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  • Author or Editor: Mohammad Moslehian x
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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.
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