Continuously decreasing solutions for polynomial-like iterative equations

Means of iterates

“…The problem still remains open even for n = 3 [7]. A partial solution in the case where n = 3 and I = R was obtained in [15] and some results in the case where n 3 and I = R can be found in [3,5,10,13,14,16].…”

Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros’ problem

“…Further research in this direction was done in [15,17,20], but still most cases remain unsolved. For some exploration of equation (1.1) on intervals see [6,11,13,16,18] and on half-lines see [4].…”

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].…”

“…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].…”