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If T is a member of the class of Lipschitz, strongly pseudocontrac-tive maps with Lipschitz constantL ≥1, then it is shown that to each Mann iteration there is a Krasnosleskij iteration

AMONG KRASNOSELSKIJ, MANN, AND ISHIKAWA

ITERATIONS IN ARBITRARY REAL BANACH SPACES

G V R BABU AND K N V V VARA PRASAD

Received 25 April 2006; Accepted 4 September 2006

LetE be an arbitrary real Banach space and K a nonempty, closed, convex (not necessarily

bounded) subset ofE If T is a member of the class of Lipschitz, strongly

pseudocontrac-tive maps with Lipschitz constantL ≥1, then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point ofT.

Copyright © 2006 G V R Babu and K N V V Vara Prasad This is an open access arti-cle distributed under the Creative Commons Attribution License, which permits unre-stricted use, distribution, and reproduction in any medium, provided the original work

is properly cited

1 Introduction

By approximation of fixed points of certain classes of operators which satisfy weak con-tractive-type conditions that do not guarantee the convergence of Picard iteration [2, Example 2.1, page 76], certain mean value fixed point iterations, namely, Krasnoselskij, Mann, and Ishikawa iteration methods are useful to approximate fixed points For more details on these iterations and further literature, see Berinde [3]

When, for a certain class of mappings, two or more fixed point iteration procedures can be used to approximate their fixed points, it is of theoretical and practical importance

to compare the rate of convergence of these iterations, and to find out, if possible, which one of them converges faster Recent works in this direction are [1,4,5]

Verma [9] approximated fixed points of Lipschitzian and generalized pseudocontrac-tive operators in Hilbert spaces by both Krasnoselskij and Mann iteration, and Berinde [4] established that, for any Mann iteration, there is a Krasnoselskij iteration which con-verges faster to the fixed point of such an operator

Chidume and Osilike [7] approximated fixed points of Lipschitzian strongly pseudo-contractive maps in Banach spaces, using both Mann and Ishikawa iterations

Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2006, Article ID 35704, Pages 1 12

DOI 10.1155/FPTA/2006/35704

Now, the interest of this paper is to compare the fastness of the convergence to the fixed point among the Krasnoselskij, Mann, and Ishikawa iterations for the class of Lipschitz, strongly pseudocontractive operators in arbitrary real Banach spaces

2 Preliminaries and known results

Suppose thatE is a real Banach space with dual E ∗, we denote byJ, the normalized duality

map fromE to 2 E ∗

defined by

J(x) =f ∗ ∈ E ∗:

x, f ∗

= x2=f ∗ 2 

where·,·denotes the generalized duality pairing

A mappingT with domain D(T) and range R(T) in E is called Lipschitz, if there exists

L > 0 such that for each x, y ∈ D(T),

A mappingT with domain D(T) and range R(T) in E is called strongly pseudocontrac-tive if and only if for any x, y ∈ D(T), there exists t > 1 such that

x − y ≤(1 +r)(x − y) − rt(Tx − T y) (2.3)

for anyr > 0 If t =1 in (2.3), thenT is called pseudocontractive.

It follows from [8, Lemma 1.1] thatT is strongly pseudocontractive if and only if the

following condition holds: there exists j(x − y) ∈ J(x − y) such that



(I − T)(x) −(I − T)(y), j(x − y)≥ kx − y2 (2.4) for eachx, y in E, where k =(t −1)/t ∈(0, 1)

Again by using [8, Lemma 1.1] and inequality (2.4) (Bogin [6]) it follows thatT is

strongly pseudocontractive if and only if the following inequality holds:

x − y ≤x − y + s

(I − T − kI)(x) −(I − T − kI)(y) (2.5)

for allx, y ∈ D(T) and s > 0.

Notation 2.1 Throughout this paper, E denotes a real Banach space, K a closed

con-vex (not necessarily bounded) subset ofE, and LS(K) the class of all Lipschitz, strongly

pseudocontractive maps onK For any T ∈LS(K), we assume that the Lipschitz constant

L ≥1 and pseudocontractive constantk ∈(0, 1)

Letx0 ∈ E be arbitrary.

(i) For anyλ ∈(0, 1), the sequence{x n } ∞

n =0⊆ E defined by

x n+1 = T λ x n =(1− λ)x n+λTx n, n =0, 1, 2, , (2.6)

is called the Krasnoselskij iteration We denote it by K(x0,λ,T).

