Volume 2007, Article ID 61434,5pages doi:10.1155/2007/61434
Review Article
Remarks of Equivalence among Picard, Mann, and Ishikawa
Iterations in Normed Spaces
Xue Zhiqun
Received 1 April 2007; Revised 16 April 2007; Accepted 21 June 2007
Recommended by J. R. L. Webb
We show that the convergence of Picard iteration is equivalent to the convergence of Mann iteration schemes for various Zamfirescu operators. Our result extends of Soltuz (2005).
Copyright © 2007 Xue Zhiqun. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetEbe a real normed space,Da nonempty convex subset ofE, andTa self-map ofD, letp0,u0,x0∈D. The Picard iteration is defined by
pn+1=T pn, n≥0. (1.1)
The Mann iteration is defined by
un+1=
1−anun+anTun, n≥0. (1.2)
The Ishikawa iteration is defined by
yn=1−bnxn+bnTxn, n≥0, xn+1=
1−anxn+anT yn, n≥0, (1.3)
where{an},{bn}are sequences of positive numbers in [0, 1]. Obviously, foran=1, the Mann iteration (1.2) reduces to the Picard iteration, and forbn=0, the Ishikawa iteration (1.3) reduces to the Mann iteration (1.2).
if, for each pair x,yin D,T satisfies at least one of the following conditions given in (1)–(3):
(1)Tx−T y ≤ax−y;
(2)Tx−T y ≤b(x−Tx+y−T y); (3)Tx−T y ≤c(x−T y+y−Tx).
It is easy to show that every Zamfirescu operatorTsatisfies the inequality
Tx−T y ≤δx−y+ 2δx−Tx (1.4)
for allx,y∈D, whereδ=max{a,b/(1−b),c/(1−c)} with 0< δ <1 (See S¸oltuz [1]). Recently, S¸oltuz [1] had studied that the equivalence of convergence for Picard, Mann, and Ishikawa iterations, and proved the following results.
Theorem 1.2 [1, Theorem 1]. Let X be a normed space,Da nonempty, convex, closed subset of X, andT :D→Dan operator satisfying condition Z (Zamfirescu operator). If u0=x0∈D, let{un}∞n=0be defined by (1.2) foru0∈D, and let{xn}∞n=0be defined by (1.3)
forx0∈Dwith{an}in[0, 1]satisfying
∞
n=0an= ∞. Then the following are equivalent:
(i)the Mann iteration (1.2) converges to the fixed point ofT; (ii)the Ishikawa iteration (1.3) converges to the fixed point ofT.
Theorem 1.3 [1, Theorem 2]. Let X be a normed space,Da nonempty, convex, closed subset of X, andT :D→Dan operator satisfying condition Z (Zamfirescu operator). If u0=p0∈D, let{pn}∞n=0be defined by (1.1) forp0∈D, and let{un}∞n=0be defined by (1.2)
foru0∈Dwith{an}in[0, 1]satisfyingn∞=0an= ∞andan→0asn→ ∞. Then
(i)if the Mann iteration (1.2) converges tox∗andlimn→∞(un+1−un/an)=0, then the Picard iteration (1.1) converges tox∗,
(ii)if the Picard iteration (1.1) converges tox∗andlimn→∞(pn+1−pn/an)=0, then
the Mann iteration (1.2) converges tox∗.
However, in the above-mentioned theorem, it is unnecessary that, for two conditions, limn→∞(un+1−un/an)=0 and limn→∞(pn+1−pn/an)=0. The aim of this paper is to show that the convergence of Picard iteration schemes is equivalent to the convergence of the Mann iteration for Zamfirescu operators in normed spaces. The result improves ones announced by S¸oltuz [1, Theorem 2]. We will use a special case of the following lemma.
Lemma1.4 [2]. Let{an}and{σn}be nonnegative real sequences satisfying the following inequality:
an+1≤
1−λnan+σn, (1.5)
whereλn∈(0, 1), for alln≥n0,
∞
n=1λn= ∞, andσn/λn→0asn→ ∞. Thenlimn→∞an=
0.
2. Main results
Theorem2.1. LetEbe a normed space,Da nonempty closed convex subset ofE, andT: D→Da Zamfirescu operator. Suppose thatThas a fixed pointq∈D. Let{pn}∞n=0be defined
by (1.1) forp0∈D, and let{un}∞n=0be defined by (1.2) foru0∈Dwith{an}in[0, 1]and satisfying∞n=0an= ∞. Then the following are equivalent:
(i)the Picard iteration (1.1) converges to the fixed point ofT; (ii)the Mann iteration (1.2) converges to the fixed point ofT.
