Abstract
In this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial {mathcal {C}}^{infty } solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented.
Highlights
We start by considering the following relation f (x + y) − f (x) − f (y) = F(x, y), (1.1)where f : R → R and we assume that f is absolutely continuous and has derivative at each point.Following [2], the first step consists in giving a suitable representation of both f and F
We assume that the triple (f, g, h) is a solution of equation (3.1), with all functions in C(R) and at least one of them, say f, real analytic
We present some other results, namely some conditions giving only trivial solutions of equation (3.1) and we prove by giving examples that without them there exist non trivial solutions
Summary
We start by considering the following relation f (x + y) − f (x) − f (y) = F(x, y),. Following [2], the first step consists in giving a suitable representation of both f and F. We conclude that if f and F satisfy (1.1), there exists a continuous function φ such that (1.5) holds and x f (x + y) − f (x) − f (y) = [φ(t + y) − φ(t)]dt + k. This simple result will be useful for solving the problem presented
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