Convex solutions to polynomial-like iterative equations on open intervals

Abstract

The paper deals with the polynomial-like iterative functional equationBy 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.Mathematics Subject Classification (2010). Primary 39B12; Secondary 26A18, 47H10.

On a Zoltán Boros’ problem connected with polynomial-like iterative equations

“…The second reason is that equation (1.1) belongs to the class of important and intensively investigated iterative functional equations; i.e., the class of polynomial-like iterative equations of the form (1.3) N n=0 a n g n (x) = F (x), where a n 's are given real numbers, F : I → I is a given function and g : I → I is the unknown function. For the theory of equation (1.3) and its generalizations we refer the readers to books [6,14], surveys [1,22], and some recent papers [3,4,5,7,10,12,13,15,16,17,20,21].…”

Convex Solutions of the Polynomial-Like Iterative Equation on Open Set

“…The study of convexity for iterative equations can be traced to 1968, when Kuczma and Smajdor [8] investigated the convexity of iterative roots. Some recent results can be found from [20,21,28] in 1-dimensional space and in high-dimensional spaces one can refer to [5]. In [28], convex solutions and concave ones of Eq.…”

“…(1.1) are completely investigated with no normalization condition and no requirement of uniform sign of coefficients on a compact interval in [21]. In [20], nondecreasing convex solutions for Eq. (1.1) on open intervals (possibly unbounded) were discussed.…”

“…In this paper, motivated by [20], we consider Eq. (1.1) on an open set (possibly unbounded) in R N .…”

Convexity of Solutions for an Iterative Equation in Banach Spaces

“…Convexity is an important property of functions and the study of convexity for iterative equations can be traced to 1968 when Kuczma and Smajdor [25] investigated the convexity of iterative roots. Some recent results can be found from [17,[26][27][28]. In [27,28], convexity of solutions for (1) was discussed on a compact interval, and in [26], nondecreasing convex solutions for (1) on open intervals were discussed.…”