Generalized Cauchy difference equations. II

Ebanks, Bruce

doi:10.1090/S0002-9939-08-09379-9

Generalized Cauchy difference equations. II
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by Bruce Ebanks

Proc. Amer. Math. Soc. 136 (2008), 3911-3919

DOI: https://doi.org/10.1090/S0002-9939-08-09379-9

Published electronically: May 20, 2008

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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 $H$.

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