Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces

Authors:
Moshe Marcus and Victor J. Mizel

Journal:
Trans. Amer. Math. Soc. **228** (1977), 1-45

MSC:
Primary 46E35; Secondary 46G99

MathSciNet review:
0454622

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Abstract: An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace of the *k*th order Sobolev space . It is shown that every mapping which is *local, continuous* and -*disjointly additive* has the form , where is a function obeying Carathéodory conditions as well as . Here -disjoint additivity means whenever . Likewise, every real functional *N* on *S* which is *continuous* and -*disjointly additive* has the form , with *T* as above. Liapunov's theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.

**[1]**Richard A. Alò and André de Korvin,*Representation of Hammerstein operators by Nemytskii measures*, J. Math. Anal. Appl.**52**(1975), no. 3, 490–513. MR**0428145****[2]**Jürgen Batt,*Nonlinear integral operators on 𝐶(𝑆,𝐸)*, Studia Math.**48**(1973), 145–177. MR**0336335****[3]**David Blackwell,*The range of certain vector integrals*, Proc. Amer. Math. Soc.**2**(1951), 390–395. MR**0041195**, 10.1090/S0002-9939-1951-0041195-0**[4]**R. V. Chacon and N. Friedman,*Additive functionals*, Arch. Rational Mech. Anal.**18**(1965), 230–240. MR**0172103****[5]**L. Drewnowski and W. Orlicz,*Continuity and representation of orthogonally additive functionals.*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**17**(1969), 647–653 (English, with Loose Russian summary). MR**0256147****[6]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869****[7]**M. A. Krasnosel’skii,*Topological methods in the theory of nonlinear integral equations*, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0159197****[8]**M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ,*Integralnye operatory v prostranstvakh summiruemykh funktsii*, Izdat. “Nauka”, Moscow, 1966 (Russian). MR**0206751****[9]**A. Liapounoff,*Sur les fonctions-vecteurs complètement additives*, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR]**4**(1940), 465–478 (Russian, with French summary). MR**0004080****[10]**A. D. Martin and V. J. Mizel,*A representation theorem for certain nonlinear functionals*, Arch. Rational Mech. Anal.**15**(1964), 353–367. MR**0162131****[11]**Victor J. Mizel,*Characterization of non-linear transformations possessing kernels*, Canad. J. Math.**22**(1970), 449–471. MR**0262890****[12]**Victor J. Mizel and K. Sundaresan,*Representation of vector valued nonlinear functions*, Trans. Amer. Math. Soc.**159**(1971), 111–127. MR**0279647**, 10.1090/S0002-9947-1971-0279647-8

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1977-0454622-4

Keywords:
-disjointly additive functional,
local -disjointly additive operator,
normalized Carathéodory function,
Liapunov vector measure,
Radon-Nikodým type theorem

Article copyright:
© Copyright 1977
American Mathematical Society