# Simpler is also better in approximating fixed points.

**ABSTRACT** In this paper we demonstrate that a number of fixed point iteration problems can be solved using a modified Krasnoselskij iteration process, which is much simpler to use than the other iteration schemes that have been defined.

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**ABSTRACT:**We show that the convergence of Mann, Ishikawa iterations are equivalent to the convergence of a multistep iteration, for various classes of operators.Nonlinear Analysis 01/2004; · 1.64 Impact Factor - SourceAvailable from: sciencedirect.com
##### Article: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings

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**ABSTRACT:**In this paper, weak and strong convergence theorems are established for a three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces. Mann-type and Ishikawa -type iterations are included by the new iterative scheme. The results obtained in this paper extend and improve the recent ones announced by Xu and Noor, Glowinski and Le Tallec, Noor, Ishikawa, and many others.Journal of Mathematical Analysis and Applications 01/2005; · 1.05 Impact Factor - SourceAvailable from: journals.cambridge.org[Show abstract] [Hide abstract]

**ABSTRACT:**Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.Bulletin of the Australian Mathematical Society 01/1991; 43(01):153 - 159. · 0.48 Impact Factor

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Simpler is also better in approximating fixed points

M. Abbasa, Safeer Hussain Khanb, B.E. Rhoadesc,*

aCentre for Advanced Studies in Mathematics and Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan

bDepartment of Mathematics and Physics, Qatar University, 2713 Doha, Qatar

cDepartment of Mathematics, Indiana University, Bloomington, IN 47405-7106, United States

a r t i c l ei n f o

Keywords:

Asymptotically nonexpansive

Fixed points

Krasnoselskij

a b s t r a c t

In this paper we demonstrate that a number of fixed point iteration problems can be solved

using a modified Krasnoselskij iteration process, which is much simpler to use than the

other iteration schemes that have been defined.

? 2008 Elsevier Inc. All rights reserved.

For maps with a slow enough growth rate, the Banach contraction principle provides the existence and uniqueness of the

fixed point, which can be obtained by repeated function iteration, or Picard iteration. For other maps, such as nonexpansive

maps, pseudocontractive maps, etc., some other iteration process is required in order to obtain weak or strong convergence

to a fixed point.

Let X denote a Banach space, T a self-map of X, or of some closed convex subset of X. In 1955 Krasnoselskij [12] used the

averaging process

x02 X;

In 1957, Schaefer [35] extended the iteration scheme of Krasnoselskij to

xnþ1¼ ðxnþ TxnÞ=2:

x02 X;

xnþ1¼ ð1 ? kÞxnþ kTxn;

ð1Þ

for some 0;< k < 1. However, this method is usually referred to as generalized Krasnoselskij, or simply Krasnoselskij.

In 1953 Mann [15] defined an iteration scheme which can be written in the form

x02 X;

xnþ1¼ ð1 ?anÞxnþanTxn;

ð2Þ

where 0 6 an6 1 andPan< 1. Thus every Krasnoselskij iteration is a Mann iteration. However, in some applications, see,

Hillam [9], has shown that, if f is a Lipschitz self-map of an interval [a,b], with Lipschitz constant L, then Krasnoselskij iter-

ation, with k = 1/(L + 1), converges monotonically to a fixed point of f.

Ishikawa [11] defined the iteration scheme

e.g., Theorem 3.2 of [4], it is also required that liman¼ 0. Hence the Mann iteration scheme is a more general one. However,

x02 X;

xnþ1¼ ð1 ?anÞxnþanTyn;

yn¼ ð1 ? bnÞxnþ bnTxn;

ð3Þ

where

0 6 an6 bn6 1;

lim bn¼ 0;

Xanbn¼ 1:

ð4Þ

Ishikawa used this method to obtain a fixed point theorem for Lipschitz pseudocontractive maps. It remained an open

question for a number of years as to whether one could use Mann iteration for these problems. In 2001 [7] an example

was provided of a Lipschitzian pseudocontractive map for which (3) converges, but no Mann iteration converges, thereby

0096-3003/$ - see front matter ? 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2008.08.021

* Corresponding author. Tel.: +1 812 855 1358; fax: +1 812 855 0046.

E-mail addresses: mujahid@lums.edu.pk (M. Abbas), Safeer@qu.edu.qa (S.H. Khan), rhoades@indiana.edu (B.E. Rhoades).

