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Convex solutions of polynomial-like iterative equations. (English) Zbl 1090.39012
The authors study the following polynomial-like iterative functional equation $$ \lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x), $$ where $F:I=[a,b] \to I$ is given, $f$ is the unknown function and $\lambda_1>0$, $\sum_{i=1}^n \lambda_i=1$. Under suitable hypothesis for $\lambda_i$’s and Lipschitzianity of $F$, they obtain existence theorems for continuous and for convex solutions.

MSC:
39B12Iterative and composite functional equations
39B22Functional equations for real functions
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References:
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