Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T07:37:51.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity properties of the superposition operator

Part of: Linear function spaces and their duals Classical measure theory Functions of several variables

Published online by Cambridge University Press:  09 April 2009

Jürgen Appell  and
Pjotr P. Zabrejko
Jürgen Appell
Affiliation:
Institut für Mathematik Universität WürzburgAm Hubland D-8700 Würzburg, West Germany
Pjotr P. Zabrejko
Affiliation:
Mech.-Mat. Fakultet Belgosuniversitet Minsk Pl. Lenina 1 SU-220080 Minsk, USSR
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26B40: Representation and superposition of functions 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 186 - 210
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Andô, T., ‘On products of Orlicz spaces’, Math. Ann. 140 (1960), 174–186.CrossRefGoogle Scholar
[2]Appell, J. and Zabrejko, P. P., ‘Boundedness properties of the superposition operator,’ Bull. Polish Acad. Sci. (to appear).Google Scholar
[3]Bondarenko, V. A., ‘Cases of degeneracy of the superposition operator in function spaces’ (Russian), Jaroslav. Gos. Univ. Kach. Priblizh. Metody Issled. Oper. Uravn. I (1976), 11–16.Google Scholar
[4]Buttazzo, G. and Maso, G. Dal, ‘On Nemyckii operators and integral representations of local functionals,’ Rend. Mat. 3 (1983), 491–509.Google Scholar
[5]Carathéodory, K., Vorlesungen über reelle Funktionen, (De Gruyter, Leipzig-Berlin, 1918).Google Scholar
[6]Dunford, N. and Pettis, B. J., ‘Linear operators on summable functions,’ Trans. Amer. Math. Soc. 47 (1940), 323–392.CrossRefGoogle Scholar
[7]Dunford, N. and Schwartz, J., Linear operators. I, (Interscience, Leyden, 1963).Google Scholar
[8]Giner, E., Etudes sur les fonctionelles intégrales, (Doctoral Dissertation, Univ. Pau. 1985).Google Scholar
[9]Grande, Z., ‘La measurabilité des fonctions de deux variables et de la superposition F(x, f(x)),’ Dissertationes Math. Warszawa 159 (1979), 1–45.Google Scholar
[10]Grande, Z. and Lipiński, J. S., ‘Un exemple d'une fonction sup-mesurable qui n'est pas mesurable,’ Colloq. Math. 39 (1978), 77–79.CrossRefGoogle Scholar
[11]Kantorovich, L. V., Vulih, B. Z. and Pinsker, A. G., Functional analysis in semi-ordered spaces (Russian), Gostehizdat, Moskva, 1950.Google Scholar
[12]Karták, K., ‘A generalization of the Carathéodory theory of differential equations,’ Czechoslovak Math. J. 17 (1967), 482–514.CrossRefGoogle Scholar
[13]Karták, K., ‘On Carathéodory operators,’ Czechoslovak Math. J. 17 (1967), 515–519.CrossRefGoogle Scholar
[14]Kirpotina, N. V., ‘On the continuity of the Nemytskij operator in Banach spaces,’ (Russian), Metody Reor. Diff. Uravn. Prilozh. Moskva 339 (1975), 76–81.Google Scholar
[15]Kozlowski, W., ‘Nonlinear operators in Banach function spaces’, Comm. Math. Prace Mat. 22 (1980), 85–103.Google Scholar
[16]Krasnosel′skij, M. A., ‘On the continuity of the operator Fu(x) = f(x, u(x)),’ (Russian), Dokl. Akad. Nauk SSSR 77 (1951), 185–188.Google Scholar
