<abstract xmlns="http://www.w3.org/1999/xhtml">Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2019-0019_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi>g</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></math><tex-math>f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = f(x) + f(y) + 2g(x) + 2g(y)</tex-math></alternatives></disp-formula>forces the unknown functions <italic>f</italic> and <italic>g</italic> to be additive and quadratic, respectively, modulo a constant.Motivated by the observation that the equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2019-0019_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mn>2</mn></msup><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mn>2</mn></msup><mo>)</mo></mrow></math><tex-math>f\left( {x + y} \right) + f({x^2}) = f(x) + f(y) + f({x^2})</tex-math></alternatives></disp-formula>implies both the additivity and multiplicativity of <italic>f</italic>, we deal also with the alienation phenomenon of equations in a single and several variables.</abstract>

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equationf(x+y)+g(x+y)+g(x-y)=f(x)+f(y)+2g(x)+2g(y)f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = f(x) + f(y) + 2g(x) + 2g(y)forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant.Motivated by the observation that the equationf(x+y)+f(x2)=f(x)+f(y)+f(x2)f\left( {x + y} \right) + f({x^2}) = f(x) + f(y) + f({x^2})implies both the additivity and multiplicativity of f, we deal also with the alienation phenomenon of equations in a single and several variables.

A Short Proof of Alienation of Additivity from Quadraticity

Institute of Mathematics Silesian University

Tatra Mountains Mathematical Publications

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

{"article-title":"A Short Proof of Alienation of Additivity from Quadraticity"}

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional...

A Short Proof of Alienation of Additivity from Quadraticity

Published Online: Nov 15, 2019

Page range: 57 - 62

Received: Dec 17, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0019

Keywords
alienation phenomenon, additive and quadratic mappings, systems of functional equations

© 2019 Roman Ger, published by Sciendo

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

A Short Proof of Alienation of Additivity from Quadraticity

Published Online: Nov 15, 2019

Page range: 57 - 62

Received: Dec 17, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0019

Keywordsalienation phenomenon, additive and quadratic mappings, systems of functional equations

© 2019 Roman Ger, published by Sciendo

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

Keywords
alienation phenomenon, additive and quadratic mappings, systems of functional equations