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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

${\lambda }_{1}f\left(x\right)+{\lambda }_{2}{f}^{2}\left(x\right)+\cdots +{\lambda }_{n}{f}^{n}\left(x\right)=F\left(x\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

($f\left(·\right):\phantom{\rule{4pt}{0ex}}S\to S$ ($S\subset ℝ$), ${f}^{0}\left(x\right)=x$, ${f}^{j}\left(x\right)=f\left({f}^{j-1}\left(x\right)\right)$). Under assumptions ${\lambda }_{1}>0$ ${\lambda }_{2},\cdots ,{\lambda }_{n}\le 0$ and the existence of continuous nondecreasing convex functions $g,h\in S$ satisfying inequalities

${\lambda }_{1}g\left(x\right)+{\lambda }_{2}{g}^{2}\left(x\right)+\cdots +{\lambda }_{n}{g}^{n}\left(x\right)\le F\left(x\right),$
${\lambda }_{1}h\left(x\right)+{\lambda }_{2}{h}^{2}\left(x\right)+\cdots +{\lambda }_{n}{h}^{n}\left(x\right)\ge F\left(x\right),$

there exists at least one continuous nondecreasing convex solution $f\left(x\right)$ such that $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$. The second part of the article presents the analoguous result for $p$-convex ($p\ge -1$) solutions of (1) (a function $f:\phantom{\rule{4pt}{0ex}}S\to ℝ$ is called convex of order $p$ or $p$-convex, if

$\left[{x}_{0},{x}_{1},\cdots ,{x}_{p+1};f\right]\ge 0,\phantom{\rule{2.em}{0ex}}{x}_{0}<{x}_{1}<\cdots <{x}_{p+1}\in S,$

($\left[{x}_{0},{x}_{1},\cdots ,{x}_{p+1};f\right]$ denotes the divided difference of $f$ at the points ${x}_{0},{x}_{1},\cdots ,{x}_{p+1}$).

##### MSC:
 39B12 Iterative and composite functional equations 39B22 Functional equations for real functions 26A18 Iteration of functions of one real variable 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
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