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Decreasing solutions and convex solutions of the polynomial-like iterative equation. (English) Zbl 1114.39007
The authors consider the polynomial-like functional equation \(\sum_{i=1}^n \lambda_i f^i(x)=F(x), x\in X\) where \(F\colon X\to X\) and \(\lambda_1,\ldots,\lambda_n\in\mathbb{R}\) are given. For \(X=I\), \(I\) a compact real interval, they give conditions on \(F\), the coefficients \(\lambda_i\) and on the interval \(I\) such that this equation has decreasing or convex increasing or convex decreasing solutions \(f\). The problem of finding such solutions is harder than the analogue question for increasing solutions. The question of uniqueness of solutions is discussed also.

MSC:
39B12 Iteration theory, iterative and composite equations
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