Aczél, János; Skof, Fulvia Local Pexider and Cauchy equations. (English) Zbl 1128.39018 Aequationes Math. 73, No. 3, 311-320 (2007). 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 \(D\subset\mathbb{R}^2\). Reviewer: Borislav Crstici (Timişoara) Cited in 5 Documents MSC: 39B22 Functional equations for real functions Keywords:extension; quasiextension; general solution; bounded on a set of positive measure; smooth; regular; linear spaces; local Pexider functional equation; restricted exponential Cauchy functional equation; counter example Citations:Zbl 0175.15305; Zbl 0625.39007 PDFBibTeX XMLCite \textit{J. Aczél} and \textit{F. Skof}, Aequationes Math. 73, No. 3, 311--320 (2007; Zbl 1128.39018) Full Text: DOI