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

Authors: A. S. Saluja; Pankaj kumar Jhade
Journal: Journal of Mathematics
 (2013)

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## 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).


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## Body

## 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(Ω,Σ) be a measurable space with Σ, a sigma algebra of Ω, and let C be a nonempty subset of a Banach space X. A mapping ξ:Ω→X is measurable if ξ-1(U)∈Σ for each open subset U of  X. The  mapping T:Ω×C→C is a random map if for each fixed  x∈C, the mapping  T(·,x):Ω→C is measurable, and it is continuous if for each ω∈Ω, the mapping T(ω,·):C→X is continuous. A measurable mapping  ξ:Ω→X is the random fixed point of the random map T:Ω×C→C if T(ω,ξ(ω))=ξ(ω), for each ω∈Ω. We denote by RF(T) and D(T) the set of all random fixed points of a random map and the domain of the random map T:Ω×C→C, respectively, and by Tn(ω,x) the nth iterate T(ω,T(ω,T(ω,…,T(ω,x))) of T. The letter I denotes the random mapping I:Ω×C→C, defined by I(ω,x)=x and T0=I.Definition 1.
LetC be a nonempty subset of a separable Banach space X and let T:D(T)⊂Ω×C→C be a random map. The map T is said to be(a)
a nonexpansive random operator ifforarbitraryx,y∈D(T), one has
(1)∥T(ω,x)-T(ω,y)∥≤∥x-y∥for eachω∈Ω and x,y∈D(T);(b)
a quasi-nonexpansive random operator ifF(T)≠∅ and
(2)∥T(ω,x)-ξ(ω)∥≤∥x-ξ(ω)∥for allx∈D(T) and ξ(ω)∈F;(c)
an asymptotically nonexpansive random operator if there exists a sequence of measurable mappingsrn:Ω→[0,∞) with limn→∞rn(ω)=0 for each ω∈Ω, such that for arbitrary x,y∈D(T),
(3)∥Tn(ω,x)-Tn(ω,y)∥≤(1+rn(ω))∥x-y∥;(d)
an asymptotically quasi-nonexpansive random operator ifF(T)≠ϕ and there exists a sequence of measurable mappings rn:Ω→[0,∞) with limn→∞rn(ω)=0 for each ω∈Ω such that for arbitrary x∈D(T)andξ(ω)∈F(T),
(4)∥Tn(ω,x)-ξ(ω)∥≤(1+rn(ω))∥x-ξ(ω)∥;(e)
an asymptotically nonexpansive type if(5)limn→∞sup{supx∈D(T)(∥Tn(ω,x)-Tn(ω,y)∥-∥x-y∥)}≤0for ally∈D(T) and n≥0;(f)
an asymptotically quasi-nonexpansive-type random operator ifF(T)≠ϕ and
(6)limn→∞sup{supx∈D(T)(∥Tn(ω,x)-ξ(ω)∥-∥x-ξ(ω)∥)}≤0for allξ(ω)∈F(T) and  n≥0;(g)
a uniformlyL-Lipschitzian random operator if for arbitrary x,y∈D(T), one has
(7)∥Tn(ω,x)-Tn(ω,y)∥≤L∥x-y∥,wheren≥0  and L is a positive constant.Remark 2.
We know that the following implications hold from the definitions:(8)Example 3.
LetC be the real line with the usual norm |·| and Ω=[0,1]. Define T:Ω×C→C by T(ω,x)=sinx,x∈[0,1] for x∈C and ω∈Ω. Obviously T(ω,0)=0; that is, 0 is a fixed point of T; that is, RF(T)={0}. Now we check that T is asymptotically quasi-nonexpansive-type random operator. In fact, if x∈[0,1] and ξ(ω)=0∈[0,1], then
(9)|T(ω,x)-ξ(ω)|=|T(ω,x)-0|=|sinx|≤|x|=|x-0|=|x-ξ(ω)|;
that is,
(10)|T(ω,x)-ξ(ω)|≤|x-ξ(ω)|.
