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Generalized Cauchy difference equations. II

Authors:	Bruce Ebanks

Institution:	Department of Mathematics and Statistics, P.O. Box MA, Mississippi State University, Mississippi State, Mississippi 39762

Abstract:	The main result is an improvement of previous results on the equation $\displaystyle f(x)+f(y)-f(x+y)=g\phi(x)+\phi(y)-\phi(x+y)]$ 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 $\displaystyle f(x)+f(y)-f(x+y)=g(H(x,y))$ for a given function . 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 .

Keywords:	Cauchy difference cocycle equation functional independence Pexider equation implicit function theorem philandering regularity properties functional equations

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