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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

λ 1 f(x)+λ 2 f 2 (x)++λ n f n (x)=F(x),

where F:I=[a,b]I is given, f is the unknown function and λ 1 >0, i=1 n λ i =1. Under suitable hypothesis for λ 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