On reduction of linear two variable functional equations to differential equations without substitutions

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Abstract

The subject of this paper is the investigation of the linear two variable functional equation {Mathematical expression} where g0, ⋯, gn, h0, ⋯, hn and F are given real valued functions on an open set Ω ⊂R2, further f0, ⋯, fn are unknown real functions. Assuming differentiability of sufficiently large order, we construct a linear partial differential operator {Mathematical expression} where αij is defined on Ω, so that {Mathematical expression} for all sufficiently smooth functions φ{symbol} defined on {Mathematical expression}. Then, applying D to (*), we obtain {Mathematical expression}, which is a k-th order linear differential-functional equation for the unknown function f0. Using Járai's regularity theorems (see [3], [4], [5]) one can see that, if the given functions gt, ht, F, are differentiable up to an order k (1 ≤k ≤ ∞), then the measurability of the unknown functions ft imply their differentiability up to the same order. In this paper we prove an analogous result, namely that the analyticity of the given functions implies that the unknown functions are also analytic provided that they are measurable.

Original languageEnglish
Pages (from-to)236-247
Number of pages12
JournalAequationes Mathematicae
Volume43
Issue number2-3
DOIs
Publication statusPublished - Apr 1 1992

Keywords

  • AMS (1990) subject classification: 39B22

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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