The stability of quadratic \(\alpha\)-functional equations

Volume 9, Issue 6, pp 3980--3991 http://dx.doi.org/10.22436/jnsa.009.06.44

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Authors

Sungsik Yun - Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Republic of Korea. Choonkill Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea.

Abstract

In this paper, we investigate the quadratic \(\alpha\)-functional equation \[2f(x) + 2f(y) = f(x - y) + \alpha^{-2}f(\alpha(x + y)); \quad(1)\] \[2f(x) + 2f(y) = f(x + y) + \alpha^{-2}f(\alpha(x - y));\quad (2)\] where \(\alpha\) is a fixed nonzero real or complex number with \(\alpha^{-1}\neq \pm\sqrt{3}\). Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic \(\alpha\)-functional equations (1) and (2) in Banach spaces.

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Keywords

• Hyers-Ulam stability
• quadratic \(\alpha\)-functional equation
• fixed point method
• direct method
• Banach space.

MSC

•  39B52
•  39B62
•  47H10

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