Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 174856, 9 pagesdoi:10.1155/2012/174856

Research Article

On Convex Total Bounded Sets in the Space of Measurable Functions

Marianna Tavernise and Alessandro Trombetta

Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende, Cosenza, Italy

Received 22 September 2010; Accepted 22 October 2011

Academic Editor: Jürgen Appell

Copyright © 2012 Marianna Tavernise and Alessandro Trombetta. 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 estimate the measure of nonconvex total boundedness in terms of simpler quantitative characteristics in the space of measurable functions  . A Fréchet-Smulian type compactness criterion for convexly totally bounded subsets of is established.

1. Introduction

In 1988, Idzik [1] proved that the answer to the well-known Schauder's problem [2, Problem 54]: does every continuous map   defined on a convex compact subset of a Hausdorff topological linear space have a fixed point? is affirmative if is convexly totally bounded.

This notion was introduced by Idzik [1]: a subset of a topological linear space is said to be convexly totally bounded (ctb for short), if for every 0-neighborhood there are and convex subsets of such that . If is locally convex, every convex compact subset of is ctb. This is not true, in general, if is nonlocally convex (see [3–5]).

In 1993, De Pascale et al. [3] defined the measure of nonconvex total boundedness, modelled on Idzik's concept, that may be regarded as the analogue of the well-known notion of Hausdorff measure of noncompactness in nonlocally convex linear spaces. The above notions of ctb set and of nonconvex total boundedness are especially useful working in the setting of nonlocally convex topological linear spaces (see, e.g., [6–8]).

Let be a normed space, a nonempty set, and the power set of . The space is an -normed linear space of -valued functions defined on which depends on an algebra in and a submeasure .

We observe that the space defined above is a generalization of the space of measurable functions introduced in [9, Chapter III], in order to develop the integration theory with respect to finitely additive measures. Recall that given , the Hausdorff measure of noncompactness of is defined by where . In [10, 11], to estimate are used two quantitative characteristics and which measure, respectively, the degree of non equi-quasiboundedness and the degree of non equi-measurability of .

The main purpose of this note is to estimate the measure of nonconvex total boundedness in and to characterize the convexly totally bounded subsets of . At this end we introduce two quantitative characteristics and involving convex sets, which measure the degree of nonconvex equi-quasiboundedness and the degree of nonconvex equi-measurability of , respectively. Then we establish some inequalities between , , , and that give, as a special case, a Fréchet-Smulian type convex total boundedness criterion in the space . This generalizes previous results of Trombetta [12]. Finally, we point out that it is not so clear if the Schauder's problem has been solved in its generality. In particular, the proof given by Cauty in [13] contains some unsolved gaps (see [14, 15], Mathscinet review of [16], and Zentralblatt Math review of [17]). However, the results of this paper are meant to be independent from the Schauder's problem.

2. Definitions and Preliminaries

For the remainder of this section we present some definitions and known results which will be needed throughout this paper.

We use the convention . Moreover, for two sets and , we denote by the set of all maps from to . We recall the definition of the space (see [11, 18]). Let be a normed space, a nonempty set, an algebra in the power set of , and a submeasure. Then defines a group pseudonorm on , that is, , , , for all . Let be the closure of the linear space of -valued -simple functions in , where is the characteristics function of . Identification of functions for which turns (the space of measurable functions) into an -normed linear space in the sense of [19, page 38], that is, , , for and .

Let . The measure of nonconvex total boundedness of is defined by Clearly is ctb if and only if .

Let be a partition of . We set In [11] the following two quantitative characteristic and are used to estimate in :

The set is called equi-quasibounded if and equi-measurable if .

Theorem 2.1 (see [11, Theorem 2.2.2]). Let . Then In particular, is totally bounded if and only if .

Moreover, in [18], it is defined the following quantitative characteristic which is useful for the calculation of : where .

The following result was established in [18, Proposition 2.1].

Proposition 2.2. Let . Then .

We omit the proof of the following proposition which is similar to the proof of Proposition 2.6 of [12].

Proposition 2.3. Let . Then

3. Inequalities in the Space

In order to estimate the measure of nonconvex total boundedness in , we introduce the following two quantitative characteristics.

Definition 3.1. Let . We define the following:

We call  convexly equi-quasibounded if and convexly equi-measurable if .

We observe that if , then the quantitative characteristics and coincide with those introduced in [12].

We point out that the request of convexity plays a crucial role in the definition of the parameters and , whereas it was not involved in the definition of and . We illustrate this with the following example.

Example 3.2. Let , where is the algebra of all Lebesgue-measurable subsets of the interval and the Lebesgue measure. Let , for , and for . If , then , , .

Proof. It is easy to check that and . We are going to prove that . On the contrary, suppose that then there exist a finite set and convex sets such that Set . We have for all . Fix , a natural number such that , and with . Put for .Then it is easy to see that there exist and a subfamily of such that for and . Moreover, a straightforward computation shows that which contradicts the convexity of the set . Since , the equality is a consequence of Theorem 3.4.

The following lemma is crucial in the proof of Theorem 3.4.

Lemma 3.3. Let be a partition of and . Then .

Proof. Obviously, . Since , it follows from Theorem 2.1 and Proposition 2.2 that . Then it is sufficient to prove the inequality which is trivial if .Assume that . By the definition of we can find a finite set , containing the origin and sets , , with for , such that(i)each for is the union of the members of a proper subfamily depending on of the partition ;(ii)for each , there is such that . For we consider the following convex subsets of : and we set We will prove that It follows that , and therefore .Let . Suppose , and , where is a proper subfamily of the partition .Put .We have Hence, for all , there is such that . Then , where Therefore, Similarly, if , we can prove that Thus, (3.7) immediately follows from (3.10) and (3.11).

