# Strong Convergence of Parallel Iterative Algorithm with Mean Errors for Two Finite Families of Ćirić Quasi-Contractive Operators

Authors: Feng Gu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to establish a strong convergence of a new parallel iterative algorithm with mean errors to a common fixed point for two finite families of Ćirić quasi-contractive operators in normed spaces. The results presented in this paper generalize and improve the corresponding results of Berinde, Gu, Rafiq, Rhoades, and Zamfirescu.


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## Body

## 1.   Introduction and Preliminaries

Let(X,d) be a metric space. A mapping T:X→X is said to be a-contraction, if d(Tx,Ty)≤ad(x,y)forallx,y∈X, where a∈(0,1).The mappingT:X→X is said to be Kannan mapping [1], if there exists b∈(0,1/2) such that d(Tx,Ty)≤b[d(x,Tx)+d(y,Ty)]for allx,y∈X.A mappingT:X→X is said to be Chatterjea mapping [2], if there exists c∈(0,1/2) such that d(Tx,Ty)≤c[d(x,Ty)+d(y,Tx)]forallx,y∈X.Combining these three definitions, Zamfirescu [3] proved the following important result.Theorem Z (see [3]).
Let(X,d) be a complete metric space and T:X→X a mapping for which there exist the real numbers a,b, and c satisfying a∈(0,1), b,c∈(0,1/2) such that for each pair x,y∈X, at least one of the following conditions holds:(z1)d(Tx,Ty)≤ad(x,y),(z2)d(Tx,Ty)≤b[d(x,Tx)+d(y,Ty)],(z3)d(Tx,Ty)≤c[d(x,Ty)+d(y,Tx)].
ThenT has a unique fixed point p and the Picard iteration {xn} defined by
(1.1)xn+1=Txn,n∈ℕ
converges to p for any arbitrary but fixed x1∈X.Remark 1.1.
An operatorT satisfying the contractive conditions (z1)–(z3) in the above theorem is called Z-operator.Remark 1.2.
The conditions(z1)–(z3) can be written in the following equivalent form:
(1.2)d(Tx,Ty)≤hmax{d(x,y),d(x,Tx)+d(y,Ty)2,d(x,Ty)+d(y,Tx)2},
for all x,y∈X,0<h<1. Thus, a class of mappings satisfying the contractive conditions (z1)–(z3) is a subclass of mappings satisfying the following condition:
(CG)d(Tx,Ty)≤hmax{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty)+d(y,Tx)2},0<h<1. The class of mappings satisfying (CG) is introduced and investigated by  C´iric´  [4] in 1971.Remark 1.3.
A mapping satisfying(CG)  is commonly called C´iric´ generalized contraction.
In 2000, Berinde [5] introduced a new class of operators on a normed space E satisfying
(1.3)∥Tx-Ty∥≤ρ∥x-y∥+L∥Tx-x∥,
for any x,y∈E, 0≤δ<1 and L≥0.Note that (1.3) is equivalent to
(1.4)∥Tx-Ty∥≤ρ∥x-y∥+Lmin{∥Tx-x∥,∥Ty-y∥},
for any x,y∈E, 0≤ρ<1 and L≥0.Berinde [5] proved that this class is wider than the class of Zamfiresu operators and used the Mann [6] iteration process to approximate fixed points of this class of operators in a normed space given in the form of following theorem.Theorem B (see [5]).
LetC be a nonempty closed convex subset of a normed space E. Let T:C→C be an operator satisfying (1.3) and F(T)≠∅. For given x0∈C, let {xn} be generated by the algorithm
(1.5)xn+1=(1-αn)xn+αnTxn,n≥0,
where {αn} be a real sequence in [0, 1]. If ∑n=1∞‍αn=∞, then {xn}n=0∞ converges strongly to the unique fixed point of T.In 2006, Rafiq [7] considered a class of mappings satisfying the following condition:
(CR)∥Tx-Ty∥≤hmax{∥x-y∥,∥x-Tx∥+∥y-Ty∥2,∥x-Ty∥,∥y-Tx∥},0<h<1. This class of mappings is a subclass of mappings satisfying the following condition:
