# zbMATH — the first resource for mathematics

On convex total bounded sets in the space of measurable functions. (English) Zbl 1242.46037
Summary: We estimate the measure of nonconvex total boundedness in terms of simpler quantitative characteristics in the space of measurable functions $$L_0$$. A Fréchet-Smulian type compactness criterion for convexly totally bounded subsets of $$L_0$$ is established.
##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
measure of nonconvex total boundedness
Full Text:
##### References:
 [1] A. Idzik, “On \gamma -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. · Zbl 0663.47036 [2] 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. [3] 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. · Zbl 0805.47055 [4] G. Trombetta, “A compact convex set not convexly totally bounded,” Polish Academy of Sciences. Bulletin. Mathematics, vol. 49, no. 3, pp. 223-228, 2001. · Zbl 1015.46003 [5] H. Weber, “Compact convex sets in non-locally convex linear spaces,” Note di Matematica, vol. 12, pp. 271-289, 1992. · Zbl 0846.46004 [6] 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. · Zbl 1106.91005 [7] 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. · Zbl 1005.47054 [8] 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. · Zbl 0912.49006 [9] N. Dunford and J. T. Schwartz, Linear Operators. Part I, Wiley-Interscience, New York, NY, USA, 1988. [10] 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. · Zbl 0507.46025 [11] G. Trombetta and H. Weber, “The Hausdorff measure of noncompactness for balls of F-normed linear spaces and for subsets of L0,” Bollettino dell’Unione Matematica Italiana C. Serie VI, vol. 5, no. 1, pp. 213-232, 1986. · Zbl 0657.47050 [12] G. Trombetta, “The measures of nonconvex total boundedness and of nonstrongly convex total boundedness for subsets of L0,” Commentationes Mathematicae. Prace Matematyczne, vol. 40, pp. 191-207, 2000. · Zbl 1047.46004 [13] R. Cauty, “Solution du problème de point fixe de Schauder,” Fundamenta Mathematicae, vol. 170, no. 3, pp. 231-246, 2001. · Zbl 0983.54045 [14] 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. · Zbl 1088.55001 [15] 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. [16] 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. · Zbl 1101.54039 [17] 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. · Zbl 1119.47054 [18] A. Avallone and G. Trombetta, “Measures of noncompactness in the space L0 and a generalization of the Arzelà-Ascoli theorem,” Bollettino dell’Unione Matematica Italiana B, vol. 5, no. 3, pp. 573-587, 1991. · Zbl 0753.47022 [19] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, Germany, 1981. · Zbl 0466.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.