The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping

Abstract

The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let be a Hilbert space, let be a nonempty bounded closed convex subset of and let be a strongly ergodic matrix. If is a lipschitzian mapping such that , then the set of fixed points is a retract of . This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1].

1. Introduction

Let be a Banach space and let be a nonempty bounded closed convex subset of . We say that a mapping is nonexpansive if

(1.1)

The result of Bruck [1] asserts that if a nonexpansive mapping has a fixed point in every nonempty closed convex subset of which is invariant under and if is convex and weakly compact, then , the set of fixed points, is nonexpansive retract of (i.e., there exists a nonexpansive mapping such that ). A few years ago, the Bruck results were extended by Domínguez Benavides and Lorenzo Ramírez [2] to the case of asymptotically nonexpansive mappings if the space was sufficiently regular.

On the other hand it is known that, the set of fixed points of -lipschitzian mapping can be very irregular for any .

Example 1.1 (Goebel [3, 4]).

Let be a nonempty closed subset of . Fix , and put

(1.2)

It is not difficult to see that and the Lipschitz constant of tends to if .

For more information on the structure of fixed point sets see [4, 5] and references therein.

In 1973, Goebel and Kirk [3] introduced the class of uniformly -lipschitzian mappings, recall that a mapping is uniformly-lipschitzian, , if

(1.3)

and proved the following theorem.

Theorem 1.2.

Let be a uniformly convex Banach space with modulus of convexity and let be a nonempty bounded closed convex subset of . Suppose that is uniformly -lipschitzian and

(1.4)

Then has a fixed point in . Note that in a Hilbert space, .

Recently Sędak and Wiśnicki [6] proved that under the assumptions of Theorem 1.2 ,is not only connected but even a retract of, and next the author proved the following theorem [7, Corollary  9].

Theorem 1.3.

Let be a Hilbert space, a nonempty bounded closed convex subset of and a uniformly - lipschitzian mapping with . Then has a fixed point in and is a retract of .

In this paper we shall continue this work. Precisely, by means of techniques of asymptotic centers and the methods of Hilbert spaces, we establish some result on the structure of fixed point sets for mappings with lipschitzian iterates in a Hilbert space. The class of mappings with lipschitzian iterates is importantly greater than the class of uniformly lipschitzian mappings; see [8, Example  1].

2. Asymptotic Center

Denote by the Lipschitz norm of :

(2.1)

Lifshitz [9] significantly extended Goebel and Kirk's result and found an example of a fixed point free uniformly lipschitzian mapping which leaves invariant a bounded closed convex subset of . The validity of Lifshitz's Theorem in a Hilbert space for remains open.

A more general approach was proposed by the present author using the methods of Hilbert spaces, asymptotic techniques, and strongly ergodic matrix. We recall that a matrix is called strongly ergodic if

(i)for all ,

(ii)for all ,

(iii)for all ,

(iv).

Then we have the following theorem.

Theorem 2.1 (see [8]).

Let be a nonempty bounded closed convex subset of a Hilbert space and let be a strongly ergodic matrix. If is a mapping such that

(2.2)

then has a fixed point in .

This result generalizes Lifshitz's Theorem (in case of a Hilbert space) and shows that the theorem admits certain perturbations in the behavior of the norm of successive iterations in infinite sets; see [8, Example  1].

Let be a Banach space. Recall that the modulus of convexity is the function defined by

(2.3)

and uniform convexity means for . A Hilbert space is uniformly convex. This fact is a direct consequence of parallelogram identity.

Now we prove some version of Sędak and Wiśnicki's result [6, Lemma  2.1]. Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a strongly ergodix matrix, and let be a mapping such that for all , and

(2.4)

Let we use

(2.5)

to denote the asymptotic radius of at and the asymptotic radius of in , respectively. It is well known in a Hilbert space [8] that the asymptotic center of in :

(2.6)

is a singleton.

