Abstract
We will investigate the superstability of the sine functional equation from the following Pexider-type functional equation (), which can be considered the mixed functional equation of the sine and cosine functions, the mixed functional equation of the hyperbolic sine and hyperbolic cosine functions, and the exponential-type functional equations.
Highlights
In 1940, Ulam 1 conjectured the stability problem
Rassias 6 which is that the condition bounded by the constant is replaced to the condition bounded by two variables, and thereafter it was improved by Gavruta 7 to the condition bounded by the function
Author Kim and Lee 18 investigated the superstability of S from the functional equation Cgh under the condition bounded by function, that is 1 if f, g, h : G → C satisfies fxyfx − y − 2g x h y ≤ φ x, 1.7 either h is bounded or g satisfies S ; 2 if f, g, h : G → C satisfies fxyfx − y − 2g x h y ≤ φ y, 1.8 either g is bounded or h satisfies S
Summary
In 1940, Ulam 1 conjectured the stability problem. year, this problem was affirmatively solved by Hyers 2 , which is through the following. In 1979, Baker et al 8 showed that if f is a function from a vector space to R satisfying f x y − f x f y ≤ ε, 1.3 either f is bounded or satisfies the exponential functional equation f x y fxf y. This method is referred to as the superstability of the functional equation 1.4. In 1980, the superstability of the cosine functional equation referred the d’Alembert functional equation fxyfx − y 2f x f y , C was investigated by Baker 9 with the following result: let ε > 0. Journal of Inequalities and Applications fxygx − y λg x h y , fxygx − y λh x g y , fxygx − y λf x g y , fxygx − y λg x f y , fxygx − y λf x f y , fxygx − y λg x g y , fxyfx − y λg x h y , fxyfx − y λg x g y , fxyfx − y λf x g y , fxyfx − y λg x f y , fxyfx − y λf x f y , fxygx − y 2h x k y , fxygx − y 2h x h y , fxygx − y 2f x h y , fxygx − y 2h x f y , fxygx − y 2g x h y , fxygx − y 2h x g y , fxygx − y 2f x g y , fxygx − y 2g x f y , fxygx − y 2f x f y , fxygx − y 2g x g y , fxyfx − y 2f x g y , fxyfx − y 2g x f y , fxyfx − y 2g x g y , fxyfx − y 2g x h y , fxyfx − y 2f x
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