On continuous solutions of a functional equation of iterative type
The authors consider the functional equation \[ \sum_{i=1}^{n} \lambda_{i} f^{2i-1}(x)=F(x), \] where the given function \(F:[a,b]\rightarrow [a,b]\) is strictly decreasing and continuous, \(n\geq 2\), \(\lambda_{1} \in (0,1)\), \(\lambda_{2}, \ldots , \lambda_{n} \geq 0\) and \(\sum_{i=1}^{n} \lambda_{i}=1\). Moreover, it is assumed that \(F(a)=b\), \(F(b)=a\) and there exist \(r, s > 0\) such that \[ r\;(y-x) \leq F(x)-F(y) \leq \lambda_{1} \;s \;(y-x) \] for all \(x, y \in [a,b]\), \(x<y\). The existence of a continuous solution \(f:[a,b]\rightarrow [a,b]\) is obtained by using Schauder's fixed point theorem. Under an additional assumption it is proved that there exists exactly one continuous solution \(f:[a,b]\rightarrow [a,b]\) of the equation such that \(f(a)=b\), \(f(b)=a\) and \(0 \leq f(x)-f(y) \leq s \;(y-x) \) for all \(x, y \in [a,b]\), \(x<y\). It is also shown that the unique solution \(f\) depends continuously on the given fuction \(F\).
scientific article; zbMATH DE number 2104677