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Local Pexider and Cauchy equations. (English) Zbl 1128.39018

If f(s+t)=f(s)+f(t) on D 2 , open and connected, there exists a unique quasiextension A on 2 so that A(x+y)=A(x)+A(y) on 2 and f=A+a on D 1 , f=A+b on D 2 , f=A+a+b on D + , where

D 1 :={st:(s,t)D},D 2 :={ts:(s,t)D},D + :={s+t(s,t)D}

[cf. Z. Daróczy and L. Losonczi, Publ. Math. 14, 239–245 (1967; Zbl 0175.15305)]. If f(s+t)=g(s)+h(t) on D there exist unique extensions F, G, H on 2 , so that F(x+y)=G(x)+f(y) on 2 and G=g on D 1 , H=h on D 2 , F=f on D + [cf. F. Radó and J. A. Baker, Aequationes Math. 32, 227–239 (1987; Zbl 0625.39007)].

In the present paper the authors study whether the similar results hold for the restricted exponential Cauchy functional equation f(s+t)=f(s)f(t) and for the Pexider variant of this equation f(s+t)=g(s)h(t) on D (both). First they show by counterexamples that in general this is not the case and further determine the general solutions, with and without regularity assumptions, of these restricted equations on D 2 .

MSC:
39B22Functional equations for real functions