Additive \(\rho\)-functional inequalities in normed spaces

Volume 9, Issue 1, pp 247--253 http://dx.doi.org/10.22436/jnsa.009.01.23

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Authors

Jiyun Choi - Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea. Juno Seong - Mathematics Branch, Seoul Science High School, Seoul 110-530, Korea. Choonkill Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea.

Abstract

In this paper, we solve the additive \(\rho\)-functional inequalities \[ \| f(x + y) - f(x) - f(y)\| \leq\left\|\rho\left(2f(\frac{x+y}{2})-f(x)-f(y)\right)\right\|\quad\quad (1)\] and \[\left\|\left(2f(\frac{x+y}{2})-f(x)-f(y)\right)\right\|\leq \| \rho (f(x + y) - f(x) - f(y))\| \quad\quad (2)\] where \(\rho\) is a number with \(|\rho|< 1\) . Using the fixed point method, we prove the Hyers-Ulam stability of the additive functional inequalities (1) and (2) in normed spaces.

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Keywords

• Additive \(\rho\)-functional inequality
• fixed point
• Hyers-Ulam stability.

MSC

•  39B52
•  47H10
•  39B72

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