Volume 2010, Article ID 169062,12pages doi:10.1155/2010/169062
Research Article
Comparison of the Rate of Convergence among
Picard, Mann, Ishikawa, and Noor Iterations
Applied to Quasicontractive Maps
B. E. Rhoades
1and Zhiqun Xue
21Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
2Department of Mathematics and Physics, Shijiazhuang Railway University, Shijiazhuang 050043, China
Correspondence should be addressed to Zhiqun Xue,xuezhiqun@126.com
Received 12 October 2010; Accepted 14 December 2010
Academic Editor: Juan J. Nieto
Copyrightq2010 B. E. Rhoades and Z. 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.
We provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij, Mann, Ishikawa, or Noor iteration for quasicontractive operators. We also compare the rates of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators.
1. Introduction
LetX, dbe a complete metric space, and letT be a self-map ofX. If T has a unique fixed point, which can be obtained as the limit of the sequence{pn}, wherepnTnp0, p0any point ofX, then T is called a Picard operatorsee, e.g., 1, and the iteration defined by{pn}is
called Picard iteration.
One of the most general contractive conditions for which a mapT is a Picard operator is that of ´Ciri´c2 see also3. A self-mapTis called quasicontractive if it satisfies
dTx, Ty≤δmaxdx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx, 1.1
for eachx, y∈X, whereδis a real number satisfying 0≤δ <1.
To obtain fixed points for some maps for which Picard iteration fails, a number of fixed point iteration procedures have been developed. LetXbe a Banach space, the corresponding quasicontractive mappingT :X → Xis defined by
Tx−Ty≤δmaxx−y,x−Tx,y−Ty,x−Ty,y−Tx. 1.2
In this paper, we will consider the following four iterations. Krasnoselskij:
∀v0∈X, vn1 1−λvnλTvn, n≥0, 1.3
where 0< λ <1. Mann:
∀u0∈X, un1 1−anunanTun, n≥0, 1.4
where 0< an≤1 forn≥0, and
∞
n0an ∞.
Ishikawa:
∀x0 ∈X,
xn1 1−anxnanTyn, n≥0,
yn 1−bnxnbnTxn, n≥0,
1.5
where{an} ⊂0,1,{bn} ⊂0,1.
Noor:
∀w0 ∈X,
zn 1−cnwncnTwn, n≥0,
yn 1−bnwnbnTzn, n≥0,
wn1 1−anwnanTyn, n≥0,
1.6
where{an} ⊂0,1,{bn},{cn} ⊂0,1.
Three of these iteration schemes have also been used to obtain fixed points for some Picard maps. Consequently, it is reasonable to try to determine which process converges the fastest.
In this paper, we will discuss this question for the above quasicontractions and for Zamfirescu operators. For this, we will need the following result, which is a special case of the Theorem in4.
Theorem 1.1. Let C be any nonempty closed convex subset of a Banach spaceX, and let T be a quasicontractive self-map ofC. Let{xn}be the Ishikawa iteration process defined by1.5, where each
an >0and
∞
2. Results for Quasicontractive Operators
To avoid trivialities, we shall always assume thatp0/q, whereqdenotes the fixed point of the mapT.
Let{fn},{gn}be two convergent sequences with the same limitq, then{fn}is said to
converge faster than{gn}see, e.g.,5if
lim
n→ ∞
fn−q
gn−q 0. 2.1
Theorem 2.1. LetEbe a Banach space,Da closed convex subset ofE, andT a quasicontractive self-map ofD, then, for0< λ <1−δ2, Picard iteration converges faster than Krasnoselskij iteration.
Proof. From Theorem 1 of2and1.2,
pn1−qTn1p0−q
≤ δn1
1−δTp0−p0
≤ δn1
1−δTp0−Tqp0−q
≤ δn1 1−δ
δmaxp0−q,p0−qTp0−Tqp0−q
≤ δn1 1−δ
δp0−q δ
1−δp0−q p0−q
≤ δn1
1−δ2
p0−q,
2.2
whereqis the fixed point ofT. From1.3, withv0/q,
vn1−q≥1−λvn−q−λTvn−Tq
≥
1− λ
1−δ vn−q ≥ · · ·
≥
1− λ 1−δ
n1
v0−q.
