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The regularity of the locally integrable and continuous solutions of nonlinear functional equations


Author: Halina Światak
Journal: Trans. Amer. Math. Soc. 221 (1976), 97-118
MSC: Primary 39A15
DOI: https://doi.org/10.1090/S0002-9947-1976-0430578-4
MathSciNet review: 0430578
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Abstract: The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class $ {C^m}$. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class $ {C^m}$ or $ {C^\infty }$. Such a property is characteristic for the solutions $ f:{R^n} \to R$ of the equations

$\displaystyle \sum\limits_{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),}$ ($ \ast$)

where $ x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1$ and where the functions $ {\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R$ satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating $ (\ast)$ and fixing t is of constant strength, hypoelliptic at a point $ {x_0}$. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of $ (\ast)$, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.

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DOI: https://doi.org/10.1090/S0002-9947-1976-0430578-4
Article copyright: © Copyright 1976 American Mathematical Society

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