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Convex solutions to polynomial-like iterative equations on open intervals. (English) Zbl 1223.39014

The first part of this article deals with nondecreasing convex solutions to the polynomial-like iterative functional equation

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

(f(·):SS (S), f 0 (x)=x, f j (x)=f(f j-1 (x))). Under assumptions λ 1 >0 λ 2 ,,λ n 0 and the existence of continuous nondecreasing convex functions g,hS satisfying inequalities

λ 1 g(x)+λ 2 g 2 (x)++λ n g n (x)F(x),
λ 1 h(x)+λ 2 h 2 (x)++λ n h n (x)F(x),

there exists at least one continuous nondecreasing convex solution f(x) such that g(x)f(x)h(x). The second part of the article presents the analoguous result for p-convex (p-1) solutions of (1) (a function f:S is called convex of order p or p-convex, if

[x 0 ,x 1 ,,x p+1 ;f]0,x 0 <x 1 <<x p+1 S,

([x 0 ,x 1 ,,x p+1 ;f] denotes the divided difference of f at the points x 0 ,x 1 ,,x p+1 ).

MSC:
39B12Iterative and composite functional equations
39B22Functional equations for real functions
26A18Iteration of functions of one real variable
47H10Fixed point theorems for nonlinear operators on topological linear spaces
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