# Faster Multistep Iterations for the Approximation of Fixed Points Applied to Zamfirescu Operators

Authors: Shin Min Kang; Ljubomir B. Ćirić; Arif Rafiq; Faisal Ali; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By taking a counterexample, we prove that the
multistep iteration process is faster than the Mann and Ishikawa iteration
processes for Zamfirescu operators.


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## Body

## 1. Introduction

LetC be a nonempty convex subset of a normed space E, and let T:C→C be a mapping.(a) For arbitraryx0∈C, the sequence {xn} defined by
(Mn)xn+1=(1-bn)xn+bnTxn,n≥0,
where {bn} is a sequence in [0,1], is known as the Mann iteration process [1].(b) For arbitraryx0∈C, the sequence {xn} defined by
(In)xn+1=(1-bn)xn+bnTyn,yn=(1-bn′)xn+bn′Txn,n≥0,
where {bn} and {bn′} are sequences in [0,1], is known as the Ishikawa iteration process [2].(c) For arbitraryx0∈C, the sequence {xn} defined by
(RSn)xn+1=(1-bn)xn+bnTyn1,yni=(1-bni)xn+bniTyni+1,ynp-1=(1-bnp-1)xn+bnp-1Tnxn,n≥0,
where {bn} and {bni},  i=1,2,…,p-2  (p≥2), are sequences in [0,1] and denoted by (RSn), is known as the multistep iteration process [3].Definition 1 (see [4]).
Suppose that{an} and {bn} are two real convergent sequences with limits a and b, respectively. Then, {an} is said to converge faster than {bn} if
(1)limn→∞|an-abn-b|=0.Definition 2 (see [4]).
Let{un} and {vn} be two fixed-point iteration procedures which, both, converge to the same fixed point p, say, with error estimates,
(2)∥un-p∥≤an,∥vn-p∥≤bn,n≥0,
where liman=0=limbn. If {an} converges faster than {bn}, then {un} is said to converge faster than {vn}.Theorem 3 (see [5]).
Let(X,d) be a complete metric space, and let T:X→X be a mapping for which there exist real numbers a, b, and c satisfying a∈(0,1) and b,c∈(0,1/2) such that, for each pair x,y∈X, at least one of the following is true:(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)].
Then,T has a unique fixed point p, and the Picard iteration {xn} defined by
(3)xn+1=Txn,n≥0,
converges to p for any x0∈X.Remark 4.
An operatorT, which satisfies the contraction conditions (z1)–(z3) of Theorem 3, will be called a Zamfirescu operator [4, 6, 7].
In [6, 7], Berinde introduced a new class of operators on a normed space E satisfying
(4)∥Tx-Ty∥≤δ∥x-y∥+L∥Tx-x∥,
for any x,y∈E,  0≤δ<1, and L≥0.
He proved that this class is wider than the class of Zamfirescu operators.
The following results are proved in [6, 7].Theorem 5 (see [7]).
LetC be a nonempty closed convex subset of a normed space E. Let T:C→C be an operator satisfying (4). Let {xn} be defined through the iterative process (Mn) and x0∈C. If F(T)≠∅ and ∑n=0∞‍bn=∞, then {xn} converges strongly to the unique fixed point of T.Theorem 6 (see [6]).
LetC be a nonempty closed convex subset of an arbitrary Banach space E, and let T:C→C be an operator satisfying (4). Let {xn} be defined through the iterative process (In) and x0∈C, where {bn} and {bn′} are sequences of positive numbers in [0,1] with {bn} satisfying ∑n=0∞‍bn=∞. Then, {xn} converges strongly to the fixed point of T.The following result can be found in [8].Theorem 7.
LetC be a closed convex subset of an arbitrary Banach space E. Let the Mann and Ishikawa iteration processes with real sequences {bn} and {bn′} satisfy 0≤bn, bn′≤1, and ∑n=0∞‍bn=∞. Then, (Mn) and (In) converge strongly to the unique fixed point of T. Let   T:C→C be a Zamfirescu operator, and, moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of T.In [4], Berinde proved the following result.Theorem 8.
LetC be a closed convex subset of an arbitrary Banach space E, and let T:C→C be a Zamfirescu operator. Let {yn} be defined by (Mn) and y0∈C with a sequence {bn} in [0,1] satisfying ∑n=0∞‍bn=∞. Then, {yn} converges strongly to the fixed point of T, and, moreover, the Picard iteration {xn} converges faster than the Mann iteration.Remark 9.
In [9], Qing and Rhoades by taking a counterexample showed that the Mann iteration process converges more slowly than the Ishikawa iteration process for Zamfirescu operators.In this paper, we establish a general theorem to approximate fixed points of quasi-contractive operators in a Banach space through the multistep iteration process. Our result generalizes and improves upon, among others, the corresponding results of Babu and Vara Prasad [8] and Berinde [4, 6, 7].We also prove that the Mann iteration process and the Ishikawa iteration process converge more slowly than the multistep iteration process for Zamfirescu operators.


