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
Trang 1Volume 2010, Article ID 169062, 12 pages
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 Rhoades1 and Zhiqun Xue2
1 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
2 Department 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
Copyrightq 2010 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, d be a complete metric space, and let T be a self-map of X If T has a unique fixed
point, which can be obtained as the limit of the sequence{p n }, where p n T n p0, p0any point
of X, then T is called a Picard operatorsee, e.g., 1, and the iteration defined by {p n} is called Picard iteration
One of the most general contractive conditions for which a map T is a Picard operator
is that of ´Ciri´c2 see also 3 A self-map T is called quasicontractive if it satisfies
d
Tx, Ty
≤ δ maxd
x, y
, d x, Tx, dy, Ty
, d
x, Ty
, d
y, Tx
, 1.1
for each x, y ∈ X, where δ is a real number satisfying 0 ≤ δ < 1.
Not every map which has a unique fixed point enjoys the Picard property For example,
let X 0, 1 with the absolute value metric, T : X → X defined by Tx 1 − x Then, T has a unique fixed point at x 1/2, but if one chooses as a starting point x0 a for any a / 1/2, then
successive function iterations generate the bounded divergent sequence{a, 1 − a, a, 1 − a, }.
Trang 2To obtain fixed points for some maps for which Picard iteration fails, a number of fixed
point iteration procedures have been developed Let X be a Banach space, the corresponding quasicontractive mapping T : X → X is 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, v n1 1 − λv n λTv n , n ≥ 0, 1.3
where 0 < λ < 1.
Mann:
∀u0 ∈ X, u n1 1 − a n u n a n Tu n , n ≥ 0, 1.4
where 0 < a n ≤ 1 for n ≥ 0, and∞
n0a n ∞
Ishikawa:
∀x0 ∈ X,
x n1 1 − a n x n a n Ty n , n ≥ 0,
y n 1 − b n x n b n Tx n , n ≥ 0,
1.5
where{a n } ⊂ 0, 1, {b n } ⊂ 0, 1.
Noor:
∀w0 ∈ X,
z n 1 − c n w n c n Tw n , n ≥ 0,
y n 1 − b n w n b n Tz n , n ≥ 0,
w n1 1 − a n w n a n Ty n , n ≥ 0,
1.6
where{a n } ⊂ 0, 1, {b n }, {c n } ⊂ 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 space X, and let T be a
quasicontractive self-map of C Let {x n } be the Ishikawa iteration process defined by 1.5, where each
a n > 0 and∞
n0a n ∞ then {x n } converges strongly to the fixed point of T.
Trang 32 Results for Quasicontractive Operators
To avoid trivialities, we shall always assume that p0 / q, where q denotes the fixed point of the map T.
Let{f n }, {g n } be two convergent sequences with the same limit q, then {f n} is said to converge faster than{g n} see, e.g., 5 if
lim
n→ ∞
f n − q
g n − q 0. 2.1
Theorem 2.1 Let E be a Banach space, D a closed convex subset of E, and T a quasicontractive
self-map of D, then, for 0 < λ < 1 − δ2, Picard iteration converges faster than Krasnoselskij iteration Proof From Theorem 1 of2 and 1.2,
p n1− q T n1p0− q
≤ δ n1
1− δTp0− p0
≤ δ n1
1− δTp0− Tq p0 − q
≤ δ n1
1− δ
δ maxp0− q,p0− q Tp0 − Tq p0 − q
≤ δ n1
1− δ
δp
0− q δ
1− δp0− q p0− q
≤ δ n1
1 − δ2p0− q,
2.2
where q is the fixed point of T.
From1.3, with v0 / q,
v n1− q ≥ 1 − λv n − q − λTv n − Tq
≥
1− λ
1− δ
v
n − q
≥ · · ·
≥
1− λ
1− δ
n1
v0− q.
2.3
By setting each β n 0 and each α n λ, it follows fromTheorem 1.1that{v n} converges
to q.
Trang 4p n1− q
v n1− q ≤1− δ − λ δ
n1
1 − δ n−1p0− q
v0− q −→ 0, 2.4
as n → ∞, since λ < 1 − δ2
Theorem 2.2 Let E, D, and T be as in Theorem 2.1 And let 0 < a n < θ 1 − δ, b n , c n ∈ 0, 1 for all
n > 0.
A If the constant 0 < θ < 1 − δ, then Picard iteration converges faster than Mann iteration.
