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

DenoteDefinition 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.