The Stability of Functional Equations with a New Direct Method

We investigate the Hyers-Ulam stability of an equation involving a single variable of the form parallel to f(x) - alpha f(k(n)(x)) - beta f(k(n+1)(x))parallel to <= u (x) where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables.

Automated summary

We investigate the Hyers-Ulam stability of an equation involving a single variable and propose a new method that does not require any restrictions on the parity, domain, and range of the function. Stability theorems are proven using this method for various functional equations involving several variables. Our findings suggest that this method is easy and appropriate for investigating the stability of functional equations, particularly for several variables.