International Journal of Mathematics and Mathematical Sciences

Volume 2008 (2008), Article ID 630589, 6 pages

http://dx.doi.org/10.1155/2008/630589

## Characterization for the Convergence of Krasnoselskij Iteration for Non-Lipschitzian Operators

^{1}Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no. 18A-10, Bogota, Colombia^{2}“Tiberiu Popoviciu” Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania^{3}Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Received 13 August 2007; Revised 2 February 2008; Accepted 24 February 2008

Academic Editor: Enrico Obrecht

Copyright © 2008 Ştefan M. Şoltuz and B. E. Rhoades. 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.

#### Abstract

We establish the convergence of Krasnoselskij iteration for various classes of non-Lipschitzian operators.

#### 1. Introduction

Let be a real Banach space; a nonempty, convex subset of and an operator. Let The following iteration is known as Krasnoselskij iteration (see [1]):

The map given by , for all is called *the
normalized duality mapping*. It is easy to see that we have

Denote*Definition 1.1. * Let
be a real
Banach space, and let
be a nonempty
subset of . A map is called uniformly pseudocontractive if there exists a map and such
that A map is called
uniformly accretive if there exists a map and such
that

Taking for all , reduces to the
usual definitions of -strongly
pseudocontractive and -strongly
accretive. Taking , , for all , we get the
usual definitions of strongly pseudocontractive and strongly accretive.
Therefore, the class of strongly pseudocontractive maps is included stricly in
the class of -strongly
pseudocontractive maps. The *example* from [2] shows that this inclusion is
proper. Remark, further, that the class of -strongly
pseudocontractive maps is also included strictly in the class of uniformly
pseudocontractive maps (see also [3]).

We will give a characterization for the convergence
of (1.1) when applied to
uniformly pseudocontractive operators. For this purpose, we need the following
lemma similar to [4, Lemma 1]. Next, denotes the set
of all natural numbers.Lemma 1.2. * Let be a positive
bounded sequence and assume that there exists such that ** where , , for all and Then **Proof. *There exists an such that , for all . Denote We will prove
that Suppose on the
contrary that Then there
exists an such
thatFrom we know that
there exists an such
thatSet Using the fact
that , we get the
following:which implies that orsince Thus which implies
that in
contradiction to Therefore, Hence there
exists a subsequence such that Fix . Then there exists an such
thatAlso there exists an such
thatDefine We claim that for each and each Suppose not.
Then there exists an and a such thatFor this let denote the
smallest positive integer for which (1.13) is true. Then

From (1.6), which implies that This leads to
the contradiction:Therefore, for all , and each hence

#### 2. Main Result

Theorem 2.1. *Let be a real
Banach space, a nonempty,
closed, convex, bounded subset of . Let be a uniformly
pseudocontractive and uniformly continuous operator with . Then for the
Krasnoselskij iteration (1.1) converges to
the fixed point of if and only if **Proof. *Since is a self-map
of which is
bounded and convex, then, from (1.1), each so is bounded for
each Uniqueness of
the fixed point follows from (1.4). If converges to
the fixed point of that is, then,
obviously, Conversely, we
will prove that if , then Suppose that for some Then from (1.1), it follows that for each and the theorem
is proved. Now suppose that for each Using (1.1) and (1.2),

HenceSince and is uniformly
continuous, it follows that Set , and use Lemma 1.2
to obtain the conlcusion.*Remark 2.2. *(1) If is not bounded,
then Theorem 2.1 holds under the assumption that is bounded.

(2) If is bounded,
then is bounded.

(3) If is strongly
pseudocontractive, then automatically

#### 3. Further Results

Let denote the
identity map. A map is called
pseudocontractive if there exists such that *Remark 3.1. *The operator is a (uniformly, strongly) pseudocontractive map if and only if is a (uniformly, strongly) accretive map.*Remark 3.2. *(1) Let and let be given. A
fixed point for the map , for all , is a solution
for

(2) Let be a given
point. If is an accretive
map, then is a strongly
pseudocontractive map.

Consider Krasnoselskij iteration with Remarks 3.1 and 3.2 and Theorem 2.1
lead to the following result.Corollary 3.3. *Let be a real
Banach space and let be a uniformly
accretive and uniformly continuous operator, with bounded.
Suppose that has a solution.
Then for any the
Krasnoselskij iteration (3.1) converges to
the solution of if and only if *

Let be an accretive
operator. The operator is strongly
pseudocontractive for a given A solution for becomes a
solution for Consider
Krasnoselskij iteration with Again, using Remarks 3.1 and 3.2
and Theorem 2.1, we obtain the following result.Corollary 3.4. *Let be a real
Banach space and let be an accretive
and uniformly continuous operator, with bounded.
Suppose that has a solution.
Then for the
Krasnoselskij iteration (3.2) converges to
the solution of if and only if **Remark 3.5. *If (1.4) holds for all and then such a map
is called *uniformly hemicontractive*. It is trivial to see that our
results hold for the uniformly hemicontractive maps.

#### Acknowledgment

The authors are indebted to referee for carefully reading the paper and for making useful suggestions.

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