Abstract
We introduce the extended Jensen equation \documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\usepackage{bbm}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$q^n f\left( {\frac{{x_1 + ....x_{q^n } }}
{{q^n }}} \right) = \sum\limits_{i = 1}^{q^n } {f(x_i )} ,$$
\end{document} where
q
> 1 and
n
are fixed positive integers. We investigate the stability and the asymptotic behavior of the above extended Jensen equation and prove that if 0 <
p
< 1 and
f
is a mapping from a normed space into a Banach space with
f
(0) = 0 which
p
-asymptotically satisfies the above equation, then it is
p
-asymptotic close to a linear mapping.