Thus, T is quasi-nonexpansive. It follows that T is asymptotically quasi-nonexpansive with limn→∞rn=0 for each n≥1, and hence it is asymptotically quasi-nonexpansive-type random operator.(h)
The mapT is a semi compact random operator if for a sequence of measurable mappings {ξn} from Ω to C, with limn→∞∥ξn(ω)-T(ω,ξn(ω))∥=0 for every ω∈Ω, one has a subsequence {ξnk} of {ξn} and a measurable mapping ξ:Ω→C such that {ξnk} converges pointwise to ξ as k→∞.(i)
A Banach spaceC is said to be uniformly convex if the modulus of convexity of C (see [15])
(11)δC(ε)=inf{1-12∥x+y∥:∥x∥=∥y∥=1,∥x-y∥=ε}>0
for all0<ε≤2 (i.e., δC(ε) is a function (0,1]→(0,1]).Definition 44 (modified general composite iteration process, cf. [14]).
Let{Ti}i=1N be a family of asymptotically quasi-nonexpansive in the intermediate sense random operators from Ω×C to C, where C is a closed, convex subset of a separable Banach space X with F=⋂i=1NRF(Ti)≠ϕ. Let ξ0:Ω→C be any fixed measurable map, then the sequence of function {ξn} defined by
(12)ξn(ω)=αnξn-1(ω)+(1-αn)Ti(n)k(n)(ω,ηn(ω)),∀n≥1,ηn(ω)=anξn(ω)+bnξn-1(ω)+cnTi(n)k(n)(ω,ξn(ω))+dnTi(n)k(n)(ω,ξn-1(ω)),∀n≥1,
where {αn},{an},{bn},{cn},{dn}⊂[0,1] with an+bn+cn+dn=1, n=(k-1)N+i, where i=i(n)∈{1,2,…,N},k=k(n)≥1 is a positive integer, and k(n)→∞ as n→∞.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 spaceC as follows.Definition 55 (two-step implicit random iterative scheme).
LetTi:Ω×C→C be random operators (i=1,2,…,N), where C is a nonempty convex subset of a separable Banach space X. Let ξ0:Ω→C be a measurable mapping from Ω to C and let {fn} and {fn′} be bounded sequences of measurable functions from Ω to C. Define sequences of functions {ξn} and {ηn} as given below:
(13)ξ1(ω)=(1-α1)ξ0(ω)+α1T1(ω,η1(ω))+f1,η1(ω)=(1-β1)ξ1(ω)+β1T1(ω,ξ1(ω))+f1′,⋮ξN(ω)=(1-αN)ξN-1(ω)+αNTN(ω,ηN(ω))+fN,ηN(ω)=(1-βN)ξN(ω)+βNTN(ω,ξN(ω))+fN′,ξN+1(ω)=(1-αN+1)ξN(ω)+αN+1TN+12(ω,ηN+1(ω))+fN+1,ηN+1(ω)=(1-βN+1)ξN+1(ω)+βN+1TN+12(ω,ξN+1(ω))+fN+1′
which can be written in the following compact form:
(14)ξn(ω)=(1-αn)ξn-1(ω)+αnTik(ω,ηn(ω))+fn,n≥1,ηn(ω)=(1-βn)ξn(ω)+βnTik(ω,ξn(ω))+fn′,n≥1,
where n=(k-1)N+i(i∈I).As a matter of fact, denote the indexing set{1,2,3,…,N} by I. Let {Ti:i∈I} be N uniformly L-Lipschitzian asymptotically quasi-nonexpansive-type random operators from Ω×C→C. We show that the scheme (14) exists.Letξ0:Ω→C and ξ1(ω)=(1-α1)ξ0(ω)α1T1(ω,(1-β1)ξ1(ω)+β1T1(ω,ξ1(ω))+f1)+f1′.DefineW(ω,x):D(T)⊂Ω×C→C, for all x∈D(T),ω∈Ω by
(15)W(ω,x)=(1-α1)ξ0(ω)α1T1×(ω,(1-β1)x+β1T1(ω,x)+f1)+f1′
for all x∈D(T),ω∈Ω. The existence of ξ1(ω) is guaranteed if W has a fixed point.For anyx,y∈D(T) and ω∈Ω, we have
(16)∥W(ω,x)-W(ω,y)∥≤α1β1L2∥x-y∥.NowW is a contraction if α1β1L2<1 or L=(1/α1β1). As α1,β1∈[0,1], therefore W is a contraction. By the Banach contraction principle, W has a unique fixed point. Thus, the existence of ξ1(ω) is established. Similarly, we can establish the existence of ξ2(ω),ξ3(ω),…. Thus, the implicit algorithm (14) is well defined.In the sequel, we will need the following lemmas.Lemma 6 (see [16]).