Applied Mathematics and Computation 205 (2008) 428–431

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amc

Page 2

reinforcing the conjecture that Ishikawa iteration is more general. Unfortunately condition (4) shows that Ishikawa iteration

does not formally include Mann iteration, since setting each bn¼ 0 forces each anto be zero. In [22] Rhoades modified the

inequality in (4) to

0 6 an;bn6 1:

ð5Þ

This modified scheme then formally contains Mann iteration as a special case.

For the past 30 years many fixed point papers have been written involving either the Mann and/or the (modified) Ishik-

awa methods. Almost every paper has used the Ishikawa iteration method with condition (5, instead of (4).

Rhoades and Soltuz [23–34] have written a series of papers establishing the equivalence of Mann and (modified) Ishikawa

iteration for various families of maps, and Chang et al. [6] have proved the equivalence of modified Mann and modified Ishi-

kwa iterations for asymptotically nonexpansive maps with bounded range.

For the past 10 years a number of papers have appeared using Mann or Ishikawa iteration with errors - either of the type

defined by Liu [13] or Xu [41]. Iterations with errors of the type of Liu require that the norms of the error terms be in ‘1, not a

realistic assumption. Using the error definition of Xu, if the bounded sequences are arbitrary, then the iteration cannot be

performed. Therefore we will not consider any iteration process with errors.

In the past few years a number of papers have appeared involving more and more complicated iteration schemes, without

any regard to the added complexity of computation that these methods require.

This paper has been written in an attempt to reverse this trend, and to underscore the fact that, from a computational

standpoint, when more than one iteration procedure is available, one should always select the simplest one.

For example, Berinde [1] has shown that Krasnoselskij iteration can be used to obtain strong convergence to the fixed

point for Lipschitzian strongly pseudocontractive maps. He has also obtained the value of k which gives the fastest such iter-

ation. (See [2].)

A map T on a space X is said to be asymptotically nonexpansive if there exists a sequence fkng ? ½1;1Þ with

Pðkn? 1Þ < 1 such that, for each n 2 N,

kTnx ? Tnyk 6 knkx ? yk:

A map T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that, for all x;y 2 X,

kTnx ? Tnyk 6 Lkx ? yk;

for all n 2 N.

Obviously every asymptotically nonexpansive map is uniformly L-Lipschitzian for some L.

In this paper we show that the modified Krasnoselskij iteration defined by (6) converges strongly or weakly to a fixed

point for asymptotically nonexpansive maps. Throughout this paper X will denote a uniformly convex Banach space, C is

a closed convex subset of X, and T is an asymptotically nonexpansive self-map of C. In some cases it will also be assumed

that C is bounded. Let 0 < k < 1;x02 C, and xnþ1defined by

xnþ1¼ ð1 ? kÞxnþ kTnxn:

ð6Þ

Theorem 1. Suppose that T is completely continuous. Then fxng, defined by (6), converges strongly to a fixed point of T.

Proof. In Theorem 4.3 of [21] set bðiÞ

n¼ aðjÞ

n ¼ bðjÞ

n¼ 0 for i P 1;j > 1;að1Þ

n ¼ k;bð1Þ

n

¼ 1 ? k.

h

In the statement of Theorem 4.3 of [21] it is also assumed that T is L-Lipschitzian, but that is automatically true, since T is

asymptotically nonexpansive. Alternatively, in Theorem 10 of [42] set an¼ bn¼ cn¼ bn¼ cn¼ 0;an¼ k. In Theorem 9 of [18]

set an¼ bn¼ cn? bn¼ cn¼ 0;an¼ k.

If C is also bounded, then, from Theorem 1 of [8], T has a fixed point.

Corollary 1. Suppose that T is completely continuous and C is bounded. Then fxng, defined by (6), converges strongly to a fixed

point of T.

Corollary 1 is a direct consequence of Theorem 1. It can be proved directly by substituting bn¼ cn¼ 0;an¼ k in Theorem

2.1 of [40]. Also it can be proved by setting an¼ bn¼ cn¼ bn¼ 0;an¼ k in Theorem 2.3 of [39], or by setting

an¼ bn¼ cn¼ bn¼ cn¼ ln¼ kn¼ 0;an¼ k in [16], or, in Theorem 1 of [10] set un¼ 0;an¼ k.

Corollary 2. Suppose that X is a Hilbert space, T is completely continuous, and C is bounded. Then fxng, defined by (6), converges

strongly to a fixed point of T.

