The Stability of Functional Equations for Trigonometric and Similar Functions

Abstract

In this section, we will be interested primarily in the stability of the following equations:

(6.1) $f\left( {x + y} \right) + f\left( {x - y} \right) = 2f\left( x \right)f\left( y \right)$ (d’Alembert)

(6.2) $f{\left( {\frac{{x + y}}{2}} \right)^2} = f\left( x \right)f\left( y \right)$ (Lobačevskiĭ)

(6.3a) $f\left( {x + y} \right)f\left( {x - y} \right) = f{\left( x \right)^2} - {\left( y \right)^2}$ (sine equation)

(6.3b) $f\left( {x + y} \right) = f\left( x \right)g\left( y \right) + g\left( x \right)f\left( y \right)$ (sine equation)

(6.4) $f\left( {x + y} \right) = f\left( x \right)f\left( y \right) - g\left( x \right)g\left( y \right)$ (cosine equation)

where f and g may be defined on a group or semigroup with values in a field K which usually is the field of real or complex numbers. Methods of solving such equations are described in the books by Aczél (1966) and Aczél and Dhombres (1989).