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The general solution of the exponential Cauchy equation on a bounded restricted domain. (English) Zbl 1121.39024

The exponential functional equation

f(x+y)=f(x)f(y)

is studied for functions f:E(a,b;r) defined on a triangle

E(a,b;r):=(x,y) 2 : x a , y b , x + y < a + b + r·

To give the general solution two cases are needed to consider.

1. If f0 on the whole E(a,b;r), then f is a restriction of a solution of an equation of type f(x+y)=Kf(x)f(y) with some K0. A related extension theorem is proved.

2. If f(x 0 ,y 0 )=0 for some (x 0 ,y 0 )E(a,b;r), then the above-mentioned behaviour fails to hold true and the general local solution consists of a function defined by means of identically zero and arbitrary functions.

MSC:
39B22Functional equations for real functions
39B82Stability, separation, extension, and related topics