[17]Krasnosel′skij, M. A., Topological methods in the theory of nonlinear integral equations (Russian), Gostehizdat, Moskva, 1956 [English transl.: Macmillian, New York, 1964].Google Scholar
[18]Krasnosel′skij, M. A. and Ladyzhenskij, L. A., ‘Conditions for the complete continuity of the P. S. Uryson operator’ (Russian), Trudy Moskov. Mat. Obshch. 3 (1954), 307–320.Google Scholar
[19]Krasnosel′skij, M. A. and Pokrovskij, A. V., ‘On a discontinuous superposition operator’ (Russian), Uspehi Mat. Nauk 32 (1977), 169–170.Google Scholar
[20]Krasnosel′skij, M. A. and Pokrovskij, A. V., ‘Equations with discontinuous nonlinearities’ (Russian), Dokl. Akad. Nauk SSSR 248 (1979), 1056–1059Google Scholar
[= Soviet Math. Dokl. 20 (1979), 1117–1120].Google Scholar
[21]Krasnosel′skij, M. A. and Rutitskij, Ja. B., Convex functions and Orlicz spaces (Russian), Fizmatgiz, Moskva, 1958 [English transl.: Noordhoff, Groningen, 1961].Google Scholar
[22]Krasnosel′skij, M. A., Rutitskij, Ja. B. and Sultanov, R. M., ‘On a nonlinear operator which acts in a space of abstract functions’ (Russian), Izv. Akad. Nauk Azerbajdzh. SSR, Ser. Fiz.-Teh. Mat. Nauk 3 (1959), 15–21.Google Scholar
[23]Krasnosel′skij, M. A., Zabrejko, P. P., Pustyl′nik, Je. I. and Sobolevskij, P. Je., Integral operators in spaces of summable functions (Russian), Nauka, Moskva, 1966 [English transl.: Noordhoff, Leyden, 1976].Google Scholar
[24]Mänz, P., ‘On the continuity of the superposition operator acting in normed product spaces’ (Russian), Litovsk. Mat. Sb. 7 (1967), 289–296.Google Scholar
[25]Makarov, A. S., ‘Continuity criteria for the superposition operator,’ (Russian), Prilozh. Funk. Anal. Priblizh. Vychisl. Kazan. Univ. (1974), 80–83.Google Scholar
[26]Obradovich, P. M., ‘On the continuity of the superposition operator’ (Russian), Voronezh. Gos. Univ. Probl. Mat. Anal. Slozhn. Sistem 2 (1968), 78–80.Google Scholar
[27]Ponosov, A. V., ‘On the Nemytskij conjecture’ (Russian), Dokl. Akad. Nauk SSSR 289 (1986), 1308–1311. [Soviet Math. Dokl. 34 (1987), 231–233.]Google Scholar
[28]Rutitskij, Ja. B., ‘On a nonlinear operator in Orlicz spaces’ (Ukrainian), Dopovidi Akad. Nauk Ukr. SSSR 3 (1952), 161–163.Google Scholar
[29]Sainte-Beuve, M. F., ‘On the extension of von Neumann-Aumann's theorem’, J. Funct. Anal. 17 (1974), 112–129.CrossRefGoogle Scholar
[30]Shragin, I. V., ‘On the weak continuity of the Nemytskij operator’ (Russian), Uchen. Zapiski Moskov. Obl. Ped. Inst. 57 (1957), 73–79.Google Scholar
[31]Shragin, I. V., ‘On the weak continuity of the Nemytskij operator in generalized Orlicz spaces’ (Russian), Uchen. Zapiski Moskov. Obl. Ped. Inst. 77 (1959), 169–179.Google Scholar
[32]Shragin, I. V., ‘On the continuity of the Nemytskij operator in Orlicz spaces’ (Russian), Uchen. Zapiski Moskov. Obl. Ped. Inst. 70 (1959), 49–51.Google Scholar
[33]Shragin, I. V., ‘On the continuity of the Nemytskij operator in Orlicz spaces’ (Russian), Dokl. Akad. Nauk SSSR 140, 3 (1961), 543–545Google Scholar
[= Soviet Math. Dokl. 2 (1961), 1246–1248].Google Scholar
