Solution and Stability of a, b, c, d Mixed Type Functional Equation in BS (Banach Space) and BA (Banach Algebra) Using Two Different Methods

Abstract

In this article, the authors introduce the general solution and generalized Ulam-Hyers stability of a generalized a, b, c, d mixed type functional equation of the form $g(\frac{a}{b} x+\frac{b}{c} y+\frac{c}{d} z)+g(\frac{a}{b} x-\frac{b}{c} y+\frac{c}{d} z)$$+g(\frac{a}{b} x+\frac{b}{c} y-\frac{c}{d} z)+g(-\frac{a}{b} x+\frac{b}{c} y+\frac{c}{d} z)$$=\frac{a}{b} [g(x)-g(-x)]+\frac{b}{c} [g(y)-g(-y)]+\frac{c}{d} [g(z)-g(-z)] +2(\frac{c}{d})^{2} [g(z)+g(-z)]$$+2(\frac{a}{b})^{2} [g(x)+g(-x)]+2(\frac{b}{c})^{2} [g(y)+g(-y)]$, where a,b,c,d are positive integers with $a\ne b\ne c\ne d\ne 0$, in BS( Banach Space) and BA (Banach Algebras )using two different methods.