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Characterization of Non-Linear Transformations Possessing Kernels

Published online by Cambridge University Press:  20 November 2018

Victor J. Mizel
Victor J. Mizel*
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
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Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form

(1.1)

with a continuous “kernel” φ: R → R,or

(1.2)

with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form

(1.3)

where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 449 - 471
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Chacon, R. V. and Friedman, N., Additive Junctionals, Arch. Rational Mech. Anal. 18 (1965), 230–240.Google Scholar
2. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations, pp. 42–48 (McGraw-Hill, New York, 1955).Google Scholar
3. Coleman, B. D. and Mizel, V. J., Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87–123.Google Scholar
4. Drewnowski, L. and Orlicz, W., On orthogonally additive junctionals, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 883–888.Google Scholar
5. Dunford, N. and Schwartz, J. T., Linear operators, Part I (Interscience, New York, 1958).Google Scholar
6. Friedman, N. and Katz, M., Additive functionals on L? spaces, Can. J. Math. 18 (1966), 1264–1271.Google Scholar
7. Friedman, N. and Katz, M., On additive functionals, Proc. Amer. Math. Soc. 21 (1969), 557–561.Google Scholar
8. Gel'fand, I. M. and N. Ya, Vilenkin, Generalized functions, Vol. 4: Applications of harmonic analysis, pp. 273–278, Translated by Feinstein, Amiel (Academic Press, New York, 1964).Google Scholar
9. M. A., Krasnosel'skiï, Topological methods in the theory of nonlinear integral equations, pp. 20–32, Translated by Armstrong, A. H. and edited by Burlak, J. (Macmillan, New York, 1964).Google Scholar
10. Martin, A. D. and Mizel, V. J., A representation theorem for certain nonlinear functionals, Arch. Rational Mech. Anal. 15 (1964), 353–367.Google Scholar
11. Mizel, V. J. and Sundaresan, K., Representation of additive and biadditive functionals, Arch. Rational Mech. Anal. 30 (1968), 102–126.Google Scholar
12. Sundaresan, K., Additive functionals on Orlicz spaces, Studia Math. 32 (1968), 270–276.Google Scholar