Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T23:25:09.197Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stability of a Functional Analogue of the Wave Equation

Published online by Cambridge University Press:  20 November 2018

Michael Bean  and
John A. Baker
Michael Bean
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For h € ℝ and ϕ : ℝ2 → ℝ define Lhϕ: ℝ2 → ℝ by (Lhϕ)(x, y) = ϕ (x + h, y) + ϕ (x — h,y) — ϕ (x,y + h) — ϕ (x,y — h) for all (x,y) € ℝ2. The aim of the paper is to establish the following "stability" theorem concerning the functional equation

if δ > 0, f : ℝ2 → ℝ and

then there exists ɛ > 0 ami ϕ : ℝ2 → ℝ such that (Lhϕ)(x, y ) = 0 for all x, y,h ∊ ℝ and

Keywords

39B40D39B5039A11
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 4 , 01 December 1990 , pp. 376 - 385
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Albert, Michael and Baker, John A. Functions with bounded nth differences, Ann. Polon. Math. XLIII, (1983), 93–103.Google Scholar
2. John Baker, A. An analogue of the wave equation and certain related functional equationsCanad. Math. Bull. Vol. 12, No 6, (1969), 837–846.Google Scholar
3. Baker, John A. Regularity properties of functional equations, Aequationes Math., 6(2/3), (1971), 243– 248.Google Scholar
4. Fenyö, I. Remark on a paper of J. A. Baker, Aequationes Math., 8(1/2), (1972), 103–108.Google Scholar
5. Haruki, H. On the functional equation f(x +t, y) +f(x — t,y) — f(x, y + t) +f(x, y — t), Aequationes Math. 5(1), (1970), 118–119.Google Scholar
6. Hyers, D. H. The stability of homomorphisms and related topics, Global analysis - analysis on manifolds, Teubner-Texte zur Math. 57, Teubner, Leipzig, (1983), 140–153.Google Scholar
7. McKiernan, M. A. The general solution of some finite difference equations analogous to the wave equation, Aequationes Math. 8(3), (1972), 263–266.Google Scholar
8. Ostrowski, A. Mathematische Miszellen, XIV: Über die Funktionalgleichung der Exponential-function und verwandte Functionalgleichungen, Jber. Deutsch. Math.-Verein 38, (1929), 54–62.Google Scholar
9. Steinhaus Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1, (1920), 93–104.Google Scholar