QUADRATIC -FUNCTIONAL INEQUALITIES IN BANACH SPACES: A FIXED POINT APPROACH

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.