Sunday, October 4, 2015

exponential additive functional equation

Let S be a semigroup with no identity element and m:SC be given function(m) satisfying the exponential functional equation
m(x+y)=m(x)m(y)
for all x, y\in S. Find all solutions f:S\to \Bbb C satisfying

the equation
\begin{equation} f(x+y)=f(x)m(y)+f(y)m(x) \tag 1 \end{equation}
for all x, y\in S.



Remark. If S is a group, then using the fact that m(x)\ne 0 for all x\in S and dividing (1) by m(x+y), we have
f(x)=m(x)A(x)

for all x\in S, where A is an additive function.

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