Let{an} and {bn} be two nonnegative sequences satisfying
(17)an+1≤an+bn∀n≥n0,
where ∑n=1∞bn<∞ and n0 is some positive integer. Then  limn→∞an exists.Lemma 7 (see [17]).
Letp>1 and r>0 be two fixed real numbers. Then a Banach space C is uniformly convex if and only if there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(18)∥λx+(1-λ)y∥p≤λ∥x∥p+(1-λ)∥y∥p-wp(λ)g(∥x-y∥)
for all x,y∈Br={x∈C:∥x∥≤r} and λ∈[0,1], where wp(λ)=λp(1-λ)+λ(1-λ)p.Lemma 8 (see [18]).
Letp>1 be a given real number. Then for any a≥0 and b≥0, there exists a nonnegative real number c between a and a+b such that (a+b)p=ap+pcp-1b.


## 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.
LetX be a uniformly convex separable Banach space and let C be a nonempty closed and convex subset of X. Let, for each i∈I,Ti:D(T)⊂Ω×C→C be asymptotically quasi-nonexpansive-type random operators. Suppose that ℱ=⋂i=1NRF(Ti)≠ϕ. Let ξ0:Ω→C be measurable mapping and let {ξn(ω)} be a sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the following conditions:(i)
s<αn<1-s for some s∈(0,1/2);(ii)
∑n=0∞αn<∞;(iii)
{fn}and{fn′} are bounded, fn=fn′′+fn′′′  (n≥0) and ∑n=0∞∥fn′′∥<∞,∥fn′′′∥=o(αn).
Note. It is pointed out that in condition (iii), ∥fn′′′∥=o(αn) for all n≥0 is equivalent to that there exists a sequence {εn} with εn≥0 and εn→0 as n→∞ such that ∥fn′′′∥=εnαn for all n≥0.
Then for any givenε>0, there exists M(ω)>0  for each ω∈Ω, such that (1)
∥ξn(ω)-ξ(ω)∥≤∥ξn-1(ω)-ξ(ω)∥+(M(ω)/s)αn+(1/s)∥fn′′∥ for all ξ(ω)∈ℱ,n≥0;(2)
∥ξn+m(ω)-ξ(ω)∥≤∥ξn(ω)-ξ(ω)∥+(M(ω)/s)∑l=n+1n+mαl+(1/s)∑l=n+1n+m∥fl′′∥ for all n≥0m≥1, where M(ω)=(2/5)ε+supn≥0{∥fn′∥+εn}<∞ and {εn} is the positive sequence appearing in the note of Lemma 9;(3)
limn→∞D(ξn(ω),ℱ) exists.Proof.
Letξ(ω)∈ℱ, from (14), where n=(k-1)N+i,Tn≡Tn(modN)=Ti(i∈I); it follows that
(19)∥ξn(ω)-ξ(ω)∥=∥(Tik(ω,ηn(ω))-ξ(ω))(1-αn)(ξn-1(ω)-ξ(ω))+αn(Tik(ω,ηn(ω))-ξ(ω))+fn∥≤(1-αn)∥ξn-1(ω)-ξ(ω)∥+αn(∥Tik(ω,ηn(ω))-ξ(ω)∥+αn-∥ηn(ω)-ξ(ω)∥∥Tik(ω,ηn(ω))-ξ(ω)∥)+αn∥ηn(ω)-ξ(ω)∥+∥fn∥,(20)∥ηn(ω)-ξ(ω)∥=∥(Tik(ω,ξn(ω))-ξ(ω))(1-βn)(ξn(ω)-ξ(ω))+βn(Tik(ω,ξn(ω))-ξ(ω))+fn′′∥≤(1-βn)∥ξn(ω)-ξ(ω)∥+βn(∥Tik(ω,ξn(ω))-ξ(ω)∥+βn-∥ξn(ω)-ξ(ω)∥∥Tik(ω,ξn(ω))-ξ(ω)∥)+βn∥ξn(ω)-ξ(ω)∥+∥fn′′∥=∥ξn(ω)-ξ(ω)∥+βn(∥Tik(ω,ξn(ω))-ξ(ω)∥-∥ξn(ω)-ξ(ω)∥∥Tik(ω,ξn(ω))-ξ(ω)∥v)+∥fn′′∥.