Proof. The result follows immediately from Corollary 1. It can also be proved directly by setting an¼ k in Theorem 1.5 of

[37].

h

Theorem 2. Suppose that C is bounded. Then fxng, defined by (6), converges strongly to a fixed point of T.

Proof. In Theorem 1.2 of [5] set cn¼ 0;an¼ k. Or, in Theorem 4 of [44], set cn¼ b0

n¼ c0

n¼ 0;a0

n¼ 1;an¼ 1 ? k;bn¼ k.

h

M. Abbas et al./Applied Mathematics and Computation 205 (2008) 428–431

429

Page 3

Let fung be a sequence in C;T is a self-map of C with FðTÞ, the fixed point set T, nonempty. Then T is said to satisfy con-

dition (A) with respect to fung [38] if there exists a nondecreasng function f : ½0;1Þ ! ½0;1Þ with fð0Þ ¼ 0;fðrÞ > 0 for

r 2 ð0;1Þ, and such that fðdðun;FðTÞÞ 6 kun? Tunk for all n P 1.

Theorem 3. Suppose that T has FðTÞ–; and T satisfies condition (A) with respect to fxng. Then fxng, defined by (6), converges

strongly to a fixed point of T.

Proof. In Theorem 2.1 of [45] set an¼ k. In Theorem 7 of [17] set an¼ bn¼ cn¼ bn¼ cn¼ 0;an¼ k.

h

Theorem 4. Let C be bounded, T is uniformly continuous. Then fxng, defined by (6), converges strongly to a fixed point of T.

Proof. In Theorem 1.1 of [19] set bn¼ cn¼ 0;an¼ k.

Note that, in the statement of Theorem 1.1 of [19], the conditionPkn< 1 should readPðkn? 1Þ < 1.

Theorem 5. Suppose that C is bounded and that Tmis compact for some m 2 N. Then fxng, defined by (6), converges strongly to a

fixed point of T.

h

Proof. In Theorem 2.2 of [36] set an¼ k. Alternatively, set bn¼ b0

A map T is said to be asymptotically quasi-nonexpansive if it satisfies the definition of being asymptotically nonexpansive

at each fixed point of T.

n¼ cn¼ c0

n¼ 0;an¼ k in Theorem 2 of [14].

h

Theorem 6. Let T be a completely continuous asymptotically quasi-nonexpansive self-map of C. Then fxng, defined by (6),

converges strongly to a fixed point of T.

Proof. In Theorem 2.3 of [20] set a00

n¼ c00

n¼ a0

n¼ c0

n;¼ 0;b00

n¼ b0

n¼ 1;an¼ k;bn¼ 1 ? k.

h

Modified Krasnoselskij iteration (6) can also be used to establish weak convergence in certain circumstances. The liter-

ature contains many such results. We list here just one.

Theorem 7. Suppose that X satisfies Opial’s condition. Then fxng, defined by (6), converges weakly to a fixed point of T.

Proof. In Theorem 3.1 of [14] set bn¼ b0

In [43] the concept of a generalized asymptotically nonexpansive mapping was defined. A map T on a space X with FðTÞ–;

is said to be a generalized asymptotically nonexpansive mapping if there exist two null sequences frng;fsng such that

kTnx ? pk 6 ð1 þ rnÞkx ? pk þ snkx ? Tnxk

for all n P 1;x 2 X and p 2 FðTÞ.

Note that the above inequality can be written in the form

n¼ cn¼ c0

n¼ 0;an¼ k.

h

kTnx ? pk 6 knkx ? pk;

where kn¼ ð1 þ rnþ s þ nÞ=ð1 ? snÞ. Since fsng is a null sequence, one may assume that each sn< 1. Therefore, if T is a gen-

eralized asymptotically nonexpansive mapping, it is quasi-asymptotically nonexpansive. Since every asymptotically nonex-

pansive mapping with a fixed point is quasi-asymptotically nonexpansive, the results of [43] are not generalizations of

known results.

The reference list is not complete. For other papers dealing with (6) and other iteration processes, the reader should con-

sult the excellent bibliography in [3].

Question. Does there exist an asymptotically nonexpansive map, with nonempty fixed point set, for which (6), for some

0 < k < 1, does not apply?

In future papers we will investigate some form of Krasnoselskij iteration for other classes of maps, and families of maps.

Acknowledgement

The authors thank the referee for his careful reading of the manuscript and for his suggestions.

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