2.3
By setting eachβn0 and eachαn λ, it follows fromTheorem 1.1that{vn}converges
Therefore,
pn1−q vn1−q
≤
δ 1−δ−λ
n1
1−δn−1p0−q v0−q
−→0, 2.4
asn → ∞, sinceλ <1−δ2.
Theorem 2.2. LetE, D, andTbe as inTheorem 2.1. And let0< an< θ1−δ, bn, cn∈0,1for all
n >0.
AIf the constant0< θ <1−δ, then Picard iteration converges faster than Mann iteration.
BIf the constant0< θ <1−δ2/1−δδ2, then Picard iteration converges faster than Ishikawa iteration.
CIf the constant0< θ <1−δ3/1−2δ2δ2, then Picard iteration converges faster than Noor iteration.
Proof. We have the following cases
Case AMann Iteration. UsingTheorem 1.1with eachβn 0,{un} converges toq. Using
1.4,
un1−q≥1−anun−q−anTun−Tq
≥
1− an
1−δ un−q≥ · · · ≥
n
i0
1− ai
1−δ u0−q.
2.5
Therefore,
pn1−q
un1−q
≤ δn1p0−q
1−δ2ni01−ai/1−δu0−q
−→0, 2.6
asn → ∞, sincean< θ1−δfor eachn >0.
Case BIshikawa Iteration. FromTheorem 1.1,{xn}converges toq. Using1.5,
xn1−q≥1−anxn−q−anTyn−Tq
≥1−anxn−q− anδ
1−δyn−q
≥1−anxn−q− anδ
≥
1−an− anδ
1−δ xn−q−
anbnδ2
1−δ2
xn−q
≥
1−an− anδ
1−δ − anδ2
1−δ2
xn−q
≥ · · ·
≥n
i0
1−ai− aiδ
1−δ − aiδ2
1−δ2
x0−q.
2.7
Hence,
pn1−q xn1−q
≤ δn1p0−q
1−δ2ni01−ai−aiδ/1−δ−aiδ2/1−δ2
x0−q −→
0, 2.8
asn → ∞, sincean< θ1−δfor eachn >0.
Case CNoor Iteration. First we must show that{wn}converges toq. The proof will follow
along the lines of that ofTheorem 1.1.
Lemma 2.3. Define
An {zi}ni0∪
yi
n
i0∪ {wi}
n
i0∪ {Tzi}ni0∪
Tyi
n
i0∪ {Twi}
n i0, αndiamAn,
βnmax
max{w0−Twi: 0≤i≤n},maxw0−Tyi: 0≤i≤n
,
max{w0−Tzi: 0≤i≤n}},
2.9
then{An}is bounded.
Proof.
Case 1. Suppose thatαnTzi−Tzjfor some 0≤i, j≤n, then, from1.2and the definition
ofαn,
αnTzi−Tzj
≤δmaxzi−zj,zi−Tzi,zj−Tzj,zi−Tzj,zj−Tzi
≤δαn,
2.10
a contradiction, sinceδ <1.
Similarly,αn/Tyi−Tyj,αn/Twi −Twj,αn/Tzi−Tyj,αn/Tzi−Twj, and
Case 2. Suppose thatαn wi−wj, without loss of generality we let 0 ≤i < j ≤ n. Then,
from1.6,
αnwi−wj
≤1−aj−1wi−wj−1aj−1wi−Tyj−1 ≤1−aj−1wi−wj−1aj−1αn.
2.11
Hence,αn ≤ wi−wj−1 ≤ αn, that is,αn wi−wj−1. By induction onj, we obtainαn
wi−wi0, a contradiction.
Case 3. Suppose thatαn wi−Twjfor some 0≤i, j≤n. Ifi >0, then we have, using1.6,
αnwi−Twj
≤1−ai−1wi−1−Twjai−1Tyi−1−Twj
≤1−ai−1wi−1−Twjai−1αn,
2.12
which implies thatαn≤ wi−1−Twj, and by induction oni, we getαnw0−Twj.
