General Solution and Stability of Quadratic Functional Equation

Abstract

In this paper, the authors present the general solution and generalized Ulam-Hyers stability of quadratic functional equation of the form $f(nx+n^{2} y+n^{3} z+n^{4} t)+f(nx-n^{2} y+n^{3} z+n^{4} t)$$+f(nx+n^{2} y-n^{3} z+n^{4} t)+f(nx+n^{2} y+n^{3} z-n^{4} t)$$+f(-nx+n^{2} y+n^{3} z+n^{4} t)$$=f(nx+n^{2} y)+f(nx+n^{3} z)+f(nx+n^{4} t)$$+f(-nx+n^{2} y+n^{3} z+n^{4} t)$$=f(nx+n^{2} y)+f(nx+n^{3} z)+f(nx+n^{4} t)$$+f(n^{2} y+n^{3} z)+f(n^{2} y+n^{4} t)$$+f(n^{3} z+n^{4} t)+2n^{2} f(x)$$+2n^{4} f(y)+2n^{6} f(z)+2n^{8} f(t)$, where n is positive integer with $n\ne 0$ in Banach algebra using direct and fixed point methods.