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

1. Introduction

Let C be a nonempty convex subset of a normed space E, and let T:C→C be a mapping.

(a) For arbitrary x0∈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 arbitrary x0∈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 arbitrary x0∈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 [<xref ref-type="bibr" rid="B2">4</xref>]).

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 [<xref ref-type="bibr" rid="B2">4</xref>]).

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 [<xref ref-type="bibr" rid="B9">5</xref>]).

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:

d(Tx,Ty)≤ad(x,y),

d(Tx,Ty)≤b[d(x,Tx)+d(y,Ty)],

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 operator T, 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 [<xref ref-type="bibr" rid="B4">7</xref>]).

Let C 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 [<xref ref-type="bibr" rid="B3">6</xref>]).

Let C 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.

Let C 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.

Let C 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.

Let C 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 that F(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, for x=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 that T:[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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the referees for their useful comments and suggestions. This study was supported by research funds from Dong-A University.

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