Sunday, October 4, 2015

exponential additive functional equation

Let S be a semigroup with no identity element and m:SC be given function(m0) satisfying the exponential functional equation
m(x+y)=m(x)m(y)


for all x,yS. Find all solutions f:SC satisfying

the equation
f(x+y)=f(x)m(y)+f(y)m(x)

for all x,yS.



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



for all xS, where A is an additive function.

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