(CQ)∥Tx-Ty∥≤hmax{∥x-y∥,∥x-Tx∥,∥y-Ty∥,∥x-Ty∥,∥y-Tx∥},0<h<1. The class of mappings satisfying (CQ) was introduced and investigated by C´iric´ [8] in 1974 and a mapping satisfying is commonly called C´iric´ quasi-contraction.Rafiq [7] proved the following result.Theorem R (see [7]).
LetC be a nonempty closed convex subset of a normed space E. Let T:C→C be an operator satisfying the condition (CR). For given x0∈C, let {xn} be generated by the algorithm
(1.6)xn+1=αnxn+βnTxn+γnun,n≥0,
where {αn},{βn}, and {γn} be three real sequences in [0, 1] satisfying αn+βn+γn=1 for all n≥1, {un} is a bounded sequences in C. If ∑n=1∞‍βn=∞ and γn=o(αn), then {xn}n=0∞ converges strongly to the unique fixed point of T.In 2007, Gu [9] proved the following theorem.Theorem G (see [9]).
LetC be a nonempty closed convex subset of a normed space E. Let {Ti}i=1N:C→C be N operators satisfying the condition (CR) with F=∩i=1NF(Ti)≠∅ (the set of common fixed points of {Ti}i=1N). Let {αn}, {βn}, and {γn} be three real sequences in [0, 1] satisfying αn+βn+γn=1 for all n≥1, {un} a bounded sequences in C satisfying the following conditions:(i)
∑n=1∞‍βn=∞;(ii)
∑n=1∞‍γn<∞ or γn=o(βn). Suppose further that x0∈C is any given point and {xn} is generated by the algorithm
(1.7)xn+1=αnxn+βnTnxn+γnun,n≥0,
where Tn=Tn(modN). Then {xn} converges strongly to a common fixed point of {Ti}i=1N.Remark 1.4.
It should be pointed out that Theorem G extends Theorem R from aC´iric´ quasi-contractive operator to a finite family of  C´iric´  quasi-contractive operators.Inspired and motivated by the facts said above, we introduced a new two-step parallel iterative algorithm with mean errors for two finite family of operators{Si}i=1m and {Tj}j=1k as follows:
(1.8)xn+1=(1-αn-γn)xn+αn∑i=1m‍λiSiyn+γnun,n≥1,yn=(1-βn-δn)xn+βn∑j=1k‍μiTjxn+δnvn,n≥1,
where {λi}i=1m,{μj}j=1k are two finite sequences of positive number such that ∑i=1m‍λi=1 and ∑j=1k‍μj=1, {αn}, {βn}, {γn} and {δn} are four real sequences in [0, 1] satisfying αn+γn≤1 and βn+δn≤1 for all n≥1, {un} and {vn} are two bounded sequences in C and x0 is a given point.Especially, if{αn}, {γn} are two sequences in [0, 1] satisfying αn+γn≤1 for all n≥1, {λi}i=1m⊂[0,1] satisfying λ1+λ2+⋯+λm=1, {un} is a bounded sequence in C and x0 is a given point in C, then the sequence {xn} defined by
(1.9)xn+1=(1-αn-γn)xn+αn∑i=1m‍λiSixn+γnun,n≥1
is called the one-step parallel iterative algorithm with mean errors for a finite family of operators {Si}i=1m.The purpose of this paper is to study the convergence of two-steps parallel iterative algorithm with mean errors defined by (1.8) to a common fixed point for two finite family of C´iric´ quasi-contractive operators in normed spaces. The results presented in this paper generalized and extend the corresponding results of Berinde [5], Gu [9], Rafiq [7], Rhoades [10], and Zamfirescu [3]. Even in the case of βn=δn=0 or γn=δn=0  for all n≥1 or m=k=1 are also new.In order to prove the main results of this paper, we need the following Lemma.Lemma 1.5 (see [11]).
Suppose that{an}, {bn}, and {cn} are three nonnegative real sequences satisfying the following condition:
(1.10)an+1≤(1-tn)an+bn+cn,∀n≥n0,
where n0 is some nonnegative integer, tn∈[0,1], ∑n=0∞‍tn=∞, bn=o(tn) and ∑n=0∞‍cn<∞. Then limn→∞an=0.


