Taylor's theorem and functional equations

Functional equations and the Cauchy mean value theorem

“…More general functional equations have been considered even in the abstract setting of groups by several authors including Kannappan [6], Ebanks [3], Fechner-Gselmann [4]. On the other hand, the result of Aczél and Haruki has been generalized for higher order Taylor expansion by Sablik [8].…”

A method of solving functional equations on convex subsets of linear spaces

“…Our aim is to solve (11) in a more abstract case that is in a linear space in which the interval is replaced by a convex set. First, let us note that equation (11) makes sense in such domains. We have to explain what the multiplications on the right-hand side mean.…”

“…But the solutions are locally polynomial functions and translations do not change this property and their degree. In the sequel, K denotes a nonempty convex subset of X with 0 ∈ alg int K. In particular, the functions in (11) are defined as f : K → Y , g i : K → SA i (X, Y ), i ∈ {0, . .…”

Solutions of two functional equations using a result of M. Sablik

“…In [9] M. Sablik obtained a polynomial analogue of L. Székelyhidi's theorem from [10] (cf. also W. H. Wilson [11]).…”