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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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