Volume 2008, Article ID 387056,5pages doi:10.1155/2008/387056
Review Article
The Comparison of the Convergence Speed between Picard, Mann, Krasnoselskij and Ishikawa Iterations in Banach Spaces
Zhiqun Xue
Department of Mathematics and Physics, Shijiazhuang Railway Institute, Shijiazhuang 050043, China
Correspondence should be addressed to Zhiqun Xue,[email protected] Received 19 April 2007; Accepted 17 January 2008
Recommended by Brailey Sims
The purpose of this paper is to compare convergence speed of the Picard and Mann iterations on one hand, Krasnoselskij and Ishikawa iterations on the other hand, for the class of Zamfirescu op- erators. The results improve corresponding results ofBerinde 2004andBabu and Vara Prasad 2006.
Copyrightq2008 Zhiqun Xue. This is an open access article distributed under the Creative 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 Banach space,Da closed convex subset ofE,andT :D →Da self-map. Let p0, v0, u0, x0∈Dbe arbitrary. The sequence{pn}∞n0⊂Ddefined by
pn1Tpn, n≥0, 1.1
is called the Picard iteration or Picard iterative procedure. Forλ∈0,1, the sequence{vn}∞n0⊂ Ddefined by
vn1 1−λvnλTvn, n≥0, 1.2
is called the Krasnoselskij iteration or Krasnoselskij iterative procedure. Let{an}be a sequence of real numbers in0,1. The sequence{un}∞n0⊂Ddefined by
un1 1−anunanTun, n≥0, 1.3
is called the Mann iteration or Mann iterative procedure. The sequence{xn}∞n0⊂Ddefined by x0∈D,
yn 1−bn
xnbnTxn, n≥0,
xn1 1−an
xnanTyn, n≥0,
1.4
is called the Ishikawa iteration or Ishikawa iterative procedure, where{an}and{bn}are se- quences of real numbers in 0,1. Obviously, for bn 0 the Ishikawa iteration 1.4 can be reduced to1.3; and forλ 1 we obtain the Picard iteration. In the last twenty years, many authors have studied the convergence of the sequence of the Picard, Krasnoselskij, Mann, and Ishikawa iterations of a mappingTto a fixed point ofT, under various contractive conditions.
In such situations, it is of theoretical and practical importance to compare these iteration meth- ods in order to establish which one converges faster if possible.
Definition 1.1see1. The operatorT : X →X satisfies condition Zamfirescu if and only if there exist real numbersa, b, csatisfying 0< a <1, 0< b,c <1/2 such that for each pairx, yin X, at least one of the following conditions is true:
1Tx−Ty ≤ax−y;
2Tx−Ty ≤bx−Txy−Ty;
3Tx−Ty ≤cx−Tyy−Tx.
Obviously, we could obtain that every Zamfirescu operatorT satisfies the inequality
Tx−Ty ≤δx−y2δx−Tx 1.5 for allx, y∈D, whereδmax{a, b/1−b, c/1−c}with 0< δ <1.
In 1972, Zamfirescu1obtained a very interesting fixed point theorem for Zamfirescu operator.
Theorem Zsee1. LetX, dbe a complete metric space andT :X → Xa Zamfirescu operator.
Then,Thas a unique fixed pointqand the Picard iteration1.1converges toq.
Later on, Berinde 3 improved and extended the above-mentioned theorem and the results in paper2with the following result.
Theorem B1see3. LetEbe an arbitrary Banach space,Da closed convex subset ofE, and T : D → D an operator satisfying condition Z. Let{un}∞n0 be the Mann iteration defined by 1.2for u0 ∈ D, with{an} ⊂ 0,1satisfying∞
n0αn ∞. Then,{un}∞n0 converges strongly to the fixed point ofT.
Theorem B2see3. LetEbe an arbitrary Banach space,Da closed convex subset ofE, andT :D→ Dan operator satisfying condition Z. Let{xn}∞n0be the Ishikawa iteration defined by1.3forx0∈D, with{an}and{bn}being sequences of positive numbers in0,1and{an}satisfying∞
n0an ∞.
Then,{xn}∞n0converges strongly to the fixed point ofT.
In order to compare the fixed point iteration procedures{pn},{un},and{xn}that con- verge to a certain fixed point of given operatorT, Berinde4provided the following defini- tions.
Definition 1.2see4. Let{an}∞n0and{bn}∞n0be two sequences of real numbers that converge toaandb, respectively, and assume that there existsllimn→∞|an−a|/|bn−b|. Ifl0, then it can be said that{an}∞n0converges faster toathan{bn}∞n0tob. If 0 < l < ∞, then it can be said that{an}∞n0and{bn}∞n0have the same rate of convergence.