[34]Shragin, I. V., On the continuity of the Nemytskij operator (Russian), Trudy 5-j Vsjesojuznoj Konf. Funk. Anal. Akad. Nauk Azerbajdzh. SSR Baku (1961), 272–277.Google Scholar
[35]Shragin, I. V., ‘On the continuity of the Nemytskij operator in Orlicz spaces’ (Russian), Kishin. Gos. Univ. Zapiski 70 (1964), 49–51.Google Scholar
[36]Shragin, I. V., ‘Conditions for the measurability of superpositions’ (Russian), Dokl. Akad. Nauk SSSR 197 (1971), 295–298Google Scholar
[= Soviet Math. Dokl. 12 (1971) 465–470],Google Scholar
Erratum: Dokl. Akad. Nauk SSSR 200 (1971), vii.Google Scholar
[37]Shragin, I. V., ‘The superposition operator in modular function spaces’ (Russian), Studia Math. 43 (1972), 61–75.Google Scholar
[38]Shragin, I. V., ‘Superpositional measurability’ (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. I (1975), 82–89.Google Scholar
[39]Shragin, I. V., ‘Abstract Nemytskij operators are locally defined operators’ (Russian), Dokl. Akad. Nauk SSSR 227 (1976), 47–49Google Scholar
[= Soviet Math. Dokl. 17 (1976), 354–357].Google Scholar
[40]Shragin, I. V., ‘On the continuity of locally defined operators’ (Russian), Dokl. Akad. Nauk SSSR 232 (1977), 292–295Google Scholar
[= Soviet Math. Dokl. 18 (1977), 75–78].Google Scholar
[41]Shragin, I. V., ‘The necessity of the Carathéodory conditions for the continuity of the Nemytskij operator’ (Russian), Perm'. Gos. Polit. Inst., Funkc.-Diff. Krajev. Zad. Mat. Fiz. (1978), 128–134.Google Scholar
[42]Shragin, I. V., ‘On the Carathéodory conditions’ (Russian), Uspehi Mat. Nauk 34 (1979), 219–220Google Scholar
[= Russian Math. Surveys 34 (1970), 220–221].Google Scholar
[43]Vajnberg, M. M., ‘On the continuity of some special operators’ (Russian), Dokl. Akad. Nauk SSSR 73 (1950), 253–255.Google Scholar
[44]Vajnberg, M. M., Variational methods in the study of nonlinear operators (Russian), Gostehizdat, Moskva, 1956Google Scholar
[English transl.: Holden-Day, San Francisco-London-Amsterdam, 1964].Google Scholar
[45]Vajnberg, M. M. and Shragin, I. V., The Nemytskij operator in generalized Orlicz spaces (Russian), Uchen. Zapiski Moskov. Obl. Ped. Inst. 77 (1959), 145–159.Google Scholar
[46]Vrkoč, I., ‘The representation of Carathéodory operators’, Czechoslovak Math. J. 19 (1969), 99–109.CrossRefGoogle Scholar
[47]Zabrejko, P. P., ‘Nonlinear integral operators’ (Russian), Voronezh. Gos. Univ. Trudy Sem. Funk. Anal. 8 (1966), 1–148.Google Scholar
[48]Zabrejko, P. P., On the theory of integral operators in ideal function spaces (Russian), (Doctoral Dissertation, Univ. Voronezh, 1968).Google Scholar
[49]Zabrejko, P. P., ‘Ideal function spaces. I’ (Russian), Jaroslav. Gos. Univ. Vestnik 8 (1974), 12–52.Google Scholar
[50]Zabrejko, P. P., Koshelev, A. I., Krasnosel′skij, M. A., Mihlin, S. G., Rakovshchik, L. S. and Stetsenko, V. Ja., Integral equations (Russian), Nauka, Moskva, 1968Google Scholar
[English transl.: Noordhoff, Leyden, 1975].Google Scholar
[51]Zabrejko, P. P. and Pustyl′nik, Je. I., ‘On the continuity and complete continuity of nonlinear integral operators in Lp spaces’ (Russian), Uspehi Mat. Nauk 19 (1964), 204–205.Google Scholar