Inner product spaces and quadratic functional equations

Advances in Difference Equations

卷:2021

Abbas Najati¹  M. B. Moghimi¹  Batool Noori¹  Jae-Hyeong Bae² 

[1] Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili;

[2] Humanitas College, Kyung Hee University;

关键词: Stability;    Quadratic functional equation;    Quadratic function;    Asymptotic behavior;   

DOI  :  10.1186/s13662-021-03307-x

来源: DOAJ

【 摘 要 】

Abstract In this paper, we introduce the functional equations 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 ) ] . $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ 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.

【 授权许可】

Unknown