<abstract xmlns="http://www.w3.org/1999/xhtml">Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formula<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0006_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>Λ</mi><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>,</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mn>2</mn><mtext>f</mtext><mrow><mo>(</mo><mtext>x</mtext><mo>)</mo></mrow><mo>+</mo><mtext>f</mtext><mrow><mo>(</mo><mtext>y</mtext><mo>)</mo></mrow><mo>+</mo><mtext>f</mtext><mrow><mo>(</mo><mrow><mo>-</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>-</mo><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>+</mo><mtext>y</mtext></mrow><mo>)</mo></mrow><mo>-</mo><mtext>f</mtext><mrow><mo>(</mo><mrow><mtext>x</mtext><mo>-</mo><mtext>y</mtext></mrow><mo>)</mo></mrow></mrow></math><tex-math>$$\Lambda {\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right): = 2{\rm{f}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{y}} \right) + {\rm{f}}\left( {{\rm{ - y}}} \right) - {\rm{f}}\left( {{\rm{x + y}}} \right) - {\rm{f}}\left( {{\rm{x - y}}} \right)$$</tex-math></alternatives></disp-formula>for all x, y ∈ G. In this article, we prove that if ˄f is Lipschitz, then there exists a Drygas function D : G → E such that f − D is Lipschitz with the same constant. Moreover, some results concerning the approximation of the Drygas functional equation in the Lipschitz norms are presented.</abstract>

Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formulaΛf(x,y):=2f(x)+f(y)+f(-y)-f(x+y)-f(x-y)$$\Lambda {\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right): = 2{\rm{f}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{y}} \right) + {\rm{f}}\left( {{\rm{ - y}}} \right) - {\rm{f}}\left( {{\rm{x + y}}} \right) - {\rm{f}}\left( {{\rm{x - y}}} \right)$$for all x, y ∈ G. In this article, we prove that if ˄f is Lipschitz, then there exists a Drygas function D : G → E such that f − D is Lipschitz with the same constant. Moreover, some results concerning the approximation of the Drygas functional equation in the Lipschitz norms are presented.

On approximate solution of Drygas functional equation according to the Lipschitz criteria

Department of Mathematics, Faculty of Sciences

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the...

On approximate solution of Drygas functional equation according to the Lipschitz criteria

Published Online: Aug 17, 2019

Page range: 66 - 77

Received: Mar 14, 2018

DOI: https://doi.org/10.2478/ausm-2019-0006

Keywords
Drygas functional equation, stability, Lipschitz space

© 2019 Iz-iddine EL-Fassi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On approximate solution of Drygas functional equation according to the Lipschitz criteria

Published Online: Aug 17, 2019

Page range: 66 - 77

Received: Mar 14, 2018

DOI: https://doi.org/10.2478/ausm-2019-0006

KeywordsDrygas functional equation, stability, Lipschitz space

© 2019 Iz-iddine EL-Fassi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Drygas functional equation, stability, Lipschitz space