From Definition1, for any given ε>0, there exists a positive integer n0 such that n≥n0; that is, k>(n0/N)+1. We have
(21)supx∈D(T){∥Tin(ω,x)-ξ(ω)∥-∥x-ξ(ω)∥}≤ε6
for all ξ(ω)∈ℱ.
Since{ξn},{ηn}⊂X, we have
(22)∥Tin(ω,ξn(ω))-ξ(ω)∥-∥ξn(ω)-ξ(ω)∥<ε5∀ξ(ω)∈ℱ,n≥n0,(23)∥Tin(ω,ηn(ω))-ξ(ω)∥-∥ξn(ω)-ξ(ω)∥<ε5∀ξ(ω)∈ℱ,n≥n0
for each i∈I.
Substituting (22) in (20), we obtain
(24)∥ηn(ω)-ξ(ω)∥≤∥ξn(ω)-ξ(ω)∥+ε5βn+∥fn′∥∀ξ(ω)∈ℱ,n≥n0.
Next  considering the fourth term in the right side of (19), we have
(25)∥fn∥≤∥fn′′∥+∥fn′′′∥
for all  n≥0.
From the Note of Lemma9 given previously, we have
(26)∥fn∥≤∥fn′′∥+εnαn
for all n≥n0, where {εn} is a positive sequence with limn→∞εn=0.
Substituting (23)–(26) into (19), we obtain
(27)∥ξn(ω)-ξ(ω)∥≤(1-αn)∥ξn-1(ω)-ξ(ω)∥+αnε5+αn{∥ξn(ω)-ξ(ω)∥+βnε5+∥fn′∥}+∥fn′′∥+εnαn
for all n≥0, ξ(ω)∈ℱ.
From condition (i), we have(28)1<1-αns,11-αn<1s
for all n≥0.
Thus, from (27) we know that
(29)(1-αn)∥ξn(ω)-ξ(ω)∥≤(1-αn)∥ξn-1(ω)-ξ(ω)∥+αn1-αns{ε5(1+βn)+∥fn′∥+εn}+∥fn′′∥,∥ξn(ω)-ξ(ω)∥≤∥ξn-1(ω)-ξ(ω)∥+αns{ε5(1+βn)+∥fn′∥+εn}+1s∥fn′′∥
for all n≥0 and for all ξ(ω)∈ℱ.
Sincelimn→∞εn=0 and {fn′} is bounded in C, {βn} is a real sequence in [0,1]; let
(30)M(ω)=25ε+supn≥0{∥fn′∥+εn}<∞.
Then we have
(31)∥ξn(ω)-ξ(ω)∥≤∥ξn-1(ω)-ξ(ω)∥+M(ω)sαn+1s∥fn′′∥
for all n≥0 and for all ξ(ω)∈ℱ.
Conclusion (1) is proved.
Again it follows from conclusion (1) that for anym≥1, we obtain
(32)∥ξn+m(ω)-ξ(ω)∥≤∥ξn+m-1(ω)-ξ(ω)∥+M(ω)sαn+m+1s∥fn+m′′∥≤∥ξn+m-2(ω)-ξ(ω)∥+M(ω)s(αn+m-1+αn+m)+1s(∥fn+m-1′′∥+∥fn+m′′∥)≤∥ξn(ω)-ξ(ω)∥+M(ω)s∑l=n+1n+mαl+1s∑l=n+1n+m∥fl′′∥
for all n≥0 and for all ξ(ω)∈ℱ.
Thus, conclusion (2) is proved.
Similarly from conclusion (1), it is easy to see that(33)D(ξn(ω),ℱ)≤D(ξn-1(ω),ℱ)+M(ω)s+αn+1s∥fn′′∥
for all n≥0.
By conditions (i),  (ii),∑n=0∞((M(ω)/s)αn+(1/s)∥fn′′∥)<∞.
Hence, from Lemma6 we know that (limn→∞D(ξn(ω),ℱ) exists.
This proves conclusion (3) and hence completes the proof of the lemma.Theorem 10.
LetX be a uniformly convex separable Banach space and let C be a nonempty closed and convex subset of X. Let, for each i∈I, Ti:D(T)⊂Ω×C→C be asymptotically quasi-nonexpansive-type random operators satisfying the following condition: there exists constant Li>0 and ai>0  such that
(34)∥Ti(ω,x)-ξ(ω)∥≤Li∥∥x-ξ(ω)∥∥ai
for all x∈D(T),ξ(ω)∈ℱ, and i∈I. Suppose that ℱ=⋂i=1NRF(Ti)≠ϕ. Let  ξ0:Ω→C be measurable mapping and let  {ξn(ω)} be sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the conditions (i)–(iii) of Lemma 9.
