Sequences which are recurrent with respect to a mean (Q1895190)

The authors deal with strict mean values, i.e. functions \(M : X^k \to X\), where \(X\) is a convex subset of a real finite dimensional linear space \(E\), which satisfy \(M(x_1, \dots, x_k) \in C_0 (x_1, \dots, x_k)\), \(C_0 (A)\) denotes \(A\), if \(A\) is a singleton, and \(\text{conv} A \backslash \text{extr} A\), otherwise. A sequence \((x_n)\) is \(M\)-recursive if \(x_{n + k} = M(x_1, \dots, x_{n + k - 1})\), and it is \(k\)-contracting, if \(x_{n + k} \in C_0 (x_n, \dots, x_{n + k - 1})\). The aim of the paper is to show that a sequence is \(k\)-contracting iff it is \(M\)-recursive for a strict mean value \(M\). Moreover, if \(M\) is a continuous strict mean value, then every \(M\)-recursive sequence is also convergent. Conversely, every \(k\)-contracting and convergent sequence is \(M\)-recursive for some \(C^\infty\) strict mean value \(M : E^k \to E\). Also, a notion of limit mean value is introduced and characterized.