## 2. Main Results

We now prove our main results.Theorem 10.
LetC be a nonempty closed convex subset of an arbitrary Banach space E, and let T:C→C be an operator satisfying (4). Let {xn} be defined through the iterative process (RSn) and x0∈C, where {bn} and {bni},  i=1,2,…,p-2  (p≥2), are sequences in [0,1] with ∑n=0∞‍bn=∞. If F(T)≠∅, then F(T) is a singleton, and the sequence {xn} converges strongly to the fixed point of T.Proof.
Assume thatF(T)≠∅ and w∈F(T). Then, using (RSn), we have
(5)∥xn+1-w∥=∥(1-bn)xn+bnTyn1-w∥=∥(1-bn)(xn-w)+bn(Tyn1-w)∥≤(1-bn)∥xn-w∥+bn∥Tyn1-w∥.
Now, forx=w and y=yn1, (4) gives
(6)∥Tyn1-w∥≤δ∥yn1-w∥.
By substituting (6) in (5), we obtain
(7)∥xn+1-w∥≤(1-bn)∥xn-w∥+δbn∥yn1-w∥.
In a similar fashion, again by using(RSn), we can get
(8)∥yni-w∥≤(1-bni)∥xn-w∥+δbni∥yni+1-w∥,
where i=1,2,…,p-2  (p≥2) and
(9)∥ynp-1-w∥≤(1-(1-δ)bnp-1)∥xn-w∥.
It can be easily seen that, for i=1,2,…,p-2  (p≥2), we have
(10)∥yn1-w∥≤(1-bn1)∥xn-w∥+δbn1∥yn2-w∥,⋮∥ynp-3-w∥≤(1-bnp-3)∥xn-w∥+δbnp-3∥ynp-2-w∥,∥ynp-2-w∥≤(1-bnp-2)∥xn-w∥+δbnp-2∥ynp-1-w∥.
Substituting (9) in (10) gives us
(11)∥ynp-2-w∥≤(1-(1-δ)bnp-2(1+δbnp-1))∥xn-w∥.
It may be noted that, for δ∈[0,1) and {ηn}∈[0,1], the following inequality is always true:
(12)1≤1+δηn≤1+δ.
From (11) and (12), we get
(13)∥ynp-2-w∥≤(1-(1-δ)bnp-2)∥xn-w∥.
By repeating the same procedure, finally from (7) and (10), we yield
(14)∥xn+1-w∥≤[1-(1-δ)bn]∥xn-w∥.
By (14), we inductively obtain
(15)∥xn+1-w∥≤∏k=0n[1-(1-δ)bk]∥x0-w∥,n≥0.
Using the fact that 0≤δ<1,  0≤bn≤1, and ∑n=0∞‍bn=∞, it results that
(16)limn→∞∏k=0n[1-(1-δ)bk]=0,
which, by (15), implies that
(17)limn→∞∥xn+1-w∥=0.
Consequently, xn→w∈F, and this completes the proof.Now, by a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.Example 11.
Suppose thatT:[0,1]→[0,1] is defined by Tx=(1/2)x;  bn=0=bnp-1=bni, i=1,2,…,p-1  (p≥2), and n=1,2,…,15;  bn=4/n=bnp-1=bni,  i=1,2,…,p-1  (p≥2), and n≥16. It is clear that T is a Zamfirescu operator with a unique fixed point 0 and that all of the conditions of Theorem 10 are satisfied. Also, Mn=x0=In=RSn,  n=1,2,…,15. Suppose that x0≠0. For the Mann and Ishikawa iteration processes, we have
(18)Mn=(1-bn)xn+bnTxn=(1-4n)xn+4n12xn=(1-2n)xn=⋯=∏i=16n‍(1-2i)x0,In=(1-bn)xn+bnT((1-bn′)xn+bn′Txn)=(1-4n)xn+4n12(1-2n)xn=(1-2n-4n)xn=⋯=∏i=16n‍(1-2i-4i)x0,RSn=(1-bn)xn+bnTyn1,yni=(1-bni)xn+bniTyni+1,ynp-1=(1-bnp-1)xn+bnp-1Txn,n≥0,
where i=1,2,…,p-2  (p≥2) implies that
(19)RSn=(1-∑j=1p‍(2n)j)xn=⋯=∏i=16n‍(1-∑j=1p‍(2i)j)x0.
Now, consider
(20)|RSn-0Mn-0|=|∏i=16n(1-∑j=1p(2/i)j)x0∏i=16n(1-(2/i))x0|=|∏i=16n(1-∑j=1p(2/i)j)∏i=16n(1-(2/i))|=|∏i=16n‍(1-∑k=2p(2/i)k1-(2/i))|=|∏i=16n‍(1-∑k=2p(2/i)kii-2)|.
It is easy to see that
(21)0≤limn→∞∏i=16n‍(1-∑k=2p(2/i)kii-2)≤limn→∞∏i=16n‍(1-1i)=limn→∞15n=0.
Hence,
(22)limn→∞|RSn-0Mn-0|=0.
Thus, the Mann iteration process converges more slowly than the multistep iteration process to the fixed point 0 of T.
Similarly,(23)|RSn-0In-0|=|∏i=16n(1-∑j=1p(2/i)j)x0∏i=16n(1-(2/i)-(4/i))x0|=|∏i=16n(1-∑j=1p(2/i)j)∏i=16n(1-(2/i)-(4/i))|=|∏i=16n‍(1-∑k=3p(2/i)k1-(2/i)-(4/i))|=|∏i=16n‍(1-∑k=3p(2/i)kiiii-2i-4i)|,
with
(24)0≤limn→∞∏i=16n‍(1-∑k=3p(2/i)kiiii-2i-4i)≤limn→∞∏i=16n‍(1-1i)=limn→∞15n=0,
implies that
(25)limn→∞|RSn-0In-0|=0.
Thus, the Ishikawa iteration process converges more slowly than the multistep iteration process to the fixed point 0 of T.



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*Source: 464593-2013-10-08.xml*