Document Type : Research Paper

Authors

• Choonkil Park ¹
• Sang Og Kim ²
• Jung Rye Lee ³
• Dong Yun Shin ⁴

¹ Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

² Department of Mathematics, Hallym University, Chuncheon 200-7021, Korea

³ Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea

⁴ Department of Mathematics, University of Seoul, Seoul 130-743, Korea

Abstract

In [12], Park introduced the quadratic $\rho$-functional inequalities
$$\begin{array}{rcl}& & \Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \\ & & \le \Vert \rho (2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)-f(x)-f(y))\Vert ,\end{array}$$
where $\rho$ is a fixed complex number with $|\rho |<1$,
and
$$\begin{array}{rcl}& & \Vert 2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)-f(x)-f(y)\Vert \\ & & \le \Vert \rho (f(x+y)+f(x-y)-2f(x)-2f(y))\Vert ,\end{array}$$
where $\rho$ is a fixed complex number with $|\rho |<\frac{1}{2}$.

In this paper, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2)  in $\beta$-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of quadratic $\rho$-functional equations associated with  the quadratic $\rho$-functional inequalities(0.1) and (0.2) in $\beta$-homogeneous complex Banach spaces.

Keywords

• Hyers-Ulam stability
• $\beta$-homogeneous space
• quadratic $\rho$-functional equation
• quadratic $\rho$-functional inequality