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

Document Type: Research Paper

Authors

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 \cite{p}, 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