Then{ξn}  converges strongly to a common random fixed point of ℱ if and only if
(35)limn→∞infD(ξn(ω),ℱ)=0.Proof.
The necessity is obvious. We will only prove the sufficiency. Suppose that condition (35) is satisfied. Then from Lemma 9(3), we have
(36)limn→∞D(ξn(ω),ℱ)=0.
Now first we prove that {ξn} is a Cauchy sequence. In fact, by conditions (ii), (iii) and (36), for given ε>0, there exists a positive integer n1 such that for any n≥n1′=max{n1,n0} (where n0 is the positive integer appeared in Lemma 9), we have
(37)D(ξn(ω),ℱ)<ε7,(38)∑i=1N∑k=[(n1′+1)/N]∞αki<sε3M(ω),∑i=1N∑k=[(n1′+1)/N]∞∥fki′′∥<sε3.
By the definition of infimum, it follows from (37) that for any given n≥n1′, there exists a ξ*(ω)∈ℱ  such that
(39)∥ξ(ω)-ξ*(ω)∥<ε6.
On the other hand, it follows from Lemma 9(2) that for the given ε>0 and for any n≥n1′, we have
(40)∥ξn+m(ω)-ξn(ω)∥≤∥ξn+m(ω)-ξ*(ω)∥+∥ξn(ω)-ξ*(ω)∥≤∥ξn(ω)-ξ*(ω)∥+M(ω)s∑l=n+1n+mαl+1s∑l=n+1n+m∥fl′′∥+∥ξn(ω)-ξ*(ω)∥<2∥ξn(ω)-ξ*(ω)∥+M(ω)s∑i=1N∑k=[(n+1)/N]∞αki+1s∑i=1N∑k=[(n+1)/N]∞∥fki′′∥,
where  M(ω)=(2/5)ε+supn≥0{∥fn′∥+εn}<∞.
Therefore, from (38)–(40), for any n≥n1′, we have
(41)∥ξn+m(ω)-ξn(ω)∥<ε3+ε3+ε3=ε∀m≥1.
This implies that {ξn} is a Cauchy sequence. Since the space is complete, the sequence {ξn} is convergent; let limn→∞ξn(ω)=ξ′(ω).
Now, we prove thatξ′(ω) is a common fixed point of Ti.
Sincelimn→∞ξn(ω)=ξ′(ω) and  limn→∞D(ξn(ω),ℱ)=0, for given ε>0, there exists a positive integer n2 such that for n≥n2=max{n2,n1′}, we have
(42)∥ξn(ω)-ξ′(ω)∥<ε5,D(ξn(ω),ℱ)<ε7.
It follows from the second inequality in (42) that there exists ξ′′(ω)∈ℱ  such that
(43)∥ξn2′(ω)-ξ′′(ω)∥<ε5.
Moreover, it follows from (21) that for any n≥n2′, we have
(44)∥Tin(ω,ξ′(ω))-ξ′′(ω)∥-∥ξ′(ω)-ξ′′(ω)∥<ε5.
Therefore, from (42)–(44), for any n≥n2′, we have
(45)∥Tin(ω,ξ′(ω))-ξ′(ω)∥≤∥Tin(ω,ξ′(ω))-ξ′′(ω)∥-∥ξ′(ω)-ξ′′(ω)∥-∥ξ′(ω)-ξ′′(ω)∥+∥ξ′(ω)-ξ′′(ω)∥≤(∥Tin(ω,ξ′(ω))-ξ′′(ω)∥-∥ξ′(ω)-ξ′′(ω)∥)+2(∥ξ′(ω)-ξn2′(ω)∥+∥ξn2′(ω)-ξ′′(ω)∥)<ε5+2(ε5+ε5)=ε.
This implies thatTin(ω,ξ′(ω))→ξ′(ω) (as n→∞).
