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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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##### References:
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