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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
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Abstract

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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

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