<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, we solve and investigate the generalized additive functional inequalities
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<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0022_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow></mrow></math>
<tex-math>\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\|</tex-math>
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and
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<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2021-0022_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo> <mrow><mi>F</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>F</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow> <mo>‖</mo></mrow><mo>.</mo></mrow></math>
<tex-math>\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\|.</tex-math>
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<p>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.</p>
</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.

Generalized functional inequalities in Banach spaces

Department of Mathematics, Faculty of Sciences Applied

Department of Mathematics and Computer Science

Department of Mathematics, Faculty of Sciences

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Generalized functional inequalities in Banach spaces"}

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\|...

Generalized functional inequalities in Banach spaces

Published Online: Jan 29, 2021

Page range: 337 - 349

Received: Aug 30, 2020

Accepted: Jan 20, 2021

DOI: https://doi.org/10.2478/mjpaa-2021-0022

Keywords
Hyers-Ulam stability, non-Archimedean normed space, functional inequality, Banach space, Cauchy functional equation

© 2021 H. Dimou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Generalized functional inequalities in Banach spaces

Published Online: Jan 29, 2021

Page range: 337 - 349

Received: Aug 30, 2020

Accepted: Jan 20, 2021

DOI: https://doi.org/10.2478/mjpaa-2021-0022

KeywordsHyers-Ulam stability, non-Archimedean normed space, functional inequality, Banach space, Cauchy functional equation

© 2021 H. Dimou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Hyers-Ulam stability, non-Archimedean normed space, functional inequality, Banach space, Cauchy functional equation