Fixed points and quadratic rho-functional equations

Volume 9, Issue 4, pp 1858--1871 http://dx.doi.org/10.22436/jnsa.009.04.39

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Authors

Choonkil Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea. Sang Og Kim - Department of Mathematics, Hallym University, Chuncheon 24252, Korea.

Abstract

In this paper, we solve the quadratic \(\rho\)-functional equations \[f(x + y) + f(x - y) - 2f(x) - 2f(y) = \rho \left( 2f (\frac{x + y}{2}) + 2f (\frac{x - y}{2}) - f(x) - f(y)\right), \qquad (1)\] where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1;2\), and \[2f (\frac{x + y}{2}) + 2f (\frac{x - y}{2}) - f(x) - f(y) = \rho \left(f(x + y) + f(x - y) - 2f(x) - 2f(y)\right); \qquad (2)\] where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1; \frac{-1}{2}\). Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic \(\rho\)-functional equations (1) and (2) in non-Archimedean Banach spaces and in Banach spaces.

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Keywords

• Hyers-Ulam stability
• non-Archimedean normed space
• fixed point
• quadratic \(\rho\)-functional equation.

MSC

•  46S10
•  39B62
•  39B52
•  47H10
•  47S10
•  12J25

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