## A generalized Pexider equation.(English)Zbl 0927.39010

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
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