| Volume 9, Number 6, 2019, Pages 2295-2307 DOI：10.11948/20190075 |
| The stability of additive $(\alpha,\beta)$-functional equations |
| Ziying Lu,Gang Lu,Yuanfeng Jin,Choonkil Park |
| Keywords:Hyers-Ulam stability, additive $(\alpha,\beta)$-functional equation, fixed point method, direct method,non-Archimedean Banach space. |
| Abstract: |
| In this paper, we investigate the following $(\alpha,\beta)$-functional equations
$$
2f(x)+2f(z)=f(x-y)+\alpha^{-1}f(\alpha
(x+z))+\beta^{-1}f(\beta(y+z)),~~~~~~~~~(0.1)
$$
$$
2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z))
+\beta^{-1}f(\beta(y-z)),~~~~~~~~~~~(0.2)
$$
where $\alpha,\beta$ are fixed nonzero real numbers with $\alpha^{-1}+\beta^{-1}\neq 3$.
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the $(\alpha,\beta)$-functional equations $(0.1)$ and $(0.2)$ in non-Archimedean Banach spaces. |
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