Stability of Maximum Functional Equation and Some Properties of Groups

• Main Authors: Muhammad Sarfraz, Qi Liu, Yongjin Li
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Symmetry
• Subjects: maximum functional equations; discretely normed abelian group; stability of functional equation
• Online Access: https://www.mdpi.com/2073-8994/12/12/1949

In this research paper, we deal with the problem of determining the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, which is the solution to the maximum functional equation (MFE) <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> when the domain is a discretely normed abelian group or any arbitrary group <i>G</i>. We also analyse the stability of the maximum functional equation <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mspace width="0.166667em"></mspace><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mo>}</mo></mrow><mo>=</mo><mi>χ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and its solutions for the function <inline-formula><math display="inline"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>G</mi><mspace width="3.33333pt"></mspace><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, where <i>G</i> be any group and also investigate the connection of the stability with commutators and free abelian group <i>K</i> that can be embedded into a group <i>G</i>.