A generalized result on the polynomial-like iterative equation
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