Case 4. Suppose thatαn wi−zjorαn zi−zj,yi−zj,zi−Tyj,yi−yjfor some
0≤i, j≤n, then
αnwi−zj
≤1−cjwi−wjcjwi−Twj
≤maxwi−wj,wi−Twj.
2.13
From Cases2and3,wi−wj < αn, andwi−Twj ≤ w0−Twmfor somem≤ j, that is,
αnw0−Twm. Ifαnzi−zj, we obtain thatαn ≤ wi−zj. Therefore;αnw0−Twm,
other cases, omitting.
Case 5. Suppose thatαnwi−Tzjorαnzi−Tzj,wi−yj,yi−Tzjfor some 0≤i, j≤n,
then ifi >0,
αnwi−Tzj
≤1−ai−1wi−1−Tzjai−1Tyi−1−Tzj
≤1−ai−1wi−1−Tzjai−1αn,
2.14
it leads toαn ≤ wi−1−Tzj. Again by induction oni, we haveαn w0−Tzj. Similarly, if
Case 6. Suppose thatαn zi−Twjorαn yi−Twjfor some 0 ≤ i, j ≤ n, then, using
Case1,
αnzi−Twj
≤1−ciwi−TwjciTwi−Twj
≤1−ciwi−Twjciαn,
2.15
or
αnyi−Twj
≤1−biwi−TwjbiTzi−Twj
≤1−biwi−Twjbiαn,
2.16
these imply thatαn≤ wi−Twj. By Case3, we obtain thatαnw0−Twj.
Case 7. Suppose thatαn wi−Tyjorαn yi−Tyjfor some 0 ≤i, j ≤ n, then ifi > 0,
using Case2,
αn wi−Tyj
≤1−ai−1wi−1−Tyjai−1Tyi−1−Tyj
≤1−ai−1wi−1−Tyjai−1αn,
2.17
which implies thatαn≤ wi−1−Tyj. Using induction oni, we haveαn w0−Tyj.
In view of the above cases, so we have shown thatαn βn. It remains to show that
{αn}is bounded.
Indeed, suppose thatαnw0−Twjfor some 0≤j ≤n, then, using Case1,
αnw0−Twj
≤ w0−Tw0Tw0−Twj
≤Bδαn,
2.18
whereB:w0−Tw0, thenαn≤B/1−δ.
Similarly, ifαn w0−Tyj, orαn w0−Tzjwe again getαn ≤B/1−δ. Hence,
{αn}is bounded, that is,{An}is bounded.
Lemma 2.4. LetE, D, and T be as in Theorem 2.1, and thatan ∞, then{wn}, as defined by
1.6, converges strongly to the unique fixed pointqofT.
Proof. From ´Ciri´c2,T has a unique fixed pointq. For eachn∈, define
Bn{wi}i≥n∪
yi
i≥n∪ {zi}i≥n∪ {Twi}i≥n∪
Tyi
Then, using the same proof as that ofLemma 2.3, it can be shown that
rn:diamBn
maxsupwn−Twj:j≥n
,supwn−Tyj:j≥n
,supwn−Tzj:j≥n
.
2.20
Using1.2and1.6,
rnwn−Twj
≤1−an−1wn−1−Twjan−1Twn−1−Twj
≤1−an−1rn−1an−1δrn−1
1−an−11−δrn−1 ≤ · · ·
≤r0
n−1
i0
1−1−δai,
2.21
limrn0, sincean∞.
For anym, n >0 withj≥0,
wn−wm ≤wn−TwjTwj−wm
rnrm,
2.22
and{wn}is Cauchy sequence. SinceDis closed, there existsw∞∈Dsuch that limwn w∞.
Also, limwn−Twn0.
Using1.2,
Tw∞−w∞Tw∞−TwnTwn−wnwn−w∞ limTw∞−Twn
≤lim supδmax{w∞−wn,w∞−Tw∞,wn−Twn,
w∞−Twn,wn−Tw∞} δw∞−Tw∞.
