On regulated functions

• Autorzy: Cichoń K. , Cichoń M. , Satco B.

Identyfikatory

DOI

10.1515/fascmath-2018-0003

• Języki publikacji: EN

Abstrakty

EN

In this paper we investigate the space of regulated functions on a compact interval [0,1]. When equipped with the topology of uniform convergence this space is isometrically isomorphic to some space of continuous functions. We study some of its properties, including the characterization of the dual space, weak and strong compactness properties of sets. Finally, we investigate some compact and weakly compact operators on the space of regulated functions. The paper is complemented by an existence result for the Hammerstein-Stieltjes integral equation with regulated solutions.

Słowa kluczowe

EN

regulated function; discontinuous functions; compactness

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2018
• Tom: Nr 60
• Strony: 37--57
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Cichoń K.

• Institute of Mathematics Faculty of Electrical Engineering Poznan University of Technology Piotrowo 3A, 60-965 Poznań, Połand

autor

Cichoń M.

• Faculty of Mathematics and Computer Science A. Mickiewicz University Umultowska 87, 61-614 Poznań, Połand

autor

Satco B.

• Stefan CelMare University of Suceava Faculty of Electrical Engineering and Computer Science Integrated Center for Research Deyelopment and Innovation in Adyanced Materials Nanotechnologies and Distributed Systems for Fabrication and Control (MANSiD) Universitatii 13 - Suceava, Romania

Bibliografia

• [1] Appell J., Chen Ch.-J., How to solve Hammerstein equations, Jour. Integral Equat. Appl., 18(2006), 287-296.
• [2] Aziz W., Merentes N., Sánchez J.L., A note on the composition of regular functions, Z. Anal. Anwend., 20(2014), 1-5.
• [3] Batt J., Berg E.J., Linear bounded transformations on the space of continuous functions, Jour. Funct. Anal., 4(1969), 215-239.
• [4] Berberian S., The character space of the algebra of regulated functions, Pacific Jour. Math., 74(1978), 15-36.
• [5] Bombal F., Rodríguez-Salinas B., Some classes of operators on C(K, E). Extension and applications, Archiv Math., 47(1986), 55-65.
• [6] Brokate M., Krejčí P., Duality in the space of regulated functions and the play operator, Math. Zeitschrift, 245(2003), 667-688.
• [7] Brooks J.K., Lewis P.W., Linear operators and vector measures, Trans. Amer. Math. Soc., 192(1974), 139-162.
• [8] Cembranos P., Mendoza J., Banach Spaces of Vector-Valued Functions, Lecture Notes in Mathematics 1676, Springer, 1997.
• [9] Cichoń M., Satco B., Measure differential inclusions-between continuous and discrete, Adv. Difference Equ., 2014(1)(2014), 1-18.
• [10] De Marco G., Representing the algebra of regulated functions as an algebra of continuous functions, Rend. Mat. Univ. Padova, 84(1990), 195-199.
• [11] Diestel J., Uhl J.J., Jr, Vector Measures, AMS, Providence, Rhode Island, 1977.
• [12] Dieudonné J., Foundation of Modern Analysis, Academic Press, New York, 1960.
• [13] Dobrakov I., On representation of linear operators on Co(T, X), Czechoslovak Math. J, 20(1971), 13-30.
• [14] Dunford N., Schwartz J.T., Linear Operators, vol. I, Interscience, New York, 1958.
• [15] Fernandes L.A.O., Arbach R., Regulated functions with values in the Banach algebra of quaternions, Proceedings of the World Congress on Engineering, Vol. 1, London, 2011.
• [16] Fernandes L.A.O., Arbach R., Integral functionals on C*-algebra of vector-valued regulated functions, Ann. Funct. Anal., 3(2012), 21-31.
• [17] Fraňková D., Regulated functions, Math. Bohem., 116(1991), 20-59.
• [18] Fraňková D., Regulated functions with values in Banach space. I. Uniform convergence, preprint.
• [19] Gordon R.A., The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. in Math., 4, Amer. Math. Soc., 1994.
• [20] Goffman C., Moran G., Waterman D., The structure of regulated functions, Proc. Amer. Math. Soc., 57(1976), 61-65.
• [21] Hönig C.S., Volterra-Stieltjes Integral Equations, North-Holland, 1975.
• [22] Kalenda O., Stegall compact spaces which are not fragmentable, Topology Appl., 96(1999), 121-132.
• [23] Kaltenborn H.S., Linear functional operations on functions having discontinuities of the first kind, Bull. Amer. Math. Soc., 40(1934), 702-708.
• [24] Lin P.-K., Kothe-Bochner Function Spaces, Springer, Springer, Berlin, 2004.
• [25] Michalak A., On superposition operators in spaces of regular and of bounded variation functions, Z. Anal. Anwend., 35(2016), 285-308.
• [26] Michalak A., On superposition operators in spaces BV_ϕ (0,1), Jour. Math. Anal. Appl., 443(2016), 1370-1388.
• [27] Monteiro G.A., Slavik A., Extremal solutions of measure differential equations, J. Math. Anal. Appl., 444(2016), 568-597.
• [28] Monteiro G.A., Tvrdý M., On Kurzweil-Stieltjes integral in a Banach space, Math. Bohemica, 137(2012), 365-381.
• [29] Nowak M., Completely continuous operators and the strict topology, Indag. Math., 28(2017), 541-555.
• [30] Saab P., Integral operators on spaces of continuous vector-valued functions, Proc. Amer. Math. Soc., 111(1991), 1003-1013.
• [31] Satco B., Continuous dependence results for set-valued measure differential problems, Electr. Jour. Qualit. Th. Diff. Equat., 79(2015), 1-15.
• [32] Schwabik Š., Generalized ordinary differential equations, World Scientific, 1992.
• [33] Schwabik Š., Linear operators in the space of regulated functions, Math. Bohemica, 117(1992), 79-92.
• [34] Schwabik Š., Linear Stieltjes integral equations in Banach spaces, Math. Bohemica, 124(1999), 433-457.
• [35] Schwabik Š., Tvrdý M., Vejvoda O., Differential and Integral Equations. Boundary Problems and Adjoints Academia and D. Reidel, Praha, Dordrecht, 1979.
• [36] Tvrdý M., Differential and Integral Equations in the Space of Regulated Functions, Mem. Differential Equations Math. Phys., 25(2002), 1-104.
• [37] Ülger A., Weak compactness in (L¹ μ, X), Proc. Amer. Math. Soc., 113(1991), 143-149.
• [38] Zavalishchin S.T., Sesekin A.N., Discontinuous Solutions to Ordinary Nonlinear Differential Equations in the Space of Functions of Bounded Variation, in: Dynamic Impulse Systems, Springer, Netherlands, 1997.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).