(ii) The sequence{x n } ∞

n =0⊆ E defined by

x n+1 = 1− α n x n+α n Tx n, n =0, 1, 2, , (2.7)

where{α n } ∞

n =0is a real sequence satisfying 0≤ α n < 1, n =0, 1, 2, , is called the Mann iteration, and is denoted by M(x0,α n,T).

(iii) The sequence{x n } ∞

n =0⊆ E defined by

x n+1 = 1− α n x n+α n T y n, n =0, 1, 2, ,

y n = 1− β n x n+β n Tx n, n =0, 1, 2, , (2.8)

where {α n } ∞

n =0, {β n } ∞

n =0 are sequences of reals satisfying 0≤ α n,β n < 1, is called the Ishikawa iteration, and is denote by I(x0,α n,β n,T).

Chidume and Osilike [7] established the strong convergence of Mann and Ishikawa it-erations to the fixed point ofT ∈LS(K) Now, the following question arises: for a member

T of LS(K) which one of the following, namely, Krasnoselskij, Mann, and Ishikawa iterations converges faster to the fixed point of T?

To answer this question, we use the following definitions introduced by Berinde [5]

Definition 2.2 [5] Let{a n } ∞

n =0and{b n } ∞

n =0be two sequences of real numbers that con-verge toa and b, respectively Assume that there exists a real number l such that

lim

n →∞

a n − a

(i) Ifl =0, then{a n } ∞

n =0is said to converge faster toa than {b n } ∞

n =0tob.

(ii) If 0< l < ∞, then{a n } ∞

n =0and{b n } ∞

n =0are said to have the same rate of convergence

Definition 2.3 [5] Suppose that for two fixed point iteration procedures{u n } ∞

n =0 and

{v n } ∞

n =0both converging to the same fixed pointp (say) with error estimates

u n − p  ≤ a n, n =0, 1, 2, ,

v n − p  ≤ b n, n =0, 1, 2, , (2.10)

where{a n } ∞

n =0and{b n } ∞

n =0are two sequences of positive numbers converging to zero If

{a n } ∞

n =0converges faster than{b n } ∞

n =0, then{u n } ∞

n =0is said to converge faster than {v n } ∞

n =0

to p.

For more details on definitions, we refer, Berinde [4]

3 Results on the comparison of fastness of the convergence

Theorem 3.1 If T ∈LS(K), then the following hold:

(a) for any x0 ∈ K and  ∈(0,k/2M] ∩ (0, 1), the Krasnoselskij iteration {x n } ∞

n =0 de-fined by K(x0,,T) converges strongly to the fixed point x ∗ of T, where M =1 + (2−

k + L)(L + 1);

(b) for any x0 ∈ K, the Mann iteration {x n } ∞

n =0defined by M(x0,α n,T) with {α n } ∞

n =0⊂ [0, 1) satisfying (i) lim n →∞ α n = 0 and (ii)Σ∞

n =0αn = ∞ converges strongly to the fixed point x ∗ of T;

(c) for any x0 ∈ K and for any Mann iteration {x n } ∞

n =0 defined by M(x0,α n,T) with {α n } ∞

n =0⊂ [0, 1) satisfying (i) and (ii) of (b), converging to the fixed point x ∗ of

T, there is an 0 ∈ (0, 1) such that the Krasnoselskij iteration K(x0,0,T) converges faster to the fixed point x ∗ of T Moreover, x ∗ is unique.

Proof From [7, Corollary 1], (b) follows In order to establish (c), we need the following estimates, through which (a) follows

Using Mann iterationM(x0,α n,T), from (2.7), we have

x n = x n+1+α n x n − α n Tx n = 1 +α n x n+1+α n(I − T − kI)x n+1

−(2− k)α n x n+1+α n x n+α n

so that

x n − x ∗ = 1 +α n x n+1 − x ∗ +α n(I − T − kI) x n+1 − x ∗

−(1− k)α n

x n − x ∗ + (2− k)α2

n

x n − Tx n +α n

Tx n+1 − Tx n (3.2)

Thus from (2.5), we get

x n − x ∗  ≥ 1 +α n x n+1 − x ∗ −(1− k)α nx n − x ∗

−(2− k)α2

nx n − Tx n − α nTx n+1 − Tx n. (3.3) Thus

1 +α n x n+1 − x ∗

≤1 + (1− k)α nx n − x ∗+ (2− k)α2

nx n − Tx n+α nTx n+1 − Tx n. (3.4)

We have

x n − Tx n  ≤  x n − x ∗+x ∗ − Tx n  ≤(1 +L)x n − x ∗,

Tx n+1 − Tx n  ≤ Lx n+1 − x n  = L 1− α n x n+α n Tx n − x n  ≤ L(1 + L)α nx n − x ∗.