## 2. Main Results

We are now in a position to prove our main results in this paper.Theorem 2.1.
LetC be a nonempty closed convex subset of a normed space E. Let {Si}i=1m:C→C be m operators satisfying the condition (CR) and {Tj}j=1k:C→C be k operators satisfying the condition (CR) with F=(⋂i=1m‍F(Si))∩(⋂j=1k‍F(Tj))≠∅, where F(Si) and F(Tj) are the set of fixed points of Si and Tj in C, respectively. Let {αn}, {βn}, {γn}, and {δn} be four real sequences in [0, 1] satisfying αn+γn≤1 and βn+δn≤1 for all n≥1, {λi}i=1m,{μj}j=1k two finite sequences of positive number such that ∑i=1m‍λi=1 and ∑j=1k‍μj=1, {un} and {vn} two bounded sequences in C satisfying the following conditions:(i)
∑n=1∞‍αn=∞;(ii)
limn→∞δn=0;(iii)
∑n=1∞‍γn<∞ or γn=o(αn). Suppose further that x0∈C is any given point and {xn} is an iteration sequence with mane errors defined by (1.8), then {xn} converges strongly to a common fixed point of {Si}i=1m and {Tj}j=1k.Proof.
Since{Si}i=1m:C→C is mC´iric´ operator satisfying the condition (CR), hence there exists 0<hi<1 (i∈I={1,2,…,m}) such that
(2.1)∥Six-Siy∥≤himax{∥x-y∥,∥x-Six∥+∥y-Siy∥2,∥x-Siy∥,∥y-Six∥}.
For each fixed i∈I={1,2,…,m}. Denote h=max{h1,h2,…,hm}, then 0<h<1 and
(2.2)∥Six-Siy∥≤hmax{∥x-y∥,∥x-Six∥+∥y-Siy∥2,∥x-Siy∥,∥y-Six∥}
hold for each fixed i∈I={1,2,…,m}. If from (2.2) we have
(2.3)∥Six-Siy∥≤h2[∥x-Six∥+∥y-Siy∥],
then
(2.4)∥Six-Siy∥≤h2[∥x-Six∥+∥y-Siy∥]≤h2[∥x-Six∥+∥y-x∥+∥x-Six∥+∥Six-Siy∥].
Hence
(2.5)(1-h2)∥Six-Siy∥≤h2∥x-y∥+h∥x-Six∥,
which yields (using the fact that 0<h<1)
(2.6)∥Six-Siy∥≤h/21-h/2∥x-y∥+h1-h/2∥x-Six∥.
Also, from (2.2), if
(2.7)∥Six-Siy∥≤hmax{∥x-Siy∥,∥y-Six∥}
holds, then
(a)∥Six-Siy∥≤h∥x-Siy∥, which implies ∥Six-Siy∥≤h∥x-Six∥+h∥Six-Siy∥ and hence, as h<1,
(2.8)∥Six-Siy∥≤h1-h∥x-Six∥,
or
(b)∥Six-Siy∥≤h∥y-Six∥, which implies
(2.9)∥Six-Siy∥≤h∥y-x∥+h∥x-Six∥.
Thus, if (2.7) holds, then from (2.8) and (2.9) we have
(2.10)∥Six-Siy∥≤h∥y-x∥+h1-h∥x-Six∥.
Denote
(2.11)ρ1=max{h,h/21-h/2}=h,L1=max{h,h1-h/2,h1-h}=h1-h.
Then we have 0<ρ1<1 and L1≥0. Combining (2.2),(2.6), and (2.10) we get
(2.12)∥Six-Siy∥≤ρ1∥x-y∥+L1∥x-Six∥
holds for all x,y∈C and i∈I.
On the other hand, since{Tj}j=1k:C→C is kC´iric´ operator satisfying the condition (CR), similarly, we can prove
(2.13)∥Tjx-Tjy∥≤ρ2∥x-y∥+L2∥x-Six∥,
for all x,y∈C and j∈J={1,2,…,k}, where 0<ρ2<1 and L2≥0.
Letp∈F=(⋂i=1m‍F(Si))∩(⋂j=1k‍F(Tj)); using (1.8) we have
(2.14)∥xn+1-p∥=∥(1-αn-γn)(xn-p)+αn∑i=1mλi(Siyn-p)+γn(un-p)∥≤(1-αn-γn)∥xn-p∥+αn∑i=1mλi∥Siyn-p∥+γn∥un-p∥≤(1-αn)∥xn-p∥+αn∑i=1mλi∥Siyn-p∥+γnM,
where M=supn≥1{||un-p||,||vn-p||}. Now for y=yn and x=p, (2.12) gives
(2.15)∥Siyn-p∥=∥Siyn-Sip∥≤ρ1∥yn-p∥.
Substituting (2.15) into (2.14), we obtain that
(2.16)∥xn+1-p∥≤(1-αn)∥xn-p∥+αnρ1∥yn-p∥+γnM.
Again it follows from (1.8) that
(2.17)∥yn-p∥=∥(1-βn-δn)(xn-p)+βn∑j=1kμj(Tjxn-p)+δn(vn-p)∥≤(1-βn-δn)∥xn-p∥+βn∑j=1kμj∥Tjxn-p∥+δn∥vn-p∥≤(1-βn)∥xn-p∥+βn∑j=1kμj∥Tjxn-p∥+δnM.
Now for y=xn and x=p, (2.13) gives
(2.18)∥Tjxn-p∥=∥Tjxn-Tjp∥≤ρ2∥xn-p∥.
Combining (2.17) and (2.18) we get
(2.19)∥yn-p∥≤[1-βn(1-ρ2)]∥xn-p∥+δnM≤∥xn-p∥+δnM.
Substituting (2.19) into (2.16), we obtain that
(2.20)∥xn+1-p∥≤(1-αn)∥xn-p∥+αnρ1(∥xn-p∥+δnM)+γnM=[1-αn(1-ρ1)]∥xn-p∥+αnδnρ1M+γnM=(1-tn)∥xn-p∥+bn+cn,
where
(2.21)tn=αn(1-ρ1),bn=αnδnρ1M,cn=γnM
or
(2.22)tn=αn(1-ρ1),bn=αnδnρ1M+γnM,cn=0.
