A universal Cauchy functional equation over the positive reals |
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Authors: | Vichian Laohakosol Watcharapon Pimsert |
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Institution: | a Department of Mathematics, Kasetsart University, Bangkok 10900, Thailand b Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand |
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Abstract: | The functional equation af(xy)+bf(x)f(y)+cf(x+y)+d(f(x)+f(y))=0 whose shape contains all the four well-known forms of Cauchy's functional equation is solved for solutions which are functions having the positive reals as their domain. This complements an earlier work of Dhombres in 1988 where the same functional equation was solved for solutions whose domains contain zero, which leaves out the logarithmic function. Here not only the logarithmic function is recovered but the analysis is entirely different and is based on solving appropriate difference equations. |
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Keywords: | Cauchy's functional equation |
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