Let S be a semigroup with no identity element and m:S→C be given function(m≢0) satisfying the exponential functional equation
m(x+y)=m(x)m(y)
for all x,y∈S. Find all solutions f:S→C satisfying
the equation
f(x+y)=f(x)m(y)+f(y)m(x)
for all x,y∈S.
Remark. If S is a group, then using the fact that m(x)≠0 for all x∈S and dividing (1) by m(x+y), we have
f(x)=m(x)A(x)
for all x∈S, where A is an additive function.
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