Solution and Stability of New Type of $\left({a_{aq}, b_{bq}, c_{aq}, d_{aq}}\right)$ Mixed Type Functional Equation in Various Normed Spaces: Using Two Different Methods

Abstract

Using two different methods, the authors are investigate the general solution and generalized Ulam - Hyers stability of new type of $\left({a_{aq}, b_{aq}, c_{aq}, d_{aq}}\right)$ mixed type functional equation of the form $g({ax_1+bx_2 + cx_3+dx_4})$$+g({-ax_1 + bx_2 + cx_3 + dx_4})$$+g({ax_1-bx_2 + cx_3 + dx_4})$$+ g({ax_1 + bx_2 - cx_3 + dx_4})$$+g({ax_1 + bx_2 + cx_3 - dx_4})$$=g({ax_1 + bx_2})+g({ax_1 + cx_3})$$+ g({ax_1 + dx_4})+g({bx_2 + cx_3})$$+g({bx_2 + dx_4})+g({cx_3 + dx_4})$$+g({ax_1})-g({-ax_1})$$+ g({bx_2})-g({-bx_2})+g({cx_3})$ $-g({-cx_3})+g({dx_4})$$-g({-dx_4})-(ag(x_1)-ag(-x_1)$$+ bg(x_2)-bg(-x_2)+cg(x_3)-cg(-x_3)+dg(x_4)-dg(-x_4) )$$+a^2({g(x_1)+g(-x_1)}) +b^2({g(x_2)$$+g(-x_2)})+c^2({g(x_3)+g(-x_3) })$$+d^2({g(x_4)+g(-x_4)})$, where $a, b, c, d$ are positive integers with $a \neq b \neq c \neq d$, in Banach space, Fuzzy Normed Space and Instuitionistic Fuzzy normed space.