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The Stability of a Functional Analogue of the Wave Equation
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- Journal:
- Canadian Mathematical Bulletin / Volume 33 / Issue 4 / 01 December 1990
- Published online by Cambridge University Press:
- 20 November 2018, pp. 376-385
- Print publication:
- 01 December 1990
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- Article
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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
