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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
$\lambda_1f(x) + \lambda_2f^2(x) + \dots +\lambda_n f^n(x) = F(x),\tag{1}$
($$f(\cdot):\;S \to S$$ ($$S \subset {\mathbb R}$$), $$f^0(x) = x$$, $$f^j(x) = f(f^{j-1}(x))$$). Under assumptions $$\lambda_1 > 0$$ $$\lambda_2, \dots, \lambda_n \leq 0$$ and the existence of continuous nondecreasing convex functions $$g,h \in S$$ satisfying inequalities
$\lambda_1g(x) + \lambda_2g^2(x) + \dots + \lambda_n g^n(x) \leq F(x),$
$\lambda_1h(x) + \lambda_2h^2(x) + \dots +\lambda_n h^n(x) \geq F(x),$
there exists at least one continuous nondecreasing convex solution $$f(x)$$ such that $$g(x) \leq f(x) \leq h(x)$$. The second part of the article presents the analoguous result for $$p$$-convex ($$p \geq -1$$) solutions of (1) (a function $$f:\;S \to {\mathbb R}$$ is called convex of order $$p$$ or $$p$$-convex, if
$[x_0,x_1,\dots,x_{p+1};f] \geq 0, \qquad x_0 < x_1 < \dots < x_{p+1} \in S,$
($$[x_0,x_1,\dots,x_{p+1};f]$$ denotes the divided difference of $$f$$ at the points $$x_0,x_1,\dots,x_{p+1}$$).

##### MSC:
 39B12 Iteration theory, iterative and composite equations 39B22 Functional equations for real functions 26A18 Iteration of real functions in one variable 47H10 Fixed-point theorems
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