Abstract
The paper deals with the polynomial-like iterative functional equation $$\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$$ By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.
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