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Published in:
2021 | OriginalPaper | Chapter
Additive-Quadratic ρ-Functional Equations in β-Homogeneous Normed Spaces
Authors : Jung Rye Lee, Choonkil Park, Themistocles M. Rassias, Sungsik Yun
Published in: Approximation Theory and Analytic Inequalities
Publisher: Springer International Publishing
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Abstract
Let \(M_1f(x,y) : = \frac {3}{4} f(x+y) - \frac {1}{4}f(-x-y) + \frac {1}{4} f(x-y) + \frac {1}{4} f(y-x) - f(x) - f(y)\) and \(M_2 f(x,y): = 2 f\left ( \frac {x+y}{2} \right ) + f\left ( \frac {x-y}{2}\right ) + f\left ( \frac {y-x}{2}\right ) - f(x) - f(y).\) We solve the additive-quadratic ρ-functional inequalities
$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_1 f(x,y)\| \le \|\rho M_2f(x,y)\|, \end{array} $$
(1)
where ρ is a fixed complex number with \(|\rho |<\frac {1}{2}\), and
$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_2 f(x,y)\| \le \|\rho M_1 f(x,y)\| , \end{array} $$
(2)
where ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers–Ulam stability of the additive-quadratic ρ-functional inequalities (1) and (2) in β-homogeneous complex Banach spaces.