A generalized Pexider equation. (English) Zbl 0927.39010

Bajger, M.

Acta Math. Hung. 75, No. 1-2, 43-54 (1997).

Summary: Let \(K\) be a groupoid and \(D(K)\) be the domain of the binary operation on \(K\) (for convenience we will not use any symbol for the operation). Consider the following generalized Pexider equation \[ F(st)=k(st) \circ H(s)\circ G(t), \quad (s,t)\in D(K), \tag{GPE} \] where the functions \(F\), \(G\), \(H\), \(k\) to be determined are defined on \(K\) and take values in some function spaces (with composition of functions as a binary operation).
In the sequel we confine ourselves to the case of three unknown functions \(F\), \(G\), \(H\) and \(k\) is supposed to be a given one. From this point of view (GPE) has been studied by the author in [Publ. Math. 44, No. 1-2, 67-77 (1994; Zbl 0827.39008) and Publ. Math. 48, No. 1-2, 77-88 (1996; Zbl 0862.39010)] where, among other things, the general solution on a groupid with a unit element has been given. In this paper we improve the result presenting the general solution of (GPE) under the basic assumption that each element of \(K\) has left and right units. Further, we apply the obtained result to describe the general solution \(F\), \(G\), \(H\) of the equation \[ F(st)=k(st)+H(s)+G(t), \qquad (s,t)\in D(K) \tag{1} \] among functions mapping \(K\) into an arbitrary group \((E,+)\).

MSC:

39B52	Functional equations for functions with more general domains and/or ranges
20N02	Sets with a single binary operation (groupoids)

Keywords:

Pexider equation; groupoid

Citations:

Zbl 0827.39008; Zbl 0862.39010