Measurability and continuity conditions for nonlinear evolutionary processes

Ball, John M.

doi:10.1090/S0002-9939-1976-0415445-X

Measurability and continuity conditions for nonlinear evolutionary processes

Author: John M. Ball
Journal: Proc. Amer. Math. Soc. 55 (1976), 353-358
MSC: Primary 47H99; Secondary 54H15
DOI: https://doi.org/10.1090/S0002-9939-1976-0415445-X
MathSciNet review: 0415445
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper generalizes to nonlinear evolutionary processes on a metric space the well-known results connecting measurability and continuity properties with respect to time of linear semigroups of continuous operators on a Banach space.

[1] A. Alexiewicz and W. Orlicz, Remarque sur l’équation fonctionelle 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), Fund. Math. 33 (1945), 314–315 (French). MR 16510, https://doi.org/10.4064/fm-33-1-314-315
[2] H. Auerbach, Sur la relation ${\lim _{{h_n} \to 0}}f(x + {h_n}) = f(x)$ , Fund. Math. 11(1928), 193-197.
[3] J. M. Ball, Continuity properties of nonlinear semigroups, J. Functional Analysis 17 (1974), 91–103. MR 0348575, https://doi.org/10.1016/0022-1236(74)90006-8
[4] S. Banach, Sur l'équation fonctionelle , Fund. Math. 1(1920), 123-124.
[5] P. R. Chernoff, A note on separate and joint continuity (to appear).
[6] P. Chernoff and J. Marsden, On continuity and smoothness of group actions, Bull. Amer. Math. Soc. 76 (1970), 1044–1049. MR 265510, https://doi.org/10.1090/S0002-9904-1970-12552-6
[7] -, Infinite-dimensional Hamiltonian systems, Lecture Notes in Math., vol. 425, Springer-Verlag, Berlin, 1974.
[8] Michael G. Crandall and Amnon Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis 3 (1969), 376–418. MR 0243383, https://doi.org/10.1016/0022-1236(69)90032-9
[9] Constantine M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory 8 (1974/75), no. 2, 142–149. MR 0445473, https://doi.org/10.1007/BF01762184
[10] M. G. Darboux, Sur le théorème fondamental de la géométrie projective, Math. Ann. 17 (1880), no. 1, 55–61 (French). MR 1510050, https://doi.org/10.1007/BF01444119
[11] A. Denjoy, Sur les fonctions dérivées sommables, Bull. Soc. Math. France 43 (1915), 161–248 (French). MR 1504743
[12] Nelson Dunford, On one parameter groups of linear transformations, Ann. of Math. (2) 39 (1938), no. 3, 569–573. MR 1503425, https://doi.org/10.2307/1968635
[13] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
[14] Georg Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), Math. Ann. 60 (1905), no. 3, 459–462 (German). MR 1511317, https://doi.org/10.1007/BF01457624
[15] H. Kestelman, On the functional equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), Fund. Math. 34 (1947), 144–147. MR 22298, https://doi.org/10.4064/fm-34-1-144-147
[16] H. Looman, Sur deux catégories remarquables de fonctions de variable réelle, Fund. Math. 5(1924), 105-111.
[17] J. von Neumann, Über einen Satz von Herrn M. H. Stone, Ann. of Math. (2) 33 (1932), no. 3, 567–573 (German). MR 1503076, https://doi.org/10.2307/1968535
[18] A. Ostrowski, Über die Funktionalgleichung der Exponential-funktion und verwandte Funktional-gleichungen, Jber. Deutsch. Math.-Verein 38 (1929), 54-62.
[19] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
[20] R. S. Phillips, On one-parameter semi-groups of linear transformations, Proc. Amer. Math. Soc. 2 (1951), 234–237. MR 39922, https://doi.org/10.1090/S0002-9939-1951-0039922-1
[21] W. Sierpiński, Sur une propriété des fonctions de M. Hamel, Fund. Math. 5 (1924), 334-335.