2.23
Sinceδ <1, it follows thatw∞Tw∞, andw∞is a fixed point ofT. But the fixed point
Returning to the proof of Case C, from1.6,
wn1−q≥1−anwn−q−anTyn−Tq
≥1−anwn−q− anδ
1−δyn−q
≥1−anwn−q− anδ
1−δwn−qbnTzn−Tq
≥
1−an−
anδ
1−δ wn−q− anδ2
1−δ2
zn−q
≥
1−an− anδ
1−δ − anδ2
1−δ2 − anδ3
1−δ3
wn−q
≥ · · ·
≥n
i0
1−ai− aiδ
1−δ − aiδ2
1−δ2 − aiδ3
1−δ3
w0−q.
2.24
So,
pn1−q wn1−q
≤ δn1p0−q
1−δ2ni01−ai−aiδ/1−δ−aiδ2/1−δ2−aiδ3/1−δ3
w0−q −→ 0,
2.25
asn → ∞, sincean< θ1−δforn >0.
It is not possible to compare the rates of convergence between the Krasnoselskij, Mann, and Noor iterations for quasicontractive maps. However, if one considers Zamfirescu maps, then some comparisons can be made.
3. Zamfirescu Maps
A selfmapT is called a Zamfirescu operator if there exist real numbersa, b, csatisfying 0 < a < 1,0 < b,c< 1/2 such that, for eachx, y ∈ Xat least one of the following conditions is true:
1dTx, Ty≤adx, y,
2dTx, Ty≤bdx, Tx dy, Ty,
In6it was shown that the above set of conditions is equivalent to
dTx, Ty≤δmax
dx, y,
dx, Tx dy, Ty
2 ,
dx, Tydy, Tx 2
, 3.1
for some 0< δ <1.
In the following results, we shall use the representation3.1.
Theorem 3.1. LetE, andDbe as inTheorem 2.1,T a Zamfirescu selfmap ofD, then ifan < λ1−
δθ/1δwith the constant0 < θ <1δfor eachn >0, Krasnoselskij iteration converges faster than Mann, Ishikawa, or Noor iteration.
Proof. Since Zamfirescu maps are special cases of quasicontractive maps, fromTheorem 1.1 {vn},{xn}, and{wn}converge to the unique fixed point ofT, which we will callq.
Using1.2,
vn1−q≤1−λvn−qλTvn−q. 3.2
Using3.1,
Tvn−q≤δmax
vn−q,
vn−Tvn0
2 ,
vn−qq−Tvn
2
δvn−q.
3.3
Therefore,
vn1−q≤1−λ1−δvn−q
≤ · · ·
≤1−λ1−δn1v0−q,
3.4
and
un1−q≥1−an1δun−q ≥ · · ·
≥n
i0
1−ai1δu0−q.
3.5
Thus,
vn1−q
un1−q ≤ 1−λ1−δn1v0−q
n
i01−ai1δu0−q
−→0, 3.6
asn → ∞, sincean< λ1−δ.
Theorem 3.2. LetE, D, andT be as inTheorem 3.1, then ifλ1δθ/1−δ < an <1 with the
constant0< θ <1−δfor anyn, Mann iteration converges faster than Krasnoselskij iteration.
Proof. Using1.4and3.1,
un1−q≤1−anun−qanTun−q
≤1−an1−δun−q
≤ · · ·
≤n
i0
1−ai1−δu0−q.
3.7
And again using1.3,3.1, we have
vn1−q≥1−λvn−q−λTvn−Tq
≥1−λ1δvn−q
≥ · · ·
≥1−λ1δn1u0−q.
3.8
Thus,
un1−q vn1−q ≤
n
i01−ai1−δu0−q
1−λ1δn1v0−q −→0, 3.9
asn → ∞, sinceλ1δθ/1−δ< an <1.
It is not possible to compare the rates of convergence for Mann, Ishikawa, and Noor iterations, even for Zamfirescu maps.
Remark 3.3. It has been noted in7that the principal result in8is incorrect.
Remark 3.4. Krasnoselskij and Mann iterations were developed to obtain fixed point iteration methods which converge for some operators, such as nonexpansive ones, for which Picard iteration fails. Ishikawa iteration was invented to obtain a convergent fixed point iteration procedure for continuous pseudocontractive maps, for which Mann iteration failed. To date, there is no example of any operator that requires Noor iteration; that is, no example of an operator for which Noor iteration converges, but for which neither Mann nor Ishikawa converges.
Acknowledgments
References
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