Local Pexider and Cauchy equations (Q2373345)

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. \textit{Z. Daróczy} and \textit{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. \textit{F. Radó} and \textit{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 \(D\subset\mathbb{R}^2\).