Let denote a mapping which associates with a given a unique , that is, . The following Lemma is a crucial tool to prove Theorem 4.1.

Lemma 2.2.

Let be a Hilbert space and let be a nonempty bounded closed convex subset of . Then the mapping is continuous.

Proof.

On the contrary, suppose that there exists and such that for all there exists such that and , where .

Fix and take such that

(2.7)

Let , and . Notice that

(2.8)

Choose . Then

(2.9)

for all but finitely many .

If, for example, for all everyone , then

(2.10)

Multiplying both sides of this inequality (for fixed ) by suitable element of the matrix and summing up such obtained inequalities for , we have for

(2.11)

Taking the limit superior as on each side, we get

(2.12)

which is contradiction.

It follows by (2.9) and the properties of that

(2.13)

Multiplying both sides of this inequality by suitable elements of the matrix and summing up such obtained inequalities for , taking the limit superior as on each side, we get

(2.14)

Moreover,

(2.15)

Multiplying both sides of this inequality by suitable elements of the matrix and summing up such obtained inequalities for , taking the limit superior as on each side, we get

(2.16)

Similarly,

(2.17)

From (2.16) and (2.17), we have

(2.18)

If , then from (2.18) it follows . This is contradiction with (2.8). If , then combining (2.18) with (2.14) and applying the monotonicity of , we obtain

(2.19)

Letting and using the continuity of , we conclude that

(2.20)

This contradiction proves the continuity of mapping .

3. The Methods of Hilbert Spaces

Let , be as above. We define functionals

(3.1)

where . Let in be an asymptotic center of with respect to and , which minimizes the functional over in (for fix ).

Lemma 3.1.

One has .

Proof.

It is consequence of the above definitions.

Lemma 3.2.

One has

Proof.

For any , we have

(3.2)

Multiplying both sides of this inequality by suitable elements of the matrix and summing up such obtained inequalities for , taking the limit superior as on each side, we get

(3.3)

Lemma 3.3.

One has for all .

Proof.

Fix , then we have

(3.4)

Since the matrix is strongly ergodic,

(3.5)

as , we get thesis.

Lemma 3.4.

One has for every .

Proof.

For and , we have

(3.6)

Multiplying both sides of this inequality by suitable elements of the matrix and summing up such obtained inequalities for , taking the limit superior as on each side, we get

(3.7)

Since , we obtain

(3.8)

Taking , we get, .

4. Main Result

We are now in position to prove our main result.

Theorem 4.1.

Let be a nonempty bounded closed convex subset of a Hilbert space and let be a strongly ergodic matrix. If is a mapping such that

(4.1)

then is a retract of .

Proof.

Let and be sequences of natural numbers such that

(4.2)

By Theorem 2.1, . For any we can inductively define a sequence in the following manner: is the unique point in that minimizes the functional

(4.3)

over and is the unique point in that minimizes the functional

(4.4)

over , that is, , First we prove the following inequality:

(4.5)

where

(4.6)

and is the asymptotic center in which minimizes the functional

(4.7)

over in .

In fact, we put in Lemma 3.4. Then by Lemma 3.3, we get

(4.8)

For we have

(4.9)

and hence

(4.10)

Next by Lemma 3.2 and inequality (4.5), we have

(4.11)

where for , Thus

(4.12)

which implies that the sequence converges uniformly to a function

(4.13)

It follows from Lemma 2.2 that is continuous. Moreover,

(4.14)

Multiplying both sides of this inequalities by suitable elements of the matrix and summing up such obtained inequalities for , taking the limit superior as on each side, we get

(4.15)

Thus, . This implies that see [8] for details. Thus for every and is a retraction of onto .

If is the Cesaro matrix, that is, for

(4.16)

then we have the following corollary.

Corollary 4.2.

Let be a nonempty bounded closed convex subset of a Hilbert space. If is a mapping such that

(4.17)

then is a retract of .