Proof. Letqbe a fixed point ofT. We will prove (ii)⇒(i). Suppose thatun−q →0 as n→ ∞. Applying (1.1) and (1.2), we have
un+1−pn+1≤
1−anun−T pn+anTun−T pn
≤1−anun−Tun+Tun−T pn. (2.1)
Using (1.4) withx=un,y=pn, we have
Tun−T pn≤δun−pn+ 2δun−Tun. (2.2)
Therefore, from (2.1), we get
un+1−pn+1≤δun−pn+
1−an+ 2δun−Tun
≤δun−pn+1−an+ 2δun−q+Tun−Tq
≤δun−pn+1−an+ 2δ(1 +δ)un−q,
(2.3)
denoted byAn= un−pn,δ=1−λ, andBn=(1−an+ 2δ)(1 +δ)un−q. ByLemma
1.4, we obtainAn= un−pn →0 asn→ ∞. Hence bypn−q ≤ un−pn+un−q, we getpn−q →0 asn→ ∞.
Next, we will prove (i)⇒(ii), that is, if the Picard iteration converges, then the Mann iteration does too. Now by using Picard iteration (1.1) and Mann iteration (1.2), we have
un+1−pn+1≤
1−anun−T pn+anTun−T pn
≤1−anun−pn+1−anpn−T pn+anTun−T pn
≤1−anun−pn+1−anpn−q+T pn−Tq+anTun−T pn. (2.4)
On using (1.4) withx=pn,y=un, we get
Tun−T pn≤anδun−pn+ 2anδpn−T pn
Again, using (1.4) withx=q,y=pn, we get
T pn−Tq≤δpn−q. (2.6)
Hence by (2.4)–(2.6), we obtain
un+1−pn+1≤
1−(1−δ)anun−pn+1−an+ 2anδ(1 +δ)pn−q
≤1−(1−δ)anun−q+1−an+ 2anδ(2 +δ)pn−q
≤1−λan1−λan−1un−1−q+
1−an+ 2anδ(2 +δ)pn−q
≤1−λan1−λan−1
···1−λa0u0−q
+1−an+ 2anδ(2 +δ)pn−q
≤exp
−λn
i=0
ai
u0−q+
1−an+ 2anδ(2 +δ)pn−q,
(2.7)
where 1−δ=λ. Since∞n=0an= ∞andpn−q →0 as n→ ∞, henceun−pn →0
asn→ ∞. And thus,un−q ≤ un−pn+pn−q →0 asn→ ∞. This completes the
proof.
Remark 2.2. Theorem 2.1improves [1, Theorem 2] in the following sense.
(1) Both hypotheses limn→∞(un+1−un/an)=0 and limn→∞(pn+1−pn/an)=0 have been removed, and the conclusion remains valid.
(2) The assumption thatu0=p0in [1] is superfluous.
Theorem2.3. LetEbe a normed space,Da nonempty closed convex subset ofE, andT: D→Da Zamfirescu operator. Suppose thatThas a fixed pointq∈D. Let{pn}∞n=0be defined
by (1.1) for p0∈D, and let{xn}∞n=0be defined by (1.3) forx0∈Dwith{an}and{bn}in [0, 1]and satisfying∞n=0an= ∞. Then the following are equivalent:
(i)the Picard iteration (1.1) converges to the fixed point ofT; (ii)the Ishikawa iteration (1.3) converges to the fixed point ofT.
Remark 2.4. As previously suggested,Theorem 2.3reproduces exactly [1, Theorem 1]. Therefore we have the following conclusion: Picard iteration converges to the fixed point ofT ⇔Mann iteration converges to the fixed point ofT ⇔Ishikawa iteration converges to the fixed point ofT.
Acknowledgments
The author would like to thank the referee and the editor for their careful reading of the manuscript and their many valuable comments and suggestions. The project is supported by the National Science Foundation of China and Shijiazhuang Railway Institute Sciences Foundation.
References
[2] F. P. Vasilev,Numerical Methods for Solving Extremal Problems, Nauka, Moscow, Russia, 2nd edi-tion, 1988.
[3] X. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,”Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.
Xue Zhiqun: Department of Mathematics and Physics, Shijiazhuang Railway Institute, Shijiazhuang 050043, China
