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

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.

### MSC:

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