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- W2015012321 abstract "Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math>be a commutative semigroup with no neutral element,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:mi>Y</mml:mi></mml:mrow></mml:math>a Banach space, and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mi>ℂ</mml:mi></mml:mrow></mml:math>the set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mfenced open=∥ close=∥ separators=|><mml:mrow><mml:mi>f</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>≤</mml:mo><mml:mi>ϵ</mml:mi></mml:math>for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>S</mml:mi><mml:mo>→</mml:mo><mml:mi>Y</mml:mi></mml:math>. Using Jung’s theorem we obtain a better bound than that usually obtained. Also, generalizing the result of Baker (1980) we prove the superstability for Pexider-exponential equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mfenced open=| close=| separators=|><mml:mrow><mml:mi>f</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mi>h</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mfenced><mml:mo>≤</mml:mo><mml:mi>ϵ</mml:mi></mml:math>for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>S</mml:mi><mml:mo>→</mml:mo><mml:mi>ℂ</mml:mi></mml:math>. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mi>f</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mfenced separators=|><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mi>h</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi></mml:math>, a closed form of which is not yet known." @default.
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- W2015012321 date "2014-01-01" @default.
- W2015012321 modified "2023-09-26" @default.
- W2015012321 title "Stability of Pexider Equations on Semigroup with No Neutral Element" @default.
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- W2015012321 doi "https://doi.org/10.1155/2014/153610" @default.
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