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

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 \begin{equation}\tag {$\ast $} \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),} \end{equation} 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.