Again since(46)∥Tin(ω,ξ′(ω))-Ti(ω,ξ′(ω))∥≤(∥Tin(ω,ξ′(ω))-ξ′′(ω)∥-∥ξ′(ω)-ξ′′(ω)∥)+∥ξ′(ω)-ξ′′(ω)∥+∥Ti(ω,ξ′(ω))-ξ′′(ω)∥
for any n≥n2′, by condition (34) and (42)–(44), we have
(47)∥Tin(ω,ξ′(ω))-Ti(ω,ξ′(ω))∥≤ε5+∥ξ′(ω)-ξ′′(ω)∥+Li∥ξ′(ω)-ξ′′(ω)∥ai≤ε5+∥ξ′(ω)-ξn2′(ω)∥+∥ξn2′(ω)-ξ′′(ω)∥+Li(∥ξ′(ω)-ξn2′(ω)∥+∥ξn2′(ω)-ξ′′(ω)∥)ai<ε+Li·εai∀i∈I.
This shows thatTin(ω,ξ′(ω))→Ti(ω,ξ′(ω)) (as n→∞).
From the uniqueness of limit, we haveξ′(ω)=Ti(ω,ξ′(ω)); that is, ξ′(ω) is a common fixed point of Ti for all i∈I.
This completes the proof of the theorem.Ifβn=0,vn=0, in Theorem 10, we obtain the following result.Corollary 11.
LetX be a uniformly convex separable Banach space and let C be a nonempty closed and convex subset of X. Let, for each i∈I, Ti:D(T)⊂Ω×C→C be asymptotically quasi-nonexpansive-type random operators satisfying the following condition: there exists constant Li>0 and ai>0 such that
(48)∥Ti(ω,x)-ξ(ω)∥≤Li∥x-ξ(ω)∥ai
for all x∈D(T), ξ(ω)∈ℱ,  and i∈I. Suppose that ℱ=⋂i=1NRF(Ti)≠ϕ. Let ξ0:Ω→C be measurable and {ξn(ω)} defined by the implicit random iterative scheme with mixed errors
(49)ξn(ω)=αnξn-1(ω)+(1-αn)Tin(ξn(ω),ω)+fn,n≥1,
where n=(k-1)N+i(i∈I) satisfying the following conditions:(i)
s<αn<1-s, for some s∈(0,1/2);(ii)
∑n=0∞αn<∞;(iii)
{fn} is bounded, fn=fn′′+fn′′′(n≥0) such that ∑n=0∞∥fn′′∥<∞∥fn′′′∥=o(αn).Then {ξn(ω)} converges strongly to a common random fixed point of ℱ if and only if limn→∞infD(ξn(ω),ℱ)=0.Theorem 12.
LetX be a uniformly convex separable Banach space and let C be a nonempty closed and convex subset of X with C+C⊂C. Let Ti:D(T)⊂Ω×C→C(i∈I={1,2,…,N}) be asymptotically quasi-nonexpansive-type random operator with ℱ=⋂i=1NRF(Ti)≠ϕ and there exists one member Tj in {Ti∣i∈I} to be semicompact. Let ξ0:Ω→C be measurable mapping and let {ξn} be the sequence defined by the implicit random iterative scheme (14) with mixed errors satisfying the following conditions:(i)
s<αn<1-s, for some  s∈(0,1/2);(ii)
∑n=0∞αn<∞, limsupn→∞Lβn<1;(iii)
{fn}, {fn′} are bounded, fn=fn′′+fn′′′ (n≥0) such that ∑n=0∞∥fn′′∥<∞, ∥fn′′′∥=o(αn) and ∑n=0∞∥fn′∥<∞.Then the sequence {ξn} converges strongly to a common random fixed point of the random operators {Ti∣i∈I}.Proof.
From Lemma9(1),
(50)∥ξn(ω)-ξ(ω)∥≤∥ξn-1(ω)-ξ(ω)∥+M(ω)sαn+12∥fn′′∥
for all n≥0, ξ(ω)∈ℱ.