B If the constant 0 < θ < 1 − δ2/ 1 − δ δ2, then Picard iteration converges faster than
Ishikawa iteration.
C If the constant 0 < θ < 1 − δ3/ 1 − 2δ 2δ2, then Picard iteration converges faster than
Noor iteration.
Proof We have the following cases
Case AMann Iteration Using Theorem 1.1with each β n 0, {u n } converges to q Using
1.4,
u n1− q ≥ 1 − a nu n − q − a nTu n − Tq
≥
1− a n
1− δ
u
n − q ≥ ··· ≥ n
i0
1− a i
1− δ
u
0− q. 2.5
Therefore,
p n1− q
u n1− q ≤ δ n1p0− q
1 − δ2 n
i01 − a i / 1 − δu0− q −→ 0, 2.6
as n → ∞, since a n < θ 1 − δ for each n > 0.
Case BIshikawa Iteration FromTheorem 1.1,{x n } converges to q Using 1.5,
x n1− q ≥ 1 − a nx n − q − a nTy n − Tq
≥ 1 − a nx n − q − a n δ
1− δy n − q
≥ 1 − a nx n − q − a n δ
1− δx n − q b nTx n − Tq
Trang 5
1− a n− a n δ
1− δ
x n − q − a n b n δ2
1 − δ2x n − q
≥
1− a n− a n δ
1− δ−
a n δ2
1 − δ2 x n − q
≥ · · ·
≥ n
i0
1− a i− a i δ
1− δ−
a i δ2
1 − δ2 x0− q.
2.7
Hence,
p n1− q
x n1− q ≤ δ n1p0− q
1 − δ2 n
i0
1− a i − a i δ/ 1 − δ − a i δ2/ 1 − δ2x0
− q −→ 0, 2.8
as n → ∞, since a n < θ 1 − δ for each n > 0.
Case C Noor Iteration First we must show that {w n } converges to q The proof will follow
along the lines of that ofTheorem 1.1
Lemma 2.3 Define
A n {z i}n
i0∪y in
i0∪ {w i}n
i0∪ {Tz i}n
i0∪Ty in
i0∪ {Tw i}n
i0,
α n diamA n ,
β n maxmax{w0− Tw i : 0 ≤ i ≤ n}, maxw0− Ty i: 0≤ i ≤ n
,
max{w0− Tz i : 0 ≤ i ≤ n}},
2.9
then {A n } is bounded.
Proof.
Case 1 Suppose that α n Tz i − Tz j for some 0 ≤ i, j ≤ n, then, from 1.2 and the definition
of α n,
α nTz i − Tz j
≤ δ maxz i − z j, z i − Tz i ,z j − Tz j,z i − Tz j,z j − Tz i
≤ δα n ,
2.10
a contradiction, since δ < 1.
Similarly, α n / Ty i − Ty j , α n / Tw i − Tw j , α n / Tz i − Ty j , α n / Tz i − Tw j, and
α n / Ty i − Tw j for any 0 ≤ i, j ≤ n.
Trang 6Case 2 Suppose that α n w i − w j , without loss of generality we let 0 ≤ i < j ≤ n Then,
from1.6,
α nw i − w j
≤1− a j−1w i − w j−1 aj−1w i − Ty j−1
≤1− a j−1w i − w j−1 aj−1α n
2.11
Hence, α n ≤ w i − w j−1 ≤ α n , that is, α n w i − w j−1 By induction on j, we obtain α n
w i − w i 0, a contradiction
Case 3 Suppose that α n w i − Tw j for some 0 ≤ i, j ≤ n If i > 0, then we have, using 1.6,
α nw i − Tw j
≤ 1 − a i−1w i−1− Tw j a i−1Ty i−1− Tw j
≤ 1 − a i−1w i−1− Tw j a i−1α n ,
2.12
which implies that α n ≤ w i−1− Tw j , and by induction on i, we get α n w0 − Tw j
Case 4 Suppose that α n w i − z j or α n z i − z j , y i − z j , z i − Ty j , y i − y j for some
0≤ i, j ≤ n, then
α nw i − z j
≤1− c jw i − w j c jw i − Tw j
≤ maxw i − w j,w i − Tw j. 2.13
From Cases2and3,w i − w j < α n, andw i − Tw j ≤ w0 − Tw m for some m ≤ j, that is,
