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

<p>The aim of this paper is to study the superstability problem of the d’Alembert type functional equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0001_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math>
<tex-math>$$f(x + y + z) + f(x + y + \sigma (z)) + f(x + \sigma (y) + z) + f(\sigma (x) + y + z) = 4f(x)f(y)f(z)$$</tex-math>
</alternatives>
</disp-formula>
for all <italic>x, y, z</italic> ∈ <italic>G</italic>, where <italic>G</italic> is an abelian group and σ : <italic>G</italic> → <italic>G</italic> is an endomorphism such that <italic>σ</italic>(<italic>σ</italic>(<italic>x</italic>)) = <italic>x</italic> for an unknown function <italic>f</italic> from <italic>G</italic> into ℂ or into a commutative semisimple Banach algebra.</p>
</abstract>

The aim of this paper is to study the superstability problem of the d’Alembert type functional equation



f(x+y+z)+f(x+y+σ(z))+f(x+σ(y)+z)+f(σ(x)+y+z)=4f(x)f(y)f(z)
$$f(x + y + z) + f(x + y + \sigma (z)) + f(x + \sigma (y) + z) + f(\sigma (x) + y + z) = 4f(x)f(y)f(z)$$


for all x, y, z ∈ G, where G is an abelian group and σ : G → G is an endomorphism such that σ(σ(x)) = x for an unknown function f from G into ℂ or into a commutative semisimple Banach algebra.

On the superstability of generalized d’Alembert harmonic functions

Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra,

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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

{"article-title":"On the superstability of generalized d\u2019Alembert harmonic functions"}

The aim of this paper is to study the superstability problem of the d’Alembert type functional...

On the superstability of generalized d’Alembert harmonic functions

Published Online: Dec 23, 2016

Page range: 5 - 13

Received: Oct 21, 2015

Accepted: Dec 08, 2015

DOI: https://doi.org/10.1515/aupcsm-2016-0001

Keywords
stability, d’Alembert functional equation

© 2016 Iz-iddine EL-Fassi, published by De Gruyter Open

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

On the superstability of generalized d’Alembert harmonic functions

Published Online: Dec 23, 2016

Page range: 5 - 13

Received: Oct 21, 2015

Accepted: Dec 08, 2015

DOI: https://doi.org/10.1515/aupcsm-2016-0001

Keywordsstability, d’Alembert functional equation

© 2016 Iz-iddine EL-Fassi, published by De Gruyter Open

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

Keywords
stability, d’Alembert functional equation