Quadratic $\rho$-functional inequalities in $\beta$-homogeneous normed spaces

Park, Choonkil; Kim, Sang Og; Lee, Jung Rye; Shin, Dong Yun

doi:10.22075/ijnaa.2015.229

Quadratic $\rho$-functional inequalities in $\beta$-homogeneous normed spaces

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

10.22075/ijnaa.2015.229

Abstract

In [12], Park introduced the quadratic $\rho$-functional inequalities
\begin{eqnarray}\label{E01}
&& \|f(x+y)+f(x-y)-2f(x)-2f(y)\| \\ && \qquad \le \left\|\rho\left(2 f\left(\frac{x+y}{2}\right) + 2 f\left(\frac{x-y}{2}\right)- f(x) - f(y)\right)\right\|, \nonumber
\end{eqnarray}
where $\rho$ is a fixed complex number with $|\rho|<1$,
and
\begin{eqnarray}\label{E02}
&& \left\|2 f\left(\frac{x+y}{2}\right) + 2 f\left(\frac{x-y}{2}\right)- f(x) - f(y)\right\| \\ && \qquad \le \|\rho(f(x+y)+f(x-y)-2f(x)-2f(y))\| , \nonumber
\end{eqnarray}
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