α n w0 − Tw m If α n z i − z j , we obtain that α n ≤ w i − z j Therefore; α n w0 − Tw m, other cases, omitting
Case 5 Suppose that α n w i −Tz j or α n z i −Tz j , w i −y j , y i −Tz j for some 0 ≤ i, j ≤ n, then if i > 0,
α nw i − Tz j
≤ 1 − a i−1w i−1− Tz j a i−1Ty i−1− Tz j
≤ 1 − a i−1w i−1− Tz j a i−1α n ,
2.14
it leads to α n ≤ w i−1− Tz j Again by induction on i, we have α n w0 − Tz j Similarly, if
α n z i − Tz j or, α n w i − y j , we also get α n w0 − Tz j; other cases, omitting
Trang 7Case 6 Suppose that α n z i − Tw j or α n y i − Tw j for some 0 ≤ i, j ≤ n, then, using
Case1,
α nz i − Tw j
≤ 1 − c iw i − Tw j c iTw i − Tw j
≤ 1 − c iw i − Tw j c i α n ,
2.15
or
α ny i − Tw j
≤ 1 − b iw i − Tw j b iTz i − Tw j
≤ 1 − b iw i − Tw j b i α n ,
2.16
these imply that α n ≤ w i − Tw j By Case3, we obtain that α n w0 − Tw j
Case 7 Suppose that α n w i − Ty j or α n y i − Ty j for some 0 ≤ i, j ≤ n, then if i > 0,
using Case2,
α n w i − Ty j
≤ 1 − a i−1w i−1− Ty j a i−1Ty i−1− Ty j
≤ 1 − a i−1w i−1− Ty j a i−1α n ,
2.17
which implies that α n ≤ w i−1− Ty j Using induction on i, we have α n w0 − Ty j
In view of the above cases, so we have shown that α n β n It remains to show that
{α n} is bounded
Indeed, suppose that α n w0 − Tw j for some 0 ≤ j ≤ n, then, using Case1,
α nw0− Tw j
≤ w0 − Tw0 Tw0− Tw j
≤ B δα n ,
2.18
where B : w0 − Tw0, then α n ≤ B/1 − δ.
Similarly, if α n w0 − Ty j , or α n w0 − Tz j we again get α n ≤ B/1 − δ Hence, {α n } is bounded, that is, {A n} is bounded
Lemma 2.4 Let E, D, and T be as in Theorem 2.1 , and that
a n ∞, then {w n }, as defined by
1.6, converges strongly to the unique fixed point q of T.
Proof From ´Ciri´c2, T has a unique fixed point q For each n ∈, define
B n {w i}i ≥n∪y i
i ≥n ∪ {z i}i ≥n ∪ {Tw i}i ≥n∪Ty i
i ≥n ∪ {Tz i}i ≥n 2.19
Trang 8Then, using the same proof as that ofLemma 2.3, it can be shown that
r n: diamB n
maxsupw n − Tw j: j ≥ n
, supw n − Ty j: j ≥ n
, supw n − Tz j: j ≥ n
.
2.20 Using1.2 and 1.6,
r nw n − Tw j
≤ 1 − a n−1w n−1− Tw j a n−1Tw n−1− Tw j
≤ 1 − a n−1rn−1 a n−1δr n−1
1 − a n−11 − δrn−1
≤ · · ·
≤ r0 n−1
i0
1 − 1 − δa i ,
2.21
lim r n 0, sincea n ∞
For any m, n > 0 with j≥ 0,
w n − w m ≤w n − Tw j Tw j − w m
r n r m , 2.22
and{w n } is Cauchy sequence Since D is closed, there exists w∞ ∈ D such that lim w n w∞ Also, limw n − Tw n 0
Using1.2,
Tw∞− w∞ Tw∞− Tw n Tw n − w n w n − w∞
limTw∞ − Tw n
≤ lim sup δ max{w∞ − w n , w∞ − Tw∞, w n − Tw n ,
w∞− Tw n , w n − Tw∞}
δw∞ − Tw∞.
2.23
Since δ < 1, it follows that w∞ Tw∞ , and w∞is a fixed point of T But the fixed point
is unique Therefore, w∞ q.