Definition 1.3 see 4. Suppose that for two fixed point iteration procedures {un}∞n0 and {vn}∞n0 both converging to the same fixed point p with the error estimatesun−p ≤ an, vn−p ≤bn,n≥0, where{an}∞n0and{bn}∞n0are two sequences of positive numbersconverg- ing to zero. If{an}∞n0converges faster than{bn}∞n0, then it can be said that{un}∞n0converges faster than{vn}∞n0top.
The purpose of this paper is to improve the results in 4,5by giving a direct rate of convergence for some fixed point procedures.
2. Main results
In the sequel, suppose thatδis a constant from1.5.
Theorem 2.1. LetEbe an arbitrary real Banach space,Da closed convex subset ofE, andT :D→D a Zamfirescu operator. Let{pn}∞n0be defined by1.1forx0 ∈D, and let{un}∞n0be defined by1.3 fory0∈Dwith{an}in0,1/1δand satisfyingi∞
n0an∞,iian →0 asn→ ∞. Then, the Picard iteration converges faster than the Mann iteration to the fixed point ofT.
Proof. By1, Theorem 2.3,Thas a unique fixed point, denote it byq. Moreover, Picard’s itera- tion{pn}∞n0defined by1.1converges toq, for anyp0∈E, and
pn1−qTpn−q. 2.1
Takexqandypnin1.5, then we get
pn1−q≤δpn−q≤δn1p0−q, n≥0. 2.2
Now, by Mann’s iteration in1.3and1.5, un1−q≥
1−anun−q−anTun−Tq
≥
1−1δanun−q
≥
1−1δan
1−1δan−1
· · ·
1−1δa0u0−q.
2.3
From2.2and2.3, it follows thatpn1−q/un1−q ≤δn1p0−q/1−1δan1−1 δan−1· · ·1−1δa0u0−q →0 asn→ ∞. Indeed, we consider∞
n0δn1p0−q/1− 1δan1−1δan−1· · ·1−1δa0u0−q. Setwnδn1p0−q/1−1δan1− 1δan−1· · ·1−1δa0u0−q, then we obtain that limn→∞wn1/wn δ <1. Applying the ratio test, we get∞
n0wn < ∞, sown → 0 asn → ∞, that is,pn−q oun−q. By Definition 1.2, we obtain the conclusion ofTheorem 2.1.
Theorem 2.2. LetEbe an arbitrary Banach space,Da closed convex subset ofE, andT :D →Da Zamfirescu operator. Let{vn}∞n0be defined by1.2forv0 ∈D, and let{xn}∞n0 be defined by1.4 forx0 ∈Dwith{an}and{bn}in0,1/1δand satisfyingi∞
n0an ∞,iian, bn →0 as n → ∞. Letqbe a fixed point of T inD. Then, the Krasnoselskij iteration converges faster than the Ishikawa iteration to the fixed pointqofT.
Proof. By Theorem B2see3, there exists a unique fixed point, denote it byq. For the Kras- noselskij iteration, by using1.2we have
vn1−q≤1−λvn−qλTvn−Tq. 2.4
Takexqandyvnin1.5to obtain
Tvn−Tq≤δvn−q, 2.5
and then
vn1−q≤
1−1−δλvn−q
≤
1−1−δλn1v0−q−→0 2.6
asn→ ∞. For the Ishikawa iterative procedure, by1.4we get xn1−q≥
1−anxn−q−anTyn−Tq. 2.7 Takexqandyynin1.5to obtain
Tyn−Tq≤δyn−q, 2.8
and again using1.4and1.5, yn−q≤
1−bnxn−qbnTxn−Tq
≤
1−1−δbnxn−q, 2.9
and hence by2.8,2.9, and2.7, we get xn1−q≥
1−an−anδ
1−1−δbnxn−q
≥
1−1δanxn−q
≥
1−1δan
1−1δan−1xn−1−q
≥
1−1δan
1−1δan−1
· · ·
1−1δa0x0−q.
2.10
On repeating the proof course ofTheorem 2.1, thenvn1−q/xn1−q ≤1−1−δλn1v0− q/1−1δan1−1δan−1· · ·1−1δa0x0−q →0 asn→ ∞. Hence,vn−q oxn−q. ByDefinition 1.2, we also obtain the conclusion ofTheorem 2.2.
Remark 2.3. Theorem 2.1provides a direct comparison of the rate of convergence of Picard and Mann iterations in the class of Zamfirescu operators, whileTheorem 2.2obtains a similar result for Krasnoselskij and Ishikawa iterations. However, we do not have a direct comparison result of the rate of convergence in the case of Mann and Ishikawa iterations in the same class of mappings. So, the best result for these two fixed point iterations remains that of5, obtained by means of the comparison sequences{an}and{bn}and not in a direct way, as in the present paperTheorems2.1and2.2.
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.
References
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