(3.5) Thus from (3.4), (3.5), we have

1 +α n x n+1 − x ∗  ≤ 1 + (1− k)α n+ (2− k)α2

n(1 +L) + α2

n L(1 + L)·x n − x ∗.

(3.6) Now

x n+1 − x ∗  ≤1 + (1− k)α n

1 +α n + (2− k)α2

n(1 +L) + α2

n L(1 + L) ·x n − x ∗

≤1− kα n+α2

n+α2

n(1 +L)(2 − k + L)·x n − x ∗

=1− kα n+α2

n

1 + (2− k + L)(1 + L) ·x n − x ∗.

(3.7)

x n+1 − x ∗  ≤ 1− kα n+α2

n M·x n − x ∗, (3.8)

whereM =1 + (2− k + L)(1 + L).

On replacingα nbyin (3.8), we get the following estimate for the Krasnoselskij iter-ationK(x0,,T):

x n+1 − x ∗  ≤ 1− k+2M·x n − x ∗. (3.9)

Here we observe that 1− k+2M < 1 for any  < k/M Thus (a) follows.

From the elementary calculus, the function f defined on [0,1] by f ()=[1− k+

2M] has the minimum value at  = 0, where0 = k/2M.

In particular, for this0 > 0, from (3.9), we have the following estimate for the Kras-noselskij iteration:

x n+1 − x ∗  ≤ θ0 ·x n − x ∗, (3.10)

whereθ0 =1−(k0/2) (< 1).

Thus, inductively it follows that

x n+1 − x ∗  ≤ θ n

0·x1 − x ∗. (3.11)

Letη =min{k/2M,k2/2}

Since α n →0 as n → ∞, then there is a positive integer N0 such that α n < η for all

n ≥ N0

Then from (3.8), we have

x n+1 − x ∗<

1− kα n+α n ηM·x n − x ∗  ∀ n ≥ N0

≤

1− kα n+α n k

2M M ·x n − x ∗  ∀ n ≥ N0

=

1− kα n

2 ·x n − x ∗  ∀ n ≥ N0.

(3.12)

On repeating this process, we get

x n+1 − x ∗< n

i = N0

1− kα i

2 ·x N

0− x ∗  ∀ n ≥ N0. (3.13)

On comparing the coefficients of the inequalities (3.11) and (3.13) obtained through

K(x0,0,T) and M(x0,α n,T), respectively, we have, for n ≥ N0,

θ n

0

n

i = N0



1− kα i /2  ≤

1

1 +k0/2 n − N0 −→0 asn −→ ∞. (3.14) Thus byDefinition 2.2, the Krasnoselskij iteration converges faster than the Mann

Table 3.1

x n K x0 ,0 ,T K x0 ,,T

Remark 3.2 From (3.10) ofTheorem 3.1, it follows that for any ∈(0, 1) with < 0, the Krasnoselskij iterationK(x0,0,T) converges faster than K(x0,,T) to the fixed point x ∗

ofT for any x0 ∈ K This observation also is numerically shown inTable 3.1

Theorem 3.3 Let E, K, and T be as in Theorem 3.1 Suppose that {α n } ∞

n =0 and {β n } ∞

n =0

are real sequences in [0, 1) such thatΣ∞

n =0αn = ∞ and lim n →∞ α n =limn →∞ β n = 0 Then (a) for any x0 ∈ K, the Ishikawa iteration I(x0,α n,β n,T) converges strongly to the fixed point x ∗ of T, and

(b) the Mann iteration M(x0,α n,T) converges faster than the Ishikawa iteration I(x0,α n,

β n,T) to the fixed point x ∗ of T.

