Generalized functional inequalities in Banach spaces

Q3 Mathematics

Moroccan Journal of Pure and Applied Analysis Pub Date : 2021-01-29 DOI:10.2478/mjpaa-2021-0022

H. Dimou, Y. Aribou, S. Kabbaj

引用次数: 0

Abstract

Abstract In this paper, we solve and investigate the generalized additive functional inequalities ‖ F(∑i=1nxi)-∑i=1nF(xi) ‖≤‖ F(1n∑i=1nxi)-1n∑i=1nF(xi) ‖ \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| and ‖ F(1n∑i=1nxi)-1n∑i=1nF(xi) ‖≤‖ F(∑i=1nxi)-∑i=1nF(xi) ‖. \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|. Using the direct method, we prove the Hyers-Ulam stability of the functional inequalities (0.1) in Banach spaces and (0.2) in non-Archimedian Banach spaces.

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Banach空间中的广义泛函不等式

摘要本文求解并研究了广义可加泛函不等式‖F(∑i=1nxi)—∑i=1nF(xi)‖≤‖F(1n∑i=1nxi)—1n∑i=1nF(xi)‖ \left\ b| {f\left（ {\sum\limits＿{I = 1}^n {{x＿i}} } \right)—— \sum\limits＿{I = 1}^n {f\left（ {{x＿i}} \right）} } \right\ b| \le \left\ b| {f\left（ {{1 \over n}\sum\limits＿{I = 1}^n {{x＿i}} } \right)—— {1 \over n}\sum\limits＿{I = 1}^n {f\left（ {{x＿i}} \right）} } \right\|和‖F(1n∑i=1nxi)-1n∑i=1nF(xi)‖≤‖F(∑i=1nxi)-∑i=1nF(xi)‖。 \left\ b| {f\left（ {{1 \over n}\sum\limits＿{I = 1}^n {{x＿i}} } \right)—— {1 \over n}\sum\limits＿{I = 1}^n {f\left（ {{x＿i}} \right）} } \right\ b| \le \left\ b| {f\left（ {\sum\limits＿{I = 1}^n {{x＿i}} } \right)—— \sum\limits＿{I = 1}^n {f\left（ {{x＿i}} \right）} } \right\|。利用直接方法证明了Banach空间中的(0.1)和非archimid Banach空间中的(0.2)泛函不等式的Hyers-Ulam稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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