Trang 9Returning to the proof of Case C, from1.6,
w n1− q ≥ 1 − a nw n − q − a nTy n − Tq
≥ 1 − a nw n − q − a n δ
1− δy n − q
≥ 1 − a nw n − q − a n δ
1− δw n − q b nTz n − Tq
≥
1− a n− a n δ
1− δ
w
n − q − a n δ2
1 − δ2z n − q
≥
1− a n− a n δ
1− δ−
a n δ2
1 − δ2 − a n δ3
1 − δ3 w n − q
≥ · · ·
≥ n
i0
1− a i− a i δ
1− δ −
a i δ2
1 − δ2 − a i δ3
1 − δ3 w0− q.
2.24
So,
p n1− q
w n1− q
≤ δ n1p0− q
1 − δ2 n
i0
1− a i − a i δ/1 − δ − a i δ2/ 1 − δ2− a i δ3/ 1 − δ3w0
− q −→ 0,
2.25
as n → ∞, since a n < θ 1 − δ for n > 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 selfmap T is called a Zamfirescu operator if there exist real numbers a, b, c satisfying 0 <
a < 1, 0 < b, c < 1/2 such that, for each x, y ∈ X at least one of the following conditions is
true:
1 dTx, Ty ≤ adx, y,
2 dTx, Ty ≤ bdx, Tx dy, Ty,
3 dTx, Ty ≤ cdx, Ty dy, Tx.
Trang 10In6 it was shown that the above set of conditions is equivalent to
d
Tx, Ty
≤ δ max
d
x, y
,
d x, Tx dy, Ty
d
x, Ty
dy, Tx
2
, 3.1
for some 0 < δ < 1.
In the following results, we shall use the representation3.1
Theorem 3.1 Let E, and D be as in Theorem 2.1 , T a Zamfirescu selfmap of D, then if a n < λ1 −
δ θ/1 δ with the constant 0 < θ < 1 δ for each n > 0, Krasnoselskij iteration converges faster
than Mann, Ishikawa, or Noor iteration.
Proof Since Zamfirescu maps are special cases of quasicontractive maps, fromTheorem 1.1
{v n }, {x n }, and {w n } converge to the unique fixed point of T, which we will call q.
Using1.2,
v n1− q≤ 1 − λv n − q λTv n − q. 3.2 Using3.1,
Tv n − q ≤ δmaxv n − q, v n − Tv n 0
2 ,
v n − q q − Tv n
2
δv n − q. 3.3
Therefore,
v n1− q ≤ 1 − λ1 − δv n − q
≤ · · ·
≤ 1 − λ1 − δ n1v0− q,
3.4
and
u n1− q ≥ 1 − a n 1 δu n − q
≥ · · ·
≥ n
i0
1 − a i 1 δu0− q. 3.5
Thus,
v n1− q
u n1− q ≤ 1 − λ1 − δ n1v0− q
n
i01 − a i 1 δu0− q −→ 0, 3.6
as n → ∞, since a n < λ 1 − δ.
The proofs for Ishikawa and Noor iterations are similar
Trang 11Theorem 3.2 Let E, D, and T be as in Theorem 3.1 , then if λ 1 δθ/1 − δ < a n < 1 with the constant 0 < θ < 1 − δ for any n, Mann iteration converges faster than Krasnoselskij iteration.
Proof Using1.4 and 3.1,
u n1− q ≤ 1 − a nu n − q a nTu n − q
≤ 1 − a n 1 − δu n − q
≤ · · ·
≤ n
i0
1 − a i 1 − δu0− q.
3.7
And again using1.3, 3.1, we have
v n1− q ≥ 1 − λv n − q − λTv n − Tq
≥ 1 − λ1 δv n − q
≥ · · ·
≥ 1 − λ1 δ n1u0− q.
3.8
Thus,
u n1− q
v n1− q ≤
n
i01 − a i 1 − δu0− q
1 − λ1 δ n1v0− q −→ 0, 3.9
as n → ∞, since λ1 δθ/1 − δ < a n < 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 in7 that the principal result in 8 is 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
The authors would like to thank the reviewers for valuable suggestions, and the National Natural Science Foundation of China Grant 10872136 for the financial support
...Since δ < 1, it follows that w∞ Tw∞ , and w∞is a fixed point of T But the fixed point< /i>
is unique Therefore, w∞ q.... E, D, and T be as in Theorem 2.1 , and that
a n ∞, then {w n }, as defined by
1.6, converges strongly to the unique fixed point. .. fromTheorem 1.1
{v n }, {x n }, and {w n } converge to the unique fixed point of T, which we will call q.
Using1.2,
v