Proof (a) follows from [7, Theorem 1]

We now prove (b) SinceT ∈LS(K), from I(x0,α n,β n,T) defined by (2.8), we have

x n = x n+1+α n x n − α n T y n = 1 +α n x n+1+α n(I − T − kI)x n+1

−(2− k)α n x n+1+α n x n+α n

Tx n+1 − T y n

= 1 +α n x n+1+α n(I − T − kI)x n+1 −(1− k)α n x n

+ (2− k)α2

n

x n − T y n +α n

Tx n+1 − T y n

(3.15)

Hence

x n − x ∗ = 1 +α n x n+1 − x ∗ +α n(I − T − kI) x n+1 − x ∗

−(1− k)α n

x n − x ∗ + (2− k)α2

n

x n − T y n +α n

Tx n+1 − T y n (3.16)

Thus from (2.5), we get

x n − x ∗  ≥ 1 +α n x n+1 − x ∗ −(1− k)α nx n − x ∗

−(2− k)α2

nx n − T y n − α nTx n+1 − T y n. (3.17)

1 +α n x n+1 − x ∗

≤1 + (1− k)α nx n − x ∗+ (2− k)α2

nx n − T y n+α nTx n+1 − T y n. (3.18)

We have the following estimates:

y n − x ∗  ≤ 1− β n x n − x ∗+β nTx n − x ∗  ≤ 1 + (L −1)β nx n − x ∗, (3.19)

x n − T y n  ≤  x n − x ∗+x ∗ − T y n  ≤  x n − x ∗+Lx ∗ − y n

≤1 +L 1 + (L −1)β n x n − x ∗. (3.20) Also,

Tx n+1 − Tx n  ≤ Lx n+1 − y n  ≤ L 1− α n x n − y n+α nT y n − y n. (3.21)

Now

T y n − y n  ≤  T y n − x ∗+x ∗ − y n  ≤(1 +L)y n − x ∗,

x n − y n  = β nx n − Tx n  ≤(1 +L)β nx n − x ∗. (3.22) Now on substituting (3.22) in (3.21) and using (3.19), we have

Tx n+1 − T y n  ≤ L(1 +L) 1− α n β n+α n(1 +L) 1 + (L −1)β n x n − x ∗

= L(1 + L) 1− α n β n+α n

1 + (L −1)β n x n − x ∗. (3.23)

On using (3.20) and (3.23) in (3.18), we get

1 +α n x n+1 − x ∗  ≤ 1 + (1− k)α n+α2

n(2− k)1 +L 1 + (L −1)β n

+α n L(1 + L) 1− α n β n+α n

1 + (L −1)β n x n − x ∗

<1 + (1− k)α n+α2

n(2− k + L)(1 + L) + γ α n,β n,L,k x n − x ∗,

(3.24) where

γ α n,β n,L,k = α n β n L(2− k)(L −1) + (L + 1) 1− α n + (1 +L)(L −1)x n − x ∗.

(3.25) Thus

x n+1 − x ∗  ≤1 + (1− k)α n

1 +α n +α2

n(2− k + L)(1 + L) + γ α n,β n,L,k x n − x ∗

=1− kα n+α2

n+α2

n(2− k + L)(1 + L) + γ α n,β n,L,k x n − x ∗

=1− kα n+α2

n

1 + (2− k + L)(1 + L) +γ α n,β n,L,k x n − x ∗.

(3.26)

DefineM1 =(3− k + L)(1 + L).

Since

1 + (2− k + L)(1 + L) ≤ M1, (2− k)(L −1) + (L + 1) 1− α n + (1 +L)(L −1)≤ M1, (3.27)

we have

γ α n,β n,L,k ≤ α n β n LM1. (3.28) Now (3.26) becomes

x n+1 − x ∗  ≤ 1− kα n+α2

n M1+α n β n LM1x n − x ∗

=1− kα n+α n

α n+β n L M1x n − x ∗. (3.29) Sinceα n →0 asn → ∞, there is a positive integerN0such that

α n < k 0

and sinceβ n →0 asn → ∞, there is a positive integerN1such that

WriteN =max{N0,N1} Now for anyn ≥ N, (3.29) becomes

x n+1 − x ∗<1− kα n+α nk0

2M1+2k0 M1LL



M1 x n − x ∗

=1− kα n

On repeating this process, we get

x n+1 − x ∗<n

i = N



1− kα i

1− 0



x N − x ∗  ∀ n ≥ N, (3.33)

which is an estimation for the Ishikawa iterationI(x0,α n,β n,T).

On choosingβ n =0 for alln, in (3.29), we get the following estimate for Mann itera-tionM(x0,α n,T):

x n+1 − x ∗  ≤ 1− kα n+α2

n M1x n − x ∗

<1− kα n+α n M12k0 M1 x n − x ∗  ∀ n ≥ N

=

1− kα n



1− 0

2 x n − x ∗  ∀ n ≥ N.