We are now in a position to prove the main result of this note.

Theorem 3.4. Let . Then,

Proof. We first prove the left inequality which is trivial if . Assume that . By Proposition 2.3, there are functions and convex sets in such that Put and let be a partition of such that is constant for and . Then, hence , and .Therefore, We now prove the right inequality. Clearly, it is true if or . Assume that . By the definition of , we can find a partition and convex sets of such that Set then we have . It easy to see that Therefore, by Lemma 3.3, we have that and so The proof is complete.

As a corollary of Theorem 3.4, we obtain the following Fréchet-Smulian type convex total boundedness criterion.

Corollary 3.5. A subset of is ctb if and only if .

Remark 3.6. In [12], Trombetta proved that for a subset . Since , Theorem 3.4 improves and generalizes to -valued case the above inequalities.

We point out that the approach used in the scalar case [12] in order to prove Theorem 3.7 cannot be used in our framework. The crucial difference is in the proof of Lemma 3.3. In fact, if , it might exist some such that the set is bounded but not necessarily totally bounded.

The following example shows how the value of the parameters and changes when passing from the scalar case to the -valued case.

Example 3.7. Let , where . Then if is finite dimensional, , otherwise. Moreover, since , we have .

Corollary 3.8. Let . Then,

Proof. It is sufficient to observe that .In particular, ctb implies , , and .

The next two corollaries of Theorem 3.4 are useful in order to compute or to estimate in particular classes of subsets of . Moreover, the second one generalizes [12, Proposition 3.10].

Corollary 3.9. Let be a convexly equi-measurable subset of . Then, .

Corollary 3.10. Let be an equi-quasibounded subset of . Then,

We observe that if is a totally bounded subset of and , then, since , the inequalities (3.21) are true for .

Example 3.11. Let be the space of Example 3.2 and assume that is the Banach space of all sequences with finite norm . If and , it is easy to prove that and . If is the closed unit ball of and , the set satisfies .

References

A. Idzik, “On γ-almost fixed point theorems. The single-valued case,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 35, no. 7-8, pp. 461–464, 1987.

A. Granas, “KKK-maps and their applications to nonlinear problems,” in The Scottish Book, R. D. Mauldin, Ed., pp. 45–61, Birkhäauser, Boston, Mass, USA, 1981.

E. De Pascale, G. Trombetta, and H. Weber, “Convexly totally bounded and strongly totally bounded sets. Solution of a problem of Idzik,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 20, no. 3, pp. 341–355, 1993.

G. Trombetta, “A compact convex set not convexly totally bounded,” Polish Academy of Sciences. Bulletin. Mathematics, vol. 49, no. 3, pp. 223–228, 2001.

H. Weber, “Compact convex sets in non-locally convex linear spaces,” Note di Matematica, vol. 12, pp. 271–289, 1992.

W. K. Kim and X. Ding, “On generalized weight Nash equilibria for generalized multiobjective games,” Journal of the Korean Mathematical Society, vol. 40, no. 5, pp. 883–899, 2003.

S. Park, “Almost fixed points of multimaps having totally bounded ranges,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 51, no. 1, pp. 1–9, 2002.

S. Park and B. G. Kang, “Generalized variational inequalities and fixed point theorems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 31, no. 1-2, pp. 207–216, 1998.

N. Dunford and J. T. Schwartz, Linear Operators. Part I, Wiley-Interscience, New York, NY, USA, 1988.

J. Appell and E. De Pascale, “Some parameters associated with the Hausdorff measure of noncompactness in spaces of measurable functions,” Bollettino dell'Unione Matematica Italiana B, vol. 3, no. 2, pp. 497–515, 1984.

G. Trombetta and H. Weber, “The Hausdorff measure of noncompactness for balls of F-normed linear spaces and for subsets of ,” Bollettino dell'Unione Matematica Italiana C. Serie VI, vol. 5, no. 1, pp. 213–232, 1986.

G. Trombetta, “The measures of nonconvex total boundedness and of nonstrongly convex total boundedness for subsets of ,” Commentationes Mathematicae. Prace Matematyczne, vol. 40, pp. 191–207, 2000.

R. Cauty, “Solution du problème de point fixe de Schauder,” Fundamenta Mathematicae, vol. 170, no. 3, pp. 231–246, 2001.

R. Cauty, “Un théorème de point fixe pour les fonctions multivoques acycliques,” in Functional Analysis and Its Applications, vol. 197 of North-Holland Mathematics Studies, pp. 71–80, Elsevier, Amsterdam, The Netherlands, 2004.

G. Isac, “Erratum: on Rothe's fixed point theorem in a general topological vector space,” Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, vol. 14, no. 1, p. 109, 2006.

Y. Askoura and C. Godet-Thobie, “Fixed points in contractible spaces and convex subsets of topological vector spaces,” Journal of Convex Analysis, vol. 13, no. 2, pp. 193–205, 2006.

S. Park, “Remarks on recent results in analytical fixed point theory,” in Nonlinear Analysis and Convex Analysis, pp. 517–525, Yokohama Publishers, Yokohama, Japan, 2007.

A. Avallone and G. Trombetta, “Measures of noncompactness in the space and a generalization of the Arzelà-Ascoli theorem,” Bollettino dell'Unione Matematica Italiana B, vol. 5, no. 3, pp. 573–587, 1991.

H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, Germany, 1981.