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Characterization of Non-Linear Transformations Possessing Kernels

  • Victor J. Mizel (a1)

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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: RR,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 × TR satisfies “Carathéodory conditions”.

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References

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1. Chacon, R. V. and Friedman, N., Additive Junctionals, Arch. Rational Mech. Anal. 18 (1965), 230240.
2. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations, pp. 4248 (McGraw-Hill, New York, 1955).
3. Coleman, B. D. and Mizel, V. J., Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87123.
4. Drewnowski, L. and Orlicz, W., On orthogonally additive junctionals, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 883888.
5. Dunford, N. and Schwartz, J. T., Linear operators, Part I (Interscience, New York, 1958).
6. Friedman, N. and Katz, M., Additive functionals on L? spaces, Can. J. Math. 18 (1966), 12641271.
7. Friedman, N. and Katz, M., On additive functionals, Proc. Amer. Math. Soc. 21 (1969), 557561.
8. Gel'fand, I. M. and N. Ya, Vilenkin, Generalized functions, Vol. 4: Applications of harmonic analysis, pp. 273278, Translated by Feinstein, Amiel (Academic Press, New York, 1964).
9. M. A., Krasnosel'skiï, Topological methods in the theory of nonlinear integral equations, pp. 2032, Translated by Armstrong, A. H. and edited by Burlak, J. (Macmillan, New York, 1964).
10. Martin, A. D. and Mizel, V. J., A representation theorem for certain nonlinear functionals, Arch. Rational Mech. Anal. 15 (1964), 353367.
11. Mizel, V. J. and Sundaresan, K., Representation of additive and biadditive functionals, Arch. Rational Mech. Anal. 30 (1968), 102126.
12. Sundaresan, K., Additive functionals on Orlicz spaces, Studia Math. 32 (1968), 270276.
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Characterization of Non-Linear Transformations Possessing Kernels

  • Victor J. Mizel (a1)

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