Accessible Requires Authentication Published by De Gruyter November 30, 2017

Monotonicity and total boundedness in spaces of “measurable” functions

Diana Caponetti, Alessandro Trombetta and Giulio Trombetta
From the journal Mathematica Slovaca

Abstract

We define and study the moduli d(x, 𝓐, D) and i(x, 𝓐,D) related to monotonicity of a given function x of the space L0(Ω) of real-valued “measurable” functions defined on a linearly ordered set Ω. We extend the definitions to subsets X of L0(Ω), and we use the obtained quantities, d(X) and i(X), to estimate the Hausdorff measure of noncompactness γ(X) of X. Compactness criteria, in special cases, are obtained.


Cordially dedicated to Professor Paolo de Lucia on the occasion of his 80th birthday anniversary

Communicated by Anna De Simone


References

[1] Akhmerov, R. R.—Kamenskii, M. I.—Potapov, A. S.—Rodkina, A. E.—Sadovskii, B. N.: Measures of Noncompactness and Condensing Operators, Birkhäuser Verlag, Basel, Boston and Berlin, 1992. Search in Google Scholar

[2] Appell, J.—Banaś, J.—Merentes, N.: Some quantities related to monotonicity and bounded variation of functions, J. Math. Anal. Appl. 367 (2010), 476–485.10.1016/j.jmaa.2010.02.004 Search in Google Scholar

[3] Appell, J.—De Pascale, E.: Some parameters associated with the Hausdorff measure of noncompactness in spaces of measurable functions, Boll. Un. Mat. Ital. (6) 3–B (1984), 497–515. Search in Google Scholar

[4] Appell, J.—Zabreiko, P. P.: Nonlinear Superposition Operators. Cambridge Tracts in Math. 95, Cambridge University Press, 1990. Search in Google Scholar

[5] Avallone, A.—Trombetta, G.: Measures of noncompactness in the space L0 and a generalization of the Arzelà-Ascoli theorem, Boll. Un. Mat. Ital. (7) 5–B (1991), 573–587. Search in Google Scholar

[6] Ayerbe Toledano, J. M.—Domínguez Benavides, T.—López Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, Basel, 1997. Search in Google Scholar

[7] Banaś, J.—Goebel, K.: Measures of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math. 60, Dekker, New York, 1980. Search in Google Scholar

[8] Banaś, J.—Olszowy, L.: Measures of noncompactness related to monotonicity, Comment. Math. 41 (2001), 13–23. Search in Google Scholar

[9] Banaś, J.—Sadarangani, K.: On some measures of noncompactness in the space of continuous functions, Nonlinear Anal. 68 (2008), 377–383.10.1016/j.na.2006.11.003 Search in Google Scholar

[10] Caponetti, D.—Lewicki, G.—Trombetta, G.: Control functions and total boundedness in the space Lo, Novi Sad J. Math. 32 (2002), 109–123. Search in Google Scholar

[11] Dunford, N.—Schwartz, J. T.: Linear Operators. Part I: General Theory, Wiley Classics Library, New York, 1988. Search in Google Scholar

[12] Tavernise, M.—Trombetta, A.: On convex total bounded sets in the space of measurable functions, J. Funct. Spaces Appl. (2012), Art. ID 174856, 9 pp. Search in Google Scholar

[13] Tavernise, M.—Trombetta, A.—Trombetta, G.: Total boundedness in vector-valued F-seminormed function spaces, Le Matematiche 66 (2011), 171–179. Search in Google Scholar

[14] Tavernise, M.—Trombetta, A.—Trombetta, G.: A remark on the lack of equi-measurability for certain sets of Lebesgue-measurable functions, Math. Slovaca 67 (2017), 1595–1601. Search in Google Scholar

[15] Trombetta, G.—Weber, H.: The Hausdorff measures of noncompactness for balls in F-normed linear spaces and for subsets of L0, Boll. Un. Mat. Ital. (6) 5–C (1986), 213–232. Search in Google Scholar

Received: 2016-6-16
Accepted: 2016-10-26
Published Online: 2017-11-30
Published in Print: 2017-11-27

© 2017 Mathematical Institute Slovak Academy of Sciences