w)∥≤(1-γn)∥ξn(w)-ξ(w)∥+γn∥T3(PT3)n-1(w,ξn(w))-ξ(w)∥≤(1-γn)∥ξn(w)-ξ(w)∥+γnϵ+γn∥ξn(w)-ξ(w)∥=∥ξn(w)-ξ(w)∥+γnϵ.
Substituting (29), (30), (31), (32), and (33) into (28) and simplifying, we obtain
(34)∥ξn+1(w)-ξ(w)∥≤(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an∥μn(w)-ξ(w)∥+σnM(w)≤(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an[∥ηn(w)-ξ(w)∥+αnϵ(PT3)n-1]+σnM(w)=(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an∥ηn(w)-ξ(w)∥+anαnϵ+σnM(w)≤(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an[(1-bn-δn)∥ξn(w)-ξ(w)∥+bnϵ+bn∥vn(w)-ξ(w)∥+δnM(w)]+anαnϵ+σnM(w)=(1-an-σn)∥ξn(w)-ξ(w)∥+anϵ+an(1-bn-δn)∥ξn(w)-ξ(w)∥+anbnϵ+anbn∥vn(w)-ξ(w)∥+anδnM(w)+anαnϵ+σnM(w)=(1-σn-anbn-anδn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbn∥vn(w)-ξ(w)∥+anδnM(w)+anαnϵ+σnM(w)≤(1-σn-anbn-anδn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbn[∥ζn(w)-ξ(w)∥+βnϵ(PT3)n-1]+anδnM(w)+anαnϵ+σnM(w)=(1-σn-anbn-anδn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbn∥ζn(w)-ξ(w)∥+anbnβnϵ+anδnM(w)+anαnϵ+σnM(w)≤(1-σn-anbn-anδn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbn(1-cn-λn)∥ξn(w)-ξ(w)∥+anbncnϵ+anbncn∥τn(w)-ξ(w)∥+anbnλnM(w)+anbnβnϵ+anδnM(w)+anαnϵ+σnM(w)=(1-σn-anδn-anbncn-anbnλn)×∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbncnϵ+anbncn∥τn(w)-ξ(w)∥+anbnλnM(w)+anbnβnϵ+anδnM(w)+anαnϵ+σnM(w)≤(1-σn-anδn-anbncn-anbnλn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbncnϵ+anbncn×∥ξn(w)-ξ(w)∥+anbncnγnϵ+anbnλnM(w)+anbnβnϵ+anδnM(w)+anαnϵ+σnM(w)=(1-σn-anδn-anbnλn)∥ξn(w)-ξ(w)∥+anϵ+anbnϵ+anbncnϵ+anbncnγnϵ+anbnλnM(w)+anbnβnϵ+anδnM(w)+anαnϵ+σnM(w)=(1-σn-anδn-anbnλn)∥ξn(w)-ξ(w)∥+[an+anbn+anbncn+anbncnγn+anbnβn+anαn]ϵ+[anbnλn+anδn+σn]M(w)≤∥ξn(w)-ξ(w)∥+6ϵ+(λn+δn+σn)M(w).
Let Rn(w)=6ϵ+(λn+δn+σn)M(w); then, ∑n=1∞‍Rn(w)<∞ for all w∈Ω.
It follows by (34) that
(35)infξ(w)∈F∥ξn+1(w)-ξ(w)∥≤infξ(w)∈F∥ξn(w)-ξ(w)∥+Rn(w),∀n≥n0,w∈Ω.
From (35) and ∑n=1∞‍Rn(w)<∞ for all w∈Ω, we have
(36)d(ξn+1(w),F)≤d(ξn(w),F)+Rn(w),∀w∈Ω.
By Lemma 7 and (36), it follows that limn→∞d(ξn(w),F) exists for all ξ(w)∈F=⋂i=13‍RF(Ti) and w∈Ω.
Sinceliminfn→∞d(ξn(w),F)=0, then we have
(37)limn→∞d(ξn(w),F)=0,w∈Ω.
Next, we prove that ξn(w) is a Cauchy sequence in E for each w∈Ω.
Forn≥n0,m≥n1, and ξ(w)∈F, we have by (35) that
(38)∥ξn+m(w)-ξ(w)∥≤∥ξn+m-1(w)-ξ(w)∥+Rn+m-1(w)≤∥ξn+m-2(w)-ξ(w)∥+Rn+m-1(w)+Rn+m-2(w)⋮≤∥ξn(w)-ξ(w)∥+∑k=nn+m-1‍Rk(w).
