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On measures of σ-noncompactess in F-spaces

  • Diana Caponetti EMAIL logo , Alessandro Trombetta and Giulio Trombetta
Published/Copyright: December 12, 2025
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Abstract

Given an F-space (X, τ) and στ a linear topology on X, this paper provides the definition of a general set function that allows us to define measures of σ-noncompactness in X. In particular, we construct measures of nonconvex σ-noncompactness that are monotonic, invariant under passage to the closed convex hull, and satisfy the Cantor intersection property. Additionally, we derive a fixed point theorem for maps that satisfy the classical Darbo condition with respect to a given measure of nonconvex σ-noncompactness.

Funding statement: This research has been accomplished within the UMI Group TAA Approximation Theory and Applications, the GNAMPA of INdAM. The first author has been supported by FFR-2024, University of Palermo.

Acknowledgement

The authors would like to thank the anonymous referee for carefully reading the manuscript and providing useful comments that improved the presentation of this paper.

  1. (Communicated by David Buhagiar)

References

[1] Ayerbe Toledano, J. M.—Dominguez Benavides, T.—Lopez Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, Basel, 1997.10.1007/978-3-0348-8920-9Search in Google Scholar

[2] Banaś, J.—Goebel, K.: Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Appl. Math., Marcel Dekker, Vol. 60, New York and Basel, 1980.Search in Google Scholar

[3] Banaś, J.—Rivero, J.: On measures of weak noncompactness, Ann. Mat. Pura Appl. (4) 151 (1988), 213–224.10.1007/BF01762795Search in Google Scholar

[4] Cauty, R.: Solution du problème de point fixe de Schauder, Fund. Math. 170(3) (2001), 231–246 (in French).10.4064/fm170-3-2Search in Google Scholar

[5] Cauty, R.: Rétractes absolus de voisinage algébriques, Serdica Math. J. 31(4) (2005), 309–354 (in French).Search in Google Scholar

[6] Cauty, R.: Le théorème de Lefschetz-Hopf pour les applications compactes des espaces ULC, J. Fixed Point Theory Appl. 1(1) (2007), 123–134 (in French).10.1007/s11784-006-0002-5Search in Google Scholar

[7] Cauty, R.: Points fixes des applications compactes dans les espaces ULC, https://arxiv.org/abs/1010.2401v1Search in Google Scholar

[8] De Blasi, F. S.: On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Math. Roum. 21 (1977), 259–262.Search in Google Scholar

[9] De Pascale, E.—Trombetta, G.—Weber, H.: Convexly totally bounded and strongly totally bounded sets. Solution of a problem of Idzik, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20(3) (1993), 341–355.Search in Google Scholar

[10] Goldstein, L. S.—Gohberg, I. T.—Markus, A. S.: Investigation of some properties of bounded linear operators in connection with their q-norms, Ucen. Zap. Kishinevskogo Univ. 29 (1957), 29–36 (in Russian).Search in Google Scholar

[11] Hadžić, O.: Some properties of measures of noncompactness in paranormed spaces, Proc. Amer. Math. Soc. 102(4) (1988), 843–849.10.1090/S0002-9939-1988-0934854-1Search in Google Scholar

[12] Idzik, A.: Almost fixed point theorems, Proc. Amer. Math. Soc. 104 (1988), 779–784.10.1090/S0002-9939-1988-0964857-2Search in Google Scholar

[13] Isac, G.: Leray–Schauder Type Alternatives, Complementarity Problems and Variational Inequalities. Non-convex Optimization and Its Applications, Vol. 87, Springer, New York, 2006.Search in Google Scholar

[14] Kalton, N. J.: Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. (2) 19(2) (1974/75), 151–167.10.1017/S0013091500010282Search in Google Scholar

[15] Kalton, N. J.–Peck, N. T.—Rogers, J. W.: An F-Space Sampler. London Math. Lecture Notes, Vol. 89, Cambridge Univ. Press, Cambridge, 1985.10.1017/CBO9780511662447Search in Google Scholar

[16] Kalton, N. J.—Shapiro, J. H.: Bases and basic sequences in F-spaces, Studia Math. 56(1) (1976), 47–61.10.4064/sm-56-1-47-61Search in Google Scholar

[17] Mauldin, R. D.: The Scottish Book, Mathematics from the Scottish Café with Selected Problems from the new Scottish Book, 2nd ed., Birkhäuser, 2015.10.1007/978-3-319-22897-6Search in Google Scholar

[18] Narici, L.—Beckenstein, E.: Topological Vector Spaces, Marcel Dekker, Inc., New York, 1985.Search in Google Scholar

[19] O’Regan, D.: A homotopy theory for maps having strongly convexly totally bounded ranges in topological vector spaces, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 32(2) (2024), 117–125.10.2478/auom-2024-0022Search in Google Scholar

[20] Park, S.—Park, J. A.: The Idzik type quasivariational inequalities and noncompact optimization problems, Colloq. Math. 71 (1996), 287–295.10.4064/cm-71-2-287-295Search in Google Scholar

[21] Rolewicz, S.: Metric Linear Spaces. Mathematics and Its Applications, 2nd ed., East European Series, Vol. 20, Reidel, Dordrecht, 1985.Search in Google Scholar

[22] Schauder, J.: Der Fixpunktsatz in Funktionalraumen, Stud. Math. 2 (1930), 171–180.10.4064/sm-2-1-171-180Search in Google Scholar

[23] Tychonoff, A.: Ein Fixpunktsatz, Math. Ann. 111 (1935), 767–776.10.1007/BF01472256Search in Google Scholar

[24] Trombetta, G.: A compact convex set not convexly totally bounded, Bull. Polish. Acad., Sci. Math. 49 (2001), 223–228.Search in Google Scholar

[25] Weber, H.: Compact convex sets in non-locally convex linear spaces, Schauder-Tychonoff fixed point theorem. In: Topology, Measure and Fractals (Warnemünde, 1991) (C. Bandt et al., eds.), Math. Res. 66 Akademie-Verlag, Berlin, 1992, pp. 37–40.Search in Google Scholar

[26] Weber, H.: Compact convex sets in non-locally-convex linear spaces, Note di Mat. 12 (1992), 271–289.Search in Google Scholar

Received: 2025-07-23
Accepted: 2025-09-29
Published Online: 2025-12-12
Published in Print: 2025-12-17

© 2025 Mathematical Institute Slovak Academy of Sciences

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