Stability of cubic and quartic rho-functional inequalities in fuzzy normed spaces

Volume 9, Issue 4, pp 1693--1701 http://dx.doi.org/10.22436/jnsa.009.04.25

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Authors

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

Abstract

In this paper, we solve the following cubic \(\rho\)-functional inequality \[N(f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x) - \rho (4f ( x +\frac{y}{2} ) + 4f ( x - \frac{y}{2}) - f(x + y) - f(x - y) - 6f(x)); t) \geq \frac {t} {t + \varphi(x; y)}\quad (1)\] and the following quartic \(\rho\)-functional inequality \[N(f(2x + y) + f(2x - y) - 4f(x + y) - 4f(x - y) - 24f(x) + 6f(y)- \rho (8f ( x +\frac{y}{2} ) + 8f ( x - \frac{y}{2}) - 2f(x + y) - 2f(x - y) - 12f(x)+ 3 f(y)); t) \geq \frac {t} {t + \varphi(x; y)}\quad (2)\] in fuzzy normed spaces, where \(\rho\) is a fixed real number with \(\rho\neq 2\). Using the direct method, we prove the Hyers-Ulam stability of the cubic \(\rho\)-functional inequality (1) and the quartic \(\rho\)-functional inequality (2) in fuzzy Banach spaces.

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Keywords

• fuzzy Banach space
• cubic \(\rho\) -functional inequality
• quartic \(\rho\) -functional inequality
• Hyers-Ulam stability.

MSC

•  39B52
•  46S40
•  39B62
•  26E50
•  47S40.

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