Therefore, by using (38), we have for each w∈Ω,
(39)∥ξn+m(w)-ξn(w)∥≤∥ξn+m(w)-ξ(w)∥+∥ξn(w)-ξ(w)∥≤2∥ξn(w)-ξ(w)∥+∑k=n∞‍Rk(w).
Since ξ(w)∈F and by (39), we have for each w∈Ω,
(40)∥ξn+m(w)-ξn(w)∥≤2d(ξn(w),F)+∑k=n∞‍Rk(w),∀n≥n0.
Since limn→∞d(ξn(w),F)=0 and ∑n=1∞‍Rn(w)<∞, for given ϵ>0, there exists a positive integer n1≥n0 such that d(ξn(w),F)<ϵ/4 and ∑n=1∞‍Rn(w)<ϵ/2. We have
(41)∥ξn+m(w)-ξn(w)∥<ϵ,∀w∈Ω,
or
(42)limn→∞∥ξn+m(w)-ξn(w)∥=0,∀w∈Ω;
this shows that ξn(w) is a Cauchy sequence in C for each w∈Ω.
SinceE is complete and C is a closed subset of E and so it is complete, then there exists a p(w)∈C such that ξn(w)→p(w) as n→∞, for all w∈Ω.
Now, we show thatp(w)∈F.
By contradiction, we assume thatp(w) does not belong to F. Since F is closed set, d(p(w),F)>0. By using the fact that limn→∞d(ξn(w),F)=0, it follows that for all ξ(w)∈F,
(43)∥p(w)-ξ(w)∥≤∥p(w)-ξn(w)∥+∥ξn(w)-ξ(w)∥.
This implies that
(44)d(p(w),F)≤∥p(w)-ξn(w)∥+d(ξn(w),F)cc≤0(asn→∞),
which is a contradiction. Hence, p(w)∈F.Corollary 9.
Suppose that the conditions in Theorem8 are satisfied. Then the random iterative sequence ξn(w) generated by (17) converges to a common random fixed point ξ(w) if and only if for all w∈Ω, there exists a subsequence ξnj(w) of ξn(w) which converges to ξ(w).Theorem 10.
LetE be a real separable Banach space and C a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,3, be three asymptotically quasi-nonexpansive nonself random mappings with F=⋂i=13‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)},{ηn(w)}, and {ζn(w)} are the sequences defined as in (17) where {an},{bn},{cn},{αn},{βn},{γn},{σn},{δn}, and {λn} are sequences in [0,1] such that an+σn≤1,bn+δn≤1,cn+λn≤1, and {fn(w)}, {gn(w)}, {hn(w)} are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn<∞,∑n=1∞‍δn<∞ and ∑n=1∞‍λn<∞. Then, {ξn(w)} converge to a common random fixed point of T1, T2, and T3 if and only if
(45)liminfn→∞d(ξn(w),F)=0,w∈Ω.Proof.
SinceTi:Ω×C→E,i=1,2,3, are three asymptotically quasi-nonexpansive nonself random mappings, by Remark 2, they are asymptotically quasi-nonexpansive-type nonself random mappings the conclusion of Theorem 10 can be proved from Theorem 8 immediately.Theorem 11.
LetE be a real separable Banach space and C be a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,3, be three asymptotically nonexpansive nonself random mappings with F=⋂i=13‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)},{ηn(w)} and {ζn(w)} are the sequences defined as in (17) where {an},{bn},{cn},{αn},{βn},{γn},{σn},{δn}, and {λn} are sequences in [0,1] such that an+σn≤1,bn+δn≤1,cn+λn≤1, and {fn(w)}, {gn(w)}, {hn(w)} are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn<∞,∑n=1∞‍δn<∞, and ∑n=1∞‍λn<∞. Then, {ξn(w)} converge to a common random fixed point of T1, T2, and T3 if and only if
(46)liminfn→∞d(ξn(w),F)=0,w∈Ω.Proof.
SinceTi:Ω×C→E,i=1,2,3, are three asymptotically nonexpansive nonself random mappings, by Remark 2, they are asymptotically nonexpansive-type nonself random mappings, and therefore they are asymptotically quasi-nonexpansive-type nonself random mappings; the conclusion of Theorem 11 can be obtained from Theorem 8 immediately.Now, we can extend and generalize Theorems8, 10, and 11 by using random iterative scheme (18) as follows.Theorem 12.
