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Zbl 1129.39013
Kim, Gwang Hui
On the stability of mixed trigonometric functional equations.
(English)
[J] Banach J. Math. Anal. 1, No. 2, 227-236, electronic only (2007). ISSN 1735-8787/e

Let $(G,+)$ be a group. The paper deals with the (super)stability of the following trigonometric functional equations with unknown mappings $f,g:G\to \text {\bf C}$: $$f(x+y)-f(x-y)=2f(x)f(y),$$ $$f(x+y)-f(x-y)=2f(x)g(y),$$ $$f(x+y)-f(x-y)=2g(x)f(y),$$ $$f(x+y)-f(x-y)=2g(x)g(y).$$ To give a sample result from the paper, let $f,g:G\to \text {\bf C}$ satisfy (with $\varepsilon\geq 0$) the inequality: $$\vert f(x+y)-f(x-y)-2g(x)f(y)\vert \leq \varepsilon,\qquad x,y\in G.$$ Then either $f$ is bounded or $g$ satisfies the cosine functional equation: $$g(x+y)+g(x-y)=2g(x)g(y),\qquad x,y\in G.$$ Moreover, either $g$ is bounded or it satisfies the cosine equation and $f,g$ satisfy the equations $$f(x+y)-f(x-y)=2g(x)f(y)\quad\text{and}\quad f(x+y)+f(x-y)=2f(x)g(y).$$ Some results are also proved in Banach spaces and algebras.
[Jacek Chmielinski (Kraków)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: trigonometric functional equations; cosine functional equation; Banach spaces

Cited in: Zbl 1159.39013

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