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Local Pexider and Cauchy equations. (English) Zbl 1128.39018

If \(f(s+ t)= f(s)+ f(t)\) on \(D\subset\mathbb{R}^2\), open and connected, there exists a unique quasiextension \(A\) on \(\mathbb{R}^2\) so that \(A(x+ y)= A(x)+ A(y)\) on \(\mathbb{R}^2\) 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. 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 \(\mathbb{R}^2\), so that \(F(x+ y)= G(x)+ f(y)\) on \(\mathbb{R}^2\) 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 \(D\subset\mathbb{R}^2\).

MSC:

39B22 Functional equations for real functions
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