The stability of additive $(\alpha,\beta)$-functional equations |
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| Authors: | Ziying Lu Gang Lu Yuanfeng Jin and Choonkil Park |
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| Institution: | Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China,Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P.R. China,Department of Mathematics, Yanbian University, Yanji 133001, P.R. China and Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea |
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| 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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| Keywords: | Hyers-Ulam stability additive $(\alpha \beta)$-functional equation fixed point method direct method non-Archimedean Banach space |
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