2 results
A Common Generalization of Functional Equations Characterizing Normed and Quasi-Inner-Product Spaces
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- Journal:
- Canadian Mathematical Bulletin / Volume 35 / Issue 3 / 01 September 1992
- Published online by Cambridge University Press:
- 20 November 2018, pp. 321-327
- Print publication:
- 01 September 1992
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We determine the general solutions of the functional equation
for ƒi: G → F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas [4] where he proved a Jordan and von Neumann type characterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions on ƒi.
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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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
