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Contributions to the theory of Lie’s functional equation of translation type. (English) Zbl 1193.39011

By a solution of Lie’s functional equation of translation type, \[ f(\omega(\xi_1,\dots, \xi_n))= \varphi_1(\xi_1)+\cdots+ \varphi_n(\xi_n),\tag{1} \] the authors consider the following structure
\(\bullet\) an additively written group \(G\) which, however, needs not to be Abelian,
\(\bullet\) sets \(S,A_1,\dots, A_n\), \(\Delta\) satisfying \(\emptyset\neq\Delta\subseteq A_1\times\cdots\times A_n\), \(S\neq \emptyset\),
\(\bullet\) functions \(f: S\to G\), \(\omega:\Delta\to S\), \(\varphi_i: B_i\to G\) \((i= l,\dots,n)\)
such that (1) holds true for all \((\xi_1,\dots,\xi_n)\in \Delta\), where \(B_i\) denotes the set of all \(\xi_i\in A_i\) such that there exists \((\eta_1,\dots, \eta_n)\in \Delta\) with \(\eta_i= \xi_i\). The functional equation is investigated in different algebraic situations, and some general solutions are presented. Also, the functional equation of twofold translation surfaces is studied in different situations.

MSC:

39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
39B52 Functional equations for functions with more general domains and/or ranges
53A05 Surfaces in Euclidean and related spaces
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