LetE be a real separable Banach space and C a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,…,N, be N asymptotically quasi-nonexpansive-type nonself random mappings with F=⋂i=1N‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)} is the sequence defined as in (18) where {an(i)},{αn(i)}, and {σn(i)}(i=1,2,…,N) are sequences in [0,1] such that an(i)+σn(i)≤1 for all i=1,2,…,N and {fn(i)(w)}(i=1,2,…,N) are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn(i)<∞, for all (i=1,2,…,N). Then {ξn(w)} converge to a common random fixed point of T1,T2,…,TN if and only if
(47)liminfn→∞d(ξn(w),F)=0,w∈Ω.Theorem 13.
LetE be a real separable Banach space and C be a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,…,N be N asymptotically quasi-nonexpansive nonself random mappings with F=⋂i=1N‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)} be the sequence defined as in (18) where {an(i)},{αn(i)}, and {σn(i)},(i=1,2,…,N) are sequences in [0,1] such that an(i)+σn(i)≤1 for all i=1,2,…,N and {fn(i)(w)},(i=1,2,…,N) are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn(i)<∞ for all (i=1,2,…,N). Then {ξn(w)} converge to a common random fixed point of T1,T2,…,TN if and only if
(48)liminfn→∞d(ξn(w),F)=0,w∈Ω.Theorem 14.
LetE be a real separable Banach space and C be a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,…,N be N asymptotically nonexpansive nonself random mappings with F=⋂i=1N‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)} is the sequence defined as in (18) where {an(i)}, {αn(i)}, and {σn(i)}(i=1,2,…,N) are sequences in [0,1] such that an(i)+σn(i)≤1 for all i=1,2,…,N and {fn(i)(w)}(i=1,2,…,N) are bounded sequences of measurable functions from Ω to C with the following restrictions ∑n=1∞‍σn(i)<∞ for all (i=1,2,…,N). Then, {ξn(w)} converge to a common random fixed point of T1,T2,…,TN if and only if
(49)liminfn→∞d(ξn(w),F)=0,w∈Ω.Senter and Dotson [40] defined Condition (A) as follows.Definition 15 (see [40]).
A mappingT:C→C where C is a subset of a Banach space E with F(T)≠ϕ is said to satisfy Condition (A) if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0,f(r)>0, for all r∈(0,∞) such that for all x∈C,
(50)∥x-Tx∥≥f(d(x,F(T))),
where d(x,F(T))=inf{∥x-p∥:p∈F(T)}.As an application, we can apply Theorem8 and Condition (A) to obtain a convergences theorem for scheme (17).Theorem 16.
LetE be a real uniformly separable Banach space and C a nonempty closed convex subset of E with P as a nonexpansive retraction. Let Ti:Ω×C→E,i=1,2,3, be three asymptotically quasi-nonexpansive-type nonself random mappings with F=⋂i=13‍RF(Ti)≠ϕ, for all w∈Ω. Suppose that {ξn(w)},{ηn(w)} and {ζn(w)} are the sequences defined as in (17) where {an}, {bn}, {cn}, {αn}, {βn}, {γn}, {σn}, {δn}, and {λn} are sequences in [0,1] such that an+σn≤1,bn+δn≤1,cn+λn≤1 and {fn(w)}, {gn(w)}, {hn(w)} are bounded sequences of measurable functions from Ω to C with the following restrictions: ∑n=1∞‍σn<∞,∑n=1∞‍δn<∞, and ∑n=1∞‍λn<∞. Suppose one of the mappings Ti,i=1,2,3, satisfying Condition (A) and the following condition: limn→∞∥ξn(w)-T(w,ξn(w))∥=0,  for all w∈Ω. Then, {ξn(w)} converge to a common random fixed point of  T1,T2, and T3.Proof.
From Theorem8, we have limn→∞∥ξn(w)-ξ(w)∥, and limn→∞d(ξn(w),F) exists. Let one of the mappings Ti, say T1 satisfy Condition (A) and limn→∞∥ξn(w)-T1(w,ξn(w))∥=0; then, we have for all w∈Ω,
(51)limn→∞f(d(ξn(w),F))≤limn→∞∥ξn(w)-T1(w,ξn(w))∥=0.
By the property of f and since limn→∞d(ξn(w),F) exists, we have that
(52)limn→∞d(ξn(w),F)=0.
By Theorem 8, {ξn(w)} converge to a common random fixed point of T1, T2, and T3.



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*Source: 312685-2013-05-08.xml*