Quadratic(ρ1, p2)-functional Equation in Fuzzy Banach Spaces
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In this paper, we consider the following quadratic $(\rho_1, \rho_2)$-functional equation \begin{eqnarray}\label{0.1} && N(2f\left(\frac{x+y}{2}\right) + 2f\left(\frac{x-y}{2}\right) - f(x) - f(y) - \rho_1 ( f(x+y) + f(x-y) \\ && \quad - 2 f(x) -2 f(y)) - \rho_2 ( 4f\left(\frac{x+y}{2}\right) + f\left(x-y\right) - f(x) - f(y),t ) \ge \frac{t}{t+\varphi(x, y)},\nonumber \end{eqnarray} where $\rho_1, \rho_2$ are fixed nonzero real numbers with $\rho_2 \ne 1$ and $2\rho_1 + 2\rho_2 \ne 1$, in fuzzy normed spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $(\rho_1, \rho_2)$-functional equation (\ref{0.1}) in fuzzy Banach spaces
Abstract
In this paper, we consider the following quadratic $(\rho_1, \rho_2)$-functional equation \begin{eqnarray}\label{0.1} && N(2f\left(\frac{x+y}{2}\right) + 2f\left(\frac{x-y}{2}\right) - f(x) - f(y) - \rho_1 ( f(x+y) + f(x-y) \\ && \quad - 2 f(x) -2 f(y)) - \rho_2 ( 4f\left(\frac{x+y}{2}\right) + f\left(x-y\right) - f(x) - f(y),t ) \ge \frac{t}{t+\varphi(x, y)},\nonumber \end{eqnarray} where $\rho_1, \rho_2$ are fixed nonzero real numbers with $\rho_2 \ne 1$ and $2\rho_1 + 2\rho_2 \ne 1$, in fuzzy normed spaces. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $(\rho_1, \rho_2)$-functional equation (\ref{0.1}) in fuzzy Banach spaces
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