From Lemma6 and conditions (ii), (iii), we have that  limn→∞∥ξn(ω)-ξ(ω)∥ exists. From Lemmas 7 and 8 for all n>n0 (where n0 is the positive integer appeared in Lemma 9),we have
(51)∥ξn(ω)-ξ(ω)∥2=∥αn(Tik(ω,ηn(ω))+fn-ξ(ω))+(1-αn)(ξn-1(ω)+fn-ξ(ω))(Tik(ω,ηn(ω))+fn-ξ(ω))∥2≤αn∥Tik(ω,ηn(ω))+fn-ξ(ω)∥2+(1-αn)∥ξn-1(ω)+fn-ξ(ω)∥2-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)≤αn(∥Tik(ω,ηn(ω))-ξ(ω)∥+∥fn∥)2+(1-αn)(∥ξn-1(ω)-ξ(ω)∥+∥fn∥)2-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)=αn{∥Tik(ω,ηn(ω))-ξ(ω)∥2+2c1∥fn∥}+(1-αn){∥ξn-1(ω)-ξ(ω)∥2+2c2∥fn∥}-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)=αn(∥Tik(ω,ηn(ω))-ξ(ω)∥2-∥ηn(ω)-ξ(ω)∥2)+αn∥ηn(ω)-ξ(ω)∥2+(1-αn)∥ξn-1(ω)-ξ(ω)∥2+2(c1αn+c2(1-αn))∥fn∥-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥),
where
(52)c1∈(∥Tik(ω,ηn(ω))-ξ(ω)∥,∥Tik(ω,ηn(ω))-ξ(ω)∥+∥fn∥),c2∈(∥ξn-1(ω)-ξ(ω)∥,∥ξn-1(ω)-ξ(ω)∥+∥fn∥).
Substituting (24) into (51), using Lemma 8, we obtain
(53)∥ξn(ω)-ξ(ω)∥2≤αn(∥Tik(ω,ηn(ω))-ξ(ω)∥2-∥ηn(ω)-ξ(ω)∥2)+αn(∥ξn(ω)-ξ(ω)∥+ε5βn+∥fn′∥)2+(1-αn)∥ξn-1(ω)-ξ(ω)∥2+2(c1αn+c2(1-αn))∥fn∥-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)=αn(∥Tik(ω,ηn(ω))-ξ(ω)∥2-∥ηn(ω)-ξ(ω)∥2)+αn(∥ξn(ω)-ξ(ω)∥+2c3(ε5βn+∥fn′∥))+(1-αn)∥ξn-1(ω)-ξ(ω)∥2+2(c1αn+c2(1-αn))∥fn∥-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥),
where c3∈(∥ξn(ω)-ξ(ω)∥,∥ξn(ω)-ξ(ω)∥+(ε/5)βn+∥fn′∥).
Substituting (23) and (26) into (53), we have
(54)(1-αn)∥ξn(ω)-ξ(ω)∥2≤αnε5(∥Tik(ω,ηn(ω))-ξ(ω)∥+∥ηn(ω)-ξ(ω)∥)+2αn(c3(ε5βn+∥fn′∥)+(c1αn+c2(1-αn))εn)+(1-αn)∥ξn-1(ω)-ξ(ω)∥2+2(c1αn+c2(1-αn))∥fn′′∥-αn(1-αn)g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥).
And so, we obtain
(55)∥ξn(ω)-ξ(ω)∥2≤αn1-αnε5(∥Tik(ω,ηn(ω))-ξ(ω)∥+∥ηn(ω)-ξ(ω)∥)+2αn1-αn(c3(ε5βn+∥fn′∥)+(c1αn+c2(1-αn))εn)+∥ξn-1(ω)-ξ(ω)∥2+21-αn(c1αn+c2(1-αn))∥fn′′∥-αng(∥Tik(ω,ηn(ω))-ξn-1(ω)∥).
Therefore, from condition (i), we have
(56)sg(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)<αng(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)≤∥ξn-1(ω)-ξn(ω)∥2-∥ξn(ω)-ξ(ω)∥2+αnε5s(∥Tik(ω,ηn(ω))-ξ(ω)∥-∥ηn(ω)-ξ(ω)∥)+αn2s(c3(ε5βn+∥fn′∥)+(c1αn+c2(1-αn))εn)+2s(c1αn+c2(1-αn))∥fn′′∥.
Consequently,
(57)limn→∞g(∥Tik(ω,ηn(ω))-ξn-1(ω)∥)=0.
Since g is strictly increasing and continuous at 0 with g(0)=0, it follows that
(58)limn→∞∥Tik(ω,ηn(ω))-ξn-1(ω)∥=0.
Hence, from (14), we have
(59)∥ξn(ω)-ξn-1(ω)∥=αn∥ξn-1(ω)-Tik(ω,ηn(ω))∥+∥fn∥≤αn(∥ξn-1(ω)-Tik(ω,ηn(ω))∥+εn)+∥fn′′∥⟶0(asn⟶∞)
as well as ∥ξn(ω)-ξn+l(ω)∥→0(as  n→∞) for all l<2N.
For convenience, letσn=∥Tik(ω,ξn(ω))-ξn-1(ω)∥.
