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Generalized Cauchy difference equations. II. (English) Zbl 1206.39022
Author’s abstract: The main result is an improvement of previous results on the equation
\[ f(x)+f(y)-f(x+y)=g[\phi(x)+\phi(y)-\phi(x+y)] \]
for a given function \(\phi\). We find its general solution assuming only continuous differentiability and local nonlinearity of \(\phi\). We also provide new results about the more general equation
\[ f(x)+f(y)-f(x+y)=g(H(x,y)) \]
for a given function \(H\). Previous uniqueness results required strong regularity assumptions on a particular solution \(f_0,g_0\). Here we weaken the assumptions on \(f_0,g_0\) considerably and find all solutions under slightly stronger regularity assumptions on \(H\).
[For part I, see Aequationes Math. 70, No. 1–2, 154–176 (2005; Zbl 1079.39017).]

MSC:
39B22 Functional equations for real functions
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[1] J. Aczél, A short course on functional equations, Theory and Decision Library. Series B: Mathematical and Statistical Methods, D. Reidel Publishing Co., Dordrecht, 1987. Based upon recent applications to the social and behavioral sciences. · Zbl 0607.39002
[2] Bruce R. Ebanks, On Heuvers’ logarithmic functional equation, Results Math. 42 (2002), no. 1-2, 37 – 41. · Zbl 1044.39018 · doi:10.1007/BF03323552 · doi.org
[3] Bruce Ebanks, Generalized Cauchy difference functional equations, Aequationes Math. 70 (2005), no. 1-2, 154 – 176. · Zbl 1079.39017 · doi:10.1007/s00010-004-2739-5 · doi.org
[4] B. R. Ebanks, P. L. Kannappan, and P. K. Sahoo, Cauchy differences that depend on the product of arguments, Glas. Mat. Ser. III 27(47) (1992), no. 2, 251 – 261 (English, with English and Serbo-Croatian summaries). · Zbl 0780.39007
[5] István Ecsedi, On the functional equation \?(\?+\?)-\?(\?)-\?(\?)=\?(\?\?), Mat. Lapok 21 (1970), 369 – 374 (1971) (Hungarian, with English summary). · Zbl 0231.39007
[6] Konrad J. Heuvers, Another logarithmic functional equation, Aequationes Math. 58 (1999), no. 3, 260 – 264. · Zbl 0939.39016 · doi:10.1007/s000100050112 · doi.org
[7] Antal Járai, Regularity properties of functional equations in several variables, Advances in Mathematics (Springer), vol. 8, Springer, New York, 2005. · Zbl 1081.39022
[8] Antal Járai, Gyula Maksa, and Zsolt Páles, On Cauchy-differences that are also quasisums, Publ. Math. Debrecen 65 (2004), no. 3-4, 381 – 398. · Zbl 1071.39026
[9] K. Lajkó, Special multiplicative deviations, Publ. Math. Debrecen 21 (1974), 39 – 45. · Zbl 0293.39004
[10] Gyula Maksa, On the functional equation \?(\?+\?)+\?(\?\?)=\?(\?)+\?(\?), Publ. Math. Debrecen 24 (1977), no. 1-2, 25 – 29. · Zbl 0375.39002
[11] Walter Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. International Series in Pure and Applied Mathematics. · Zbl 0346.26002
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