From the conditions (i)–(iii) it is easy to see that tn∈[0,1], ∑n=1∞‍tn=∞, bn=o(tn), and ∑n=1∞‍cn<∞. Thus using (2.20) and Lemma 1.5 we have limn→∞||xn-p||=0, and so limn→∞xn=p. This completes the proof of Theorem 2.1.Theorem 2.2.
LetC be a nonempty closed convex subset of a normed space E. Let {Si}i=1m:C→C be m operators satisfying the condition (2.12) and let {Tj}j=1k:C→C be k operators satisfying the condition (2.13) with F=(⋂i=1m‍F(Si))∩(⋂j=1k‍F(Tj))≠∅, where F(Si) and F(Tj) are the set of fixed points of Si and Tj in C, respectively. Let {αn}, {βn}, {γn}, and {δn} be four real sequences in [0, 1] satisfying αn+γn≤1 and βn+δn≤1 for all n≥1, {λi}i=1m,{μj}j=1k two finite sequences of positive number such that ∑i=1m‍λi=1, and ∑j=1k‍μj=1, {un} and {vn} two bounded sequences in C satisfying the following conditions:(i)
∑n=1∞‍αn=∞;(ii)
limn→∞δn=0;(iii)
∑n=1∞‍γn<∞ or  γn=o(αn). Suppose further that x0∈C is any given point and {xn} is an iteration sequence defined by (1.8), then {xn} converges strongly to a common fixed point of {Si}i=1m and {Tj}j=1k.Theorem 2.3.
LetC be a nonempty closed convex subset of a normed space E. Let {Si}i=1m:C→C be m operators satisfying the condition (CR) with F=⋂i=1m‍F(Si)≠∅ (the set of common fixed points of {Si}i=1m). Let {αn} and {γn} be two real sequences in [0, 1] satisfying αn+γn≤1 for all n≥1, {λi}i=1m  a finite sequence of positive number such that ∑i=1m‍λi=1, and {un} a bounded sequence in C satisfying the following conditions:(i)
∑n=1∞‍αn=∞;(ii)
∑n=1∞‍γn<∞ or γn=o(αn). Suppose further that x0∈C is any given point and {xn} is an iteration sequence with mane errors defined by (1.9), then {xn} converges strongly to a common fixed point of {Si}i=1m.Theorem 2.4.
LetC be a nonempty closed convex subset of a normed space E. Let {Si}i=1m:C→C be m operators satisfying the condition (2.12) with F=⋂i=1m‍F(Si)≠∅ (the set of common fixed points of {Si}i=1m). Let {αn} and {γn} be two real sequences in [0, 1] satisfying αn+γn≤1 for all n≥1, {λi}i=1m a finite sequence of positive number such that ∑i=1m‍λi=1, and {un} a bounded sequence in C satisfying the following conditions:(i)
∑n=1∞‍αn=∞;(ii)
∑n=1∞‍γn<∞ or γn=o(αn). Suppose further that x0∈C is any given point and {xn} is an iteration sequence defined by (1.9), then {xn} converges strongly to a common fixed point of {Si}i=1m.Corollary 2.5 (see [7]).
LetC be a nonempty closed convex subset of a normed space E. Let T:C→C be an operators satisfying the condition (CR). Let {αn}, {βn}, and {γn} be three real sequences in [0, 1] satisfying αn+βn+γn=1 for all n≥1 and  {un}  a bounded sequences in C satisfying the following conditions:(i)
∑n=1∞‍βn=∞;(ii)
∑n=1∞‍γn<∞ or γn=o(βn). Suppose further that x0∈C is any given point and {xn} is an explicit iteration sequence as follows:
(2.23)xn+1=αnxn+βnTxn+γnun,n≥1,
then {xn} converges strongly to the unique fixed point of T.Proof.
ByC´iric´ [8], we know that T has a unique fixed point in C. Taking m=1 in Theorem 2.3, then the conclusion of Corollary 2.5 can be obtained from Theorem 2.3 immediately. This completes the proof of Corollary 2.5.Remark 2.6.
Theorems2.2–2.4 and Corollary 2.5 improve and extend the corresponding results of Berinde [5], Gu [9], Rafiq [7], Rhoades [10], and Zamfirescu [3].



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