Inner product spaces and quadratic functional equations

Abstract

In this paper, we introduce the functional equations $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ f ( 2 x − y ) + f ( x + 2 y ) = 5 [ f ( x ) + f ( y ) ] , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + 4 f ( y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + f ( 2 y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 4 [ f ( x ) + f ( y ) ] + [ f ( − x ) + f ( − y ) ] . We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

Generalized Quadratic Functional Equation and Its Stability over Non-Archimedean Normed Space

“…Hyer's Ulam-Rassias stability problem of the Q.F equations in NAN spaces has been proved by many researchers ( [8][9][10][11]). Bae et al [12] introduced different types of Q.F equations in 2021.…”

On the stability of the Pexider system of bi-additive and bi-quadratic equations

“…During the last years, a number of results concerning the stability have been obtained by different ways [5,6,7,8,9,10,11], and been applied to a number of functional equations, functional inequalities and mappings.…”