Quadratic $\rho$-functional inequalities in Banach spaces: a fixed point approach
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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. \\
&& \left. \qquad \qquad
+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) \\
&&\qquad\qquad \ \nonumber +f(z-x-y)-4f(x)-4f(y)-4f(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)\\
&& \qquad\qquad +f(z-x-y)-4f(x)-4f(y) -4f(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 \\
&& \qquad \qquad \left. \left. +f\left(\frac{z-x-y}{2}\right) -f(x)-f(y)-f(z) \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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