Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces

Abstract

Abstract This article presents the general solution f : G → V f:{\mathcal{G}}\to {\mathcal{V}} of the following functional equation: f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G , f\left(x)-4f\left(x+y)+6f\left(x+2y)-4f\left(x+3y)+f\left(x+4y)=0,\hspace{1.0em}x,y\in {\mathcal{G}}, where ( G , + ) \left({\mathcal{G}},+) is an abelian group and V {\mathcal{V}} is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.