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S¸oltuz Received 20 June 2007; Accepted 14 September 2007 Recommended by Hichem Ben-El-Mechaiekh We prove the equivalence between theT-stabilities of the Krasnoselskij and the Mann itera

Volume 2007, Article ID 60732, 7 pages

doi:10.1155/2007/60732

Research Article

The Equivalence between T-Stabilities of

The Krasnoselskij and The Mann Iterations

S¸tefan M S¸oltuz

Received 20 June 2007; Accepted 14 September 2007

Recommended by Hichem Ben-El-Mechaiekh

We prove the equivalence between theT-stabilities of the Krasnoselskij and the Mann

iterations; a consequence is the equivalence with theT-stability of the Picard-Banach

iteration

Copyright © 2007 S¸tefan M S¸oltuz 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

LetX be a normed space and T a selfmap of X Let x0be a point ofX, and assume that

x n+1 = f (T,x n) is an iteration procedure, involvingT, which yields a sequence { x n }of points fromX Suppose { x n }converges to a fixed pointx ∗ofT Let { ξ n }be an arbitrary sequence inX, and set  n =  ξ n+1 − f (T,ξ n)for alln ∈ N

Definition 1.1 [1] If (limn→∞  n =0)⇒(limn→∞ ξ n = p), then the iteration procedure

x n+1 = f (T,x n) is said to beT-stable with respect to T.

Remark 1.2 [1] In practice, such a sequence{ ξ n }could arise in the following way Letx0

be a point inX Set x n+1 = f (T,x n) Letξ0= x0 Nowx1= f (T,x0) Because of rounding

or discretization in the functionT, a new value ξ1approximately equal tox1might be obtained instead of the true value of f (T,x0) Then to approximatex2, the value f (T,ξ1)

is computed to yieldξ2, an approximation of f (T,ξ1) This computation is continued to obtain{ ξ n }an approximate sequence of{ x n }

LetX be a normed space, D a nonempty, convex subset of X, and T a selfmap of D, let

p0= e0∈ D The Mann iteration (see [2]) is defined by

e n+1 =1− α n

e n+α n Te n, (1.1)

where{ α n } ⊂(0, 1) The Ishikawa iteration is defined (see [3]) by

x n+1 =1− α n

x n+α n T y n,

y n =1− β n

x n+β n Tx n, (1.2) where{ α n } ⊂(0, 1),{ β n } ⊂[0, 1) The Krasnoselskij iteration (see [4]) is defined by

p n+1 =(1− λ)p n+λT p n, (1.3) whereλ ∈(0, 1) Recently, the equivalence between theT-stabilities of Mann and Ishikawa

iterations, respectively, for modified Mann-Ishikawa iterations was shown in [5] In the present paper, we shall prove the equivalence between theT-stabilities of the

Krasnosel-skij and the Mann iterations Next,{ u n },{ v n } ⊂ X are arbitrary.

Definition 1.3.

(i) The Mann iteration (1.1) is said to beT-stable if and only if for all { α n } ⊂(0, 1) and for every sequence{ u n } ⊂ X,

lim

n→∞ ε n =0=⇒lim

n→∞ u n = x ∗, (1.4) whereε n:=  u n+1 −(1− α n)u n − α n Tu n 

(ii) The Krasnoselskij iteration (1.3) is said to beT-stable if and only if for all λ ∈

(0, 1), and for every sequence{ v n } ⊂ X,

lim

n→∞ δ n =0=⇒lim

n→∞ v n = x ∗, (1.5) whereδ n:=  v n+1 −(1− λ)v n − λTv n 

2 Main results

Theorem 2.1 Let X be a normed space and T : X → X a map with bounded range and

{ α n } ⊂ (0, 1) satisfy lim n→∞ α n = λ, λ ∈ (0, 1) Then the following are equivalent:

(i) the Mann iteration is T-stable,

(ii) the Krasnoselskij iteration is T-stable.

Proof We prove that (i) ⇒(ii) If limn→∞ δ n =0, then{ v n }is bounded Set

M1:=max



sup

x∈X { T(x) }, v0, u0



Observe that v1 ≤δ0+ (1− λ)  v0+λ  Tv0 ≤δ0+M1 SetM : = M1+ 1/λ Suppose

that v n  ≤ M to prove that  v n+1  ≤ M Remark that

v n+1  ≤ δ n+ (1− λ)δ n−1+···+ (1− λ) n δ0+M1

≤1 + (1− λ) + ···+ (1− λ) n+M1

1−(1− λ)+M1= M.

