Hyers-Ulam stability of an additive-quadratic functional equation

Abstract

In this paper, we introduce the following $(a,b,c)$-mixed type functional equation of the form \\$g(ax_1+bx_2+cx_3 )-g(-ax_1+bx_2+cx_3 )+g(ax_1-bx_2+cx_3 )-g(ax_1+bx_2-cx_3 ) +2a^2 [g(x_1 )+g(-x_1)]+2b^2 [g(x_2 )+g(-x_2)]+2c^2 [g(x_3 )+g(-x_3)]+a[g(x_1 )-g(-x_1)]+b[g(x_2 )-g(-x_2)]+c[g(x_3 )-g(-x_3)]=4g(ax_1+cx_3 )+2g(-bx_2)+ 2g(bx_2)$\\ where $a,b,c$ are positive integers with $a>1$, and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.

Keywords

• hyers-ulam stability
• mixed type functional equation
• banach space
• fixed point