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A functional equation involving f and \(f^{-1}\). (English) Zbl 0731.39006

Motivated by a more special problem, the authors show first that for an arbitrary bijection f on a compact interval (1) \(f(2x-f(x))=x\) implies that \(f(x)=x\). Equivalent forms of (1) are (2) \(1/2(f(x)+f^{-1}(x))=x\) and (3) \(f^ 2(x)-2f(x)+x=0.\) Generalizing (3), it is shown that \((3)'\exists n\sum^{n}_{k=0}(-1)^ k\left( \begin{matrix} n\\ k\end{matrix} \right)f^{n-k}(x)=0\) also implies \(f(x)=x.\) The same result is shown to be true for the following generalization of (2): \((2)'\exists \alpha \quad \quad \alpha f(x)+(1-\alpha)f^{-1}(x)=x\), \(0<\alpha <1.\)
So far f was a real function. In another direction the authors prove a result motivated by the result on (3), about linear self-mappings A of a vector space. A is invertible and \(\sum^{n}_{k=0}(-1)^ k\left( \begin{matrix} n\\ k\end{matrix} \right)A^{n-k}=0\) if and only if \(A=I+T+...+T^{n-1}\) and \(T^ n=0\) for some linear self-map T of the vector space.

MSC:

39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
39B42 Matrix and operator functional equations
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