(2.2)

Suppose that limn→∞ δ n =0 to note that

ε n =v n+1 −

1− α n

v n − α n Tv n

=v n+1 − v n+λv n − λv n+α n v n − λTv n+λTv n − α n Tv n

≤v n+1 −(1− λ)v n − λTv n+λ − α nv n − Tv n

≤v n+1 −(1− λ)v n − λTv n+ 2Mλ − α n

= δ n+ 2Mλ − α n  −→0 asn −→ ∞

(2.3)

Condition (i) assures that if limn→∞ ε n =0, then limn→∞ v n = x ∗ Thus, for a{ v n } satisfy-ing

lim

n→∞ δ n =lim

n→∞v n+1 −(1− λ)v n − λTv n  =0, (2.4)

we have shown that limn→∞ v n = x ∗

Conversely, we prove (ii)⇒(i) First, we prove that{ u n }is bounded Since limn→∞ α n =

λ, for β ∈(0, 1) given, there existsn0∈ N, such that 1 − α n ≤ β, for all n ≥ n0 SetM1:=

max{supx∈X  Tx , u0}andM : = n0+ 1 +β/(1 − β) + M1to obtain

u n+1  ≤  ε n+

1− α1



ε n−1+

1− α1



1− α2



ε n−2

+···+

1− α1



1− α2



···1− α n0



ε n−n0

+

1− α1



1− α2



···1− α n0



1− α n0 +1



ε n−n0−1

+···+

1− α1



1− α2



···1− α n

ε0+M1

≤n0+ 1

+

1− α n0 +1 

+

1− α n0 +1 

1− α n0 +2 

···

+

1− α n0 +1



···1− α n

ε0+M1

≤ n0+ 1 +β + β2+···+β n−n0+M1< M.

(2.5)

Suppose limn→∞ ε n =0 Observe that

δ n =u n+1 −(1− λ)u n − λTu n

=u n+1 − u n+λu n − λTu n+α n u n − α n u n − α n Tu n+α n Tu n

≤u n+1 −

1− α n

u n − α n Tu n+λ − α nu n − Tu n

≤u n+1 −

1− α n

u n − α n Tu n+ 2Mλ − α n

= ε n+ 2Mλ − α n  −→0 asn −→ ∞

(2.6)

Condition (ii) assures that if limn→∞ δ n =0, then limn→∞ v n = x ∗ Thus, for a{ u n } satis-fying

lim

n→∞ ε n =lim

n→∞u n+1 −

1− α n

u n − α n Tu n  =0, (2.7)

we have shown that limn→∞ u n = x ∗ 

Remark 2.2 Let X be a normed space and T : X → X a map with bounded range and

{ α n } ⊂(0, 1) satisfy limn→∞ α n = λ, λ ∈(0, 1) If the Mann iteration is notT-stable, then

the Krasnoselskij iteration is notT-stable, and conversely.

Example 2.3 Let T : [0,1) →[0, 1) be given byTx = x2, andλ =1/2 Then the

Krasnosel-skij iteration converges to the unique fixed pointx ∗ =0, and it is notT-stable.

The Krasnoselskij iteration converges because, supposingF : =supn p n <1, the sequence

p n →0, as we can see from

p n+1 = 1−1

2

p n+1

2p2

n =1

2p n+1

2p2

n

=1

2p n

1 +p n

≤1 +F

2 p n = 1 +F

2

n

p0−→0;

(2.8)

setv n = n/(n + 1) and note that v ndoes not converge to zero, whileδ ndoes:

δ n =

n + 1 n + 2 −1

2

n

n + 1 −

1 2

n2

(n + 1)2



 =2(n + 1) n2+ 42n + 2(n + 2) −→0. (2.9)

The Mann iteration also converges because (supposingE : =supn e n < 1) one has

e n+1 =1− α n

e n+α n e2

n =1−(1− E)α n

e n

≤n

k=1



1−(1− E)α k

e0≤exp −(1− E) n

k=1

α k

e0−→0; (2.10) the last inequality is true because 1− x ≤exp(− x), ∀ x ≥0, and

α n =+∞ Takeu n = n/(n + 1) →1, and note thatε n →0 because

ε n =

n + 1 n + 2 −1− α n n

n + 1 − α n n

2

(n + 1)2



 = α n n2+

2α n+ 1

n + 1

(n + 1)2(n + 2) . (2.11)

So the Mann iteration is notT-stable Actually, by use ofTheorem 2.1, one can easily obtain the non-T-stability of the other iteration, provided that the previous one is not

stable

The following result takes in consideration the case in which no condition on{ α n }are imposed

Theorem 2.4 Let X be a normed space and T : X → X a map, and { α n } ⊂ (0, 1) If

lim

n→∞v n − Tv n  =0, lim

n→∞u n − Tu n  =0, (2.12)

then the following are equivalent:

(i) the Mann iteration is T-stable,

(ii) the Krasnoselskij iteration is T-stable.

Proof We prove that (i) ⇒(ii) Suppose limn→∞ δ n =0, to note that,

ε n =v n+1 −

1− α n

v n − α n Tv n

=v n+1 − v n+λv n − λv n+α n v n − λTv n+λTv n − α n Tv n

≤v n+1 −(1− λ)v n − λTv n+λ − α nv n − Tv n

≤ δ n+ 2v n − Tv n  −→0 asn −→ ∞

(2.13)

