Pexiderization of some logarithmic functional equations

Abstract

We study some new logarithmic functional equations and their Pexiderizations on different structures.

Variations on a theme

“…2. If 𝐴 = 𝔽 + , where 𝔽 is an ordered field then the general solution of equation ( 1) are also the constant functions (Glavosits et al, 2016). The proof requires only one substitution.…”

“…In (Glavosits et al, 2016) we have shown that if 𝔽(+,⋅, ≤ ) is an ordered field, 𝑌 is an arbitrary nonempty set, then the functional equation ( 1) has only constant function solutions on 𝔽 + . In (Glavosits et al, 2016) equation ( 1) is necessary to show that a Pexider type functional equation has only quasi logarithmic function solutions.…”

“…It is only a Hint. The proof in more details can be found in the article(Glavosits et al, 2016).Take the substitution in the equation(1)…”

Decomposition of functional equations with applications

“…This result could perhaps be improved writing simply ordered field instead of Archimedean ordered field, that is, The above two problems have probability background see [18], [2], [13], and [11]. Two pairs of new equations can be derived from the above two equations, the Pexider versions of the second of these equations (which contain the unknown gamma function) characterize the logarithmic functions see [12].…”

Some remark on the functional equation

“…In papers (Davison, 2001;Davison, 1974a;Davison, 1974b; Davison and Redlin, 1980) the equation (1.1) was investigated on various abstract structures. In paper (Fenyő, 1970) a Pexider version (that is functional equation with more unknown function [Aczél, 1983;Aczél and Dhombres, 1989;Kuczma, 1985;Glavosits and Lajkó, 2016]) of equation (1.1) and its locally integrable function solutions can be found. See also (Lajkó et al, 2009;Lajkó and Mészáros, 2015).…”