SinceTn=Ti for n=(k-1)N+i, we have  σn=∥Tnk(ω,ξn(ω))-ξn-1(ω)∥.
Notice that(60)∥Tnk(ω,ξn(ω))-ξn-1(ω)∥≤∥Tnk(ω,ξn(ω))-Tnk(ω,ηn(ω))∥+∥Tnk(ω,ηn(ω))-ξn-1(ω)∥≤L∥ξn(ω)-ηn(ω)∥+∥Tnk(ω,ηn(ω))-ξn-1(ω)∥≤L∥βnξn(ω)-βnTnk(ω,ξn(ω))-fn′∥+∥Tnk(ω,ηn(ω))-ξn-1(ω)∥≤Lβn(∥ξn-1(ω)-Tnk(ω,ξn(ω))∥∥ξn(ω)-ξn-1(ω)∥Lβn+∥ξn-1(ω)-Tnk(ω,ξn(ω))∥)+L∥fn′∥+∥Tnk(ω,ηn(ω))-ξn-1(ω)∥∀n>n0.
Thus, by limsupn→∞Lβn<1, we have
(61)limn→∞σn=limn→∞∥Tnk(ω,ξn(ω))-ξn-1(ω)∥=0.
Therefore, for n>N, we have
(62)∥ξn-1(ω)-Tn(ω,ξn(ω))∥≤∥ξn-1(ω)-Tnk(ω,ξn(ω))∥+∥Tnk(ω,ξn(ω))-Tn(ω,ξn(ω))∥≤σn+L∥Tnk-1(ω,ξn(ω))-ξn(ω)∥≤σn+L{∥Tnk-1(ω,ξn(ω))-Tn-Nk-1(ω,ξn-N(ω))∥σn+L+∥Tn-Nk-1(ω,ξn-N(ω))-ξ(n-N)-1(ω)∥σn+L+∥ξ(n-N)-1(ω)-ξn(ω)∥∥Tnk-1(ω,ξn(ω))-Tn-Nk-1(ω,ξn-N(ω))∥}.
Notice that n≡(n-N)(modN). Thus, Tn=Tn-N and the previous inequality becomes
(63)∥ξn-1(ω)-Tn(ω,ξn(ω))∥≤σ2+L2∥ξn(ω)-ξn-N(ω)∥+Lσn-N+L∥ξn(ω)-ξ(n-N)-1(ω)∥
which yields that
(64)limn→∞∥ξn-1(ω)-Tn(ω,ξn(ω))∥=0.
From
(65)∥ξn(ω)-Tn(ω,ξn(ω))∥≤∥ξn(ω)-ξn-1(ω)∥+∥ξn-1(ω)-Tn(ω,ξn(ω))∥,
it follows that
(66)limn→∞∥ξn(ω)-Tn(ω,ξn(ω))∥=0.
Furthermore, for all l∈I, we obtain
(67)∥ξn(ω)-Tn+l(ω,ξn(ω))∥≤∥ξn(ω)-ξn+l(ω)∥+∥ξn+l(ω)-Tn+l(ω,ξn+l(ω))∥+∥Tn+l(ω,ξn+l(ω))-Tn+l(ω,ξn(ω))∥≤(1+L)∥ξn(ω)-ξn+l(ω)∥+∥ξn+l(ω)-Tn+l(ω,ξn+l(ω))∥
which implies that
(68)limn→∞∥ξn(ω)-Tn+l(ω,ξn(ω))∥=0∀l∈I.
Thus,
(69)limn→∞∥ξn(ω)-Tn(ω,ξl(ω))∥=0∀l∈I.
By the hypothesis that there exists Tj in {Ti:i∈I} to be semicompact, we may assume that Tl is semicompact without loss of generality. By the definition of semicompactness, there exists a subsequence {ξnj} of {ξn} such that ξnj→ξ*∈C. By (69) again, we have
(70)∥ξ*(ω)-Tl(ω,ξ*(ω))∥=limj→∞∥ξnj(ω)-Tl(ω,ξnj(ω))∥=0(∀1≤l≤N).
It shows that ξ*(ω)∈ℱ. Replace ξ(ω) by ξ*(ω) in (50); from Lemma 6, we easily know that limn→∞∥ξn(ω)-ξ*(ω)∥ exists so that  {ξn} converges strongly to a common random fixed point ξ*(ω)∈ℱ.
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.



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