(3.34)

On repeating process, we get

x n+1 − x ∗<n

i = N

1− kα i



1− 0

2 x N − x ∗. (3.35)

On comparing the coefficients of the inequalities (3.33) and (3.35), we get that for any

n ≥ N,

n

i = N

1− kα i

1− 0/2

n

i = N

1− kα i

1− 0

n



i = N

1− kα i 0

Since Σ∞

n =0αn = ∞, we have limn →∞n

i = N[1− kα i(0/2)] =0 Thus the Mann iteration

M(x0,α n,T) converges faster than the Ishikawa iteration I(x0,α n,β n,T) to the fixed point

Remark 3.4 Under the assumptions ofTheorem 3.1, it follows that for any Mann iter-ationM(x0,α n,T) there is a Krasnoselskij iteration K(x0,0,T) converges faster to the

fixed point ofT; and fromTheorem 3.3it follows that the Mann iterationM(x0,α n,T)

converges faster than the Ishikawa iterationI(x0,α n,β n,T) to the fixed point of T Hence

we conclude that the Krasnoselskij iteration converges faster than both the Mann and Ishikawa iterations to the fixed point ofT ∈LS(K).

4 Numerical examples

The following examples show the fastness of the movement of the first 10 iterates towards the fixed point

Example 4.1 [4] LetX =[1/2,2] and T : X → X given by Tx =1/x for all x ∈ X Then T

is Lipschitz with Lipschitzian constantL =4; and is strongly pseudocontractive with any positive constantk ∈(0, 1)

We note that Picard iteration does not converge for anyx0 =1 inX.

From Theorems3.1and3.3, we have the following

(i) The Krasnoselskij iterationK(x0,0,T) converges to the fixed point x ∗ =1, where

0 = k/2M, in which k ∈(0, 1) andM =31−5k Choosing k =62/67, we have 0 =1/57.

For this0, the Krasnoselskij iterationK(x0,0,T) is given by

x n+1 = 1

57

56x n+x −1

which converges to the fixed pointx ∗ =1

(ii) Also withα n =1/(n + 58), n =0, 1, 2, , the corresponding Mann iteration M(x0,

α n,T) is given by

x n+1 = n + 581



(n + 57)x n+x −1

n 

, n =0, 1, 2, , (4.2) which converges tox ∗ =1

Table 4.1

x n K x0 ,0 ,T M x0 ,α n,T I x0 ,α n,α n,T

(iii) The Ishikawa iterationI(x0,α n,β n,T) converges to x ∗ =1 withα n =β n =1/(n + 58),

n ≥0 In this case, the sequenceI(x n,α n,α n,T) is given by

x n+1 = n + 57

n + 58x n+(n + 57)x x n 2

n+ 1, n =0, 1, 2, (4.3) (iv) The comparison of the fastness of first 10 iterates of the Krasnoselskij, Mann, and Ishikawa iterations to the fixed pointx ∗ =1 is given inTable 4.1withx0 =1.9, and

α n =1/(n + 58) with 0 =1/57.

FromTable 4.1, we observe that the Krasnoselskij iteration moves faster towards the fixed pointx ∗ =1

(v)Table 3.1shows the comparison of first 10 iterates of Krasnoselskij iterationsK(x0,

,T) and K(x0,0,T), where  =1/114, 0 =1/57, and x0 =1.9 Here we observe that K(x0,0,T) moves faster than K(x0,,T) to the fixed point x ∗ =1 ofT (seeRemark 3.2)

Example 4.2 Let X =[0, 1] andT : X → X given by Tx =1− x2for allx ∈ X Then T is

Lipschitz, with Lipschitzian constantL =2, and is strongly pseudocontractive with any positive constantk ∈(0, 1)

(i) From Theorem 3.1, the Krasnoselskij iteration K(x0,0,T) converges to x ∗ =

(√

5−1)/2, where 0 = k/2M, k ∈(0, 1), andM =13−3k.

Letx0 =0.9 Now for k =26/27, we have 0 =1/21; thus the Krasnoselskij

iter-ationK(x0,0,T) is given by

x n+1 = 1

21



1 +x n

20− x n , n =0, 1, 2, (4.4) (ii) The Mann iterationM(x0,α n,T) converges to x ∗ =(√

5−1)/2, where α n =1/(n +

22),n =0, 1, 2, , and the Mann iteration M(x0,α n,T) is given by

x n+1 = n + 221



1 +x n

n + 21 − x n , n =0, 1, 2, (4.5)

we conclude that the Krasnoselskij iteration converges faster than both the Mann and Ishikawa iterations to the fixed point of< i>T ∈LS(K).