Korean J. Math. Vol. 23 No. 2 (2015) pp.231-248
DOI: https://doi.org/10.11568/kjm.2015.23.2.231

# Quadratic $\rho$-functional inequalities in Banach spaces: a fixed point approach

## Abstract

In this paper, we solve the following quadratic $\rho$-functional inequalities
\begin{eqnarray}
&&\nonumber \left\| f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right)+f\left(\frac{y-x-z}{2}\right) \right. \\
+f\left(\frac{z-x-y}{2}\right) -f(x) -f(y) -f(z) \right\| \\
&&\nonumber \leq \| \rho (f(x+y+z) + f(x-y-z) +f(y-x-z) \\
\end{eqnarray}
where $\rho$ is a fixed complex number with $|\rho|<\frac{1}{8}$,
and
\begin{eqnarray}
&& \nonumber \| f(x+y+z) + f(x-y-z)+f(y-x-z)\\
&& \leq
\left \| \rho \left( f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right) +f\left(\frac{y-x-z}{2}\right)\right.\right.\nonumber \\
\end{eqnarray}
where $\rho$ is a fixed complex number with $|\rho|<4$.

Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.

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