Hyers–Ulam Stability of Additive Functional Equation Using Direct and Fixed-Point Methods

Abstract

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation $\chi \left({\displaystyle {\sum }_{g=1}^l{v}_g}\right)={\displaystyle {\sum }_{1\le g<h<i\le l}\chi \left(v_g+v_h+v_i\right)}-{\displaystyle {\sum }_{1\le g<h\le l}\chi \left(v_g+v_h\right)}-\left(\left(l^2-5l+2\right)/2\right){\displaystyle {\sum }_{g=1}^l\left(\chi \left(v_g\right)-\chi \left(-v_g\right)/2\right)}$ . where $l$ is a nonnegative integer with $\mathbb{N}-\left\{\mathrm{0,1,2,3,4}\right\}$ in Banach spaces.