On the stability of mixed trigonometric functional equations. (English) Zbl 1129.39013

Kim, Gwang Hui

Banach J. Math. Anal. 1, No. 2, 227-236 (2007).

Let \((G,+)\) be a group. The paper deals with the (super)stability of the following trigonometric functional equations with unknown mappings \(f,g:G\to \mathbf{C}\):
\[ f(x+y)-f(x-y)=2f(x)f(y), \]
\[ f(x+y)-f(x-y)=2f(x)g(y), \]
\[ f(x+y)-f(x-y)=2g(x)f(y), \]
\[ f(x+y)-f(x-y)=2g(x)g(y). \]
To give a sample result from the paper, let \(f,g:G\to \mathbf{C}\) satisfy (with \(\varepsilon\geq 0\)) the inequality: \[ | f(x+y)-f(x-y)-2g(x)f(y)| \leq \varepsilon,\qquad x,y\in G. \] 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.

Reviewer: Jacek Chmielinski (Kraków)

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

trigonometric functional equations; cosine functional equation; Banach spaces