Condition (i) assures that if limn→∞ ε n =0, then limn→∞ v n = x ∗ Thus, for a{ v n } satisfy-ing

lim

n→∞ δ n =lim

n→∞v n+1 −(1− λ)v n − λTv n  =0, (2.14)

we have shown that limn→∞ v n = x ∗

Conversely, we prove (ii)⇒(i) Suppose limn→∞ ε n =0 Observe that

δ n =u n+1 −(1− λ)u n − λTu n

=u n+1 − u n+λu n − λTu n+α n u n − α n u n − α n Tu n+α n Tu n

≤u n+1 −(1− α n)u n − α n Tu n+λ − α nu n − Tu n

≤ ε n+ 2u n − Tu n  −→0 asn −→ ∞

(2.15)

Condition (ii) assures that if limn→∞ δ n =0, then limn→∞ v n = x ∗ Thus, for a{ u n } satis-fying

lim

n→∞ ε n =lim

n→∞u n+1 −

1− α n

u n − α n Tu n  =0, (2.16)

we have shown that limn→∞ u n = x ∗ 

Remark 2.5 Let X be a normed space and T : X → X a map, { α n }⊂(0, 1) and limn→∞  v n −

Tv n  =0, limn→∞  u n − Tu n  =0 If the Mann iteration is notT-stable, then the

Kras-noselskij iteration is notT-stable, and conversely.

Note that one can consider the usual conditionsλ =1/2, limα n =0, and

α n = ∞in

Theorem 2.4andRemark 2.5

Example 2.6 Again, let T : [0,1) →[0, 1) be given byTx = x2, andλ =1/2, α n =1/n Set

v n = u n = n/(n + 1), to note that lim n→∞ u n =1, and

lim

n→∞v n − Tv n  =lim

n→∞

n

Hence, neither the Mann nor the Krasnoselskij iteration isT-stable, as we can see from

Example 2.3

3 Further results

Letq0∈ X be fixed, and let q n+1 = Tq nbe the Picard-Banach iteration

Definition 3.1 The Picard iteration is said to be T-stable if and only if for every sequence

{ q n } ⊂ X given,

lim

n→∞Δn =0=⇒lim

n→∞ q n = x ∗, (3.1) whereΔn:=  q n+1 − Tq n 

In [6], the equivalence between theT-stabilities of Picard-Banach iteration and Mann

iteration is given, that is, the following holds

Theorem 3.2 [6] Let X be a normed space and T : X → X a map If

lim

n→∞q n − Tq n  =0, lim

n→∞u n − Tu n  =0, (3.2)

then the following are equivalent:

(i) for all { α n } ⊂ (0, 1), the Mann iteration is T- stable,

(ii) the Picard iteration is T-stable.

Theorems2.4and3.2lead to the following conclusion

Corollary 3.3 Let X be a normed space and T : X → X a map If

lim

n→∞q n − Tq n  =0, lim

n→∞v n − Tv n  =0, lim

n→∞u n − Tu n  =0, (3.3)

then the following are equivalent:

(i) for all { α n } ⊂ (0, 1), the Mann iteration is T-stable,

(ii) the Picard-Banach iteration is T-stable,

(iii) the Krasnoselskij iteration is T-stable.

Remark 3.4 Let X be a normed space and T : X → X a map, { α n } ⊂(0, 1) and limn→∞  q n − Tq n  =0, limn→∞  v n − Tv n  =0, limn→∞  u n − Tu n  =0 If the Mann or Krasnoselskij iteration is notT-stable, then the Picard-Banach iteration is not T-stable,

and conversely

Example 3.5 To see that the Picard-Banach iteration is also not T-stable, consider T :

[0, 1)→[0, 1), byTx = x2

Indeed, settingq n = n/(n + 1), we have

lim

n→∞ q n =lim

n→∞

n

n + 1 =1,

lim

n→∞



n + 1 n − n

n + 1

2 

 =(n + 1) n 2 =0.

(3.4)

The author is indebted to referee for carefully reading the paper and for making useful suggestions

References

[1] A M Harder and T L Hicks, “Stability results for fixed point iteration procedures,” Mathemat-ica JaponMathemat-ica, vol 33, no 5, pp 693–706, 1988.

[2] W R Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical So-ciety, vol 4, no 3, pp 506–510, 1953.

[3] S Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol 44, no 1, pp 147–150, 1974.

[4] M A Krasnosel’ski˘ı, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol 10, no 1(63), pp 123–127, 1955.

[5] B E Rhoades and S¸ M S¸oltuz, “The equivalence between the T-stabilities of Mann and Ishikawa iterations,” Journal of Mathematical Analysis and Applications, vol 318, no 2, pp 472–

475, 2006.

[6] S¸ M S¸oltuz, “The equivalence between theT-stabilities of Picard-Banach and Mann-Ishikawa iterations,” to appear in Applied Mathematics E—Notes.

S¸tefan M S¸oltuz: Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no 18A-10, Bogota, Colombia

Current address: Tiberiu Popoviciu Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania Email address:smsoltuz@gmail.com

then the following are equivalent:

(i) the Mann iteration is T-stable,

(ii) the Krasnoselskij. .. the equivalence between the< i >T-stabilities of Picard-Banach iteration and Mann< /i>

iteration is given, that is, the following holds

Theorem 3.2 [6] Let X be a normed space and. .. neither the Mann nor the Krasnoselskij iteration isT-stable, as we can see from

Example 2.3