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Local Pexider and Cauchy equations. (English) Zbl 1128.39018
If $f(s+ t)= f(s)+ f(t)$ on $D\subset\bbfR^2$, open and connected, there exists a unique quasiextension $A$ on $\bbfR^2$ so that $A(x+ y)= A(x)+ A(y)$ on $\bbfR^2$ and $f= A+ a$ on $D_1$, $f= A+ b$ on $D_2$, $f= A+ a+ b$ on $D_+$, where $$D_1:= \{s\mid\exists t: (s,t)\in D\},\quad D_2:= \{t\mid\exists s: (s,t)\in D\},\quad D_+:= \{s+ t\mid (s,t)\in D\}$$ [cf. {\it Z. Daróczy} and {\it 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 $\bbfR^2$, so that $F(x+ y)= G(x)+ f(y)$ on $\bbfR^2$ and $G= g$ on $D_1$, $H= h$ on $D_2$, $F= f$ on $D_+$ [cf. {\it F. Radó} and {\it 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 $D\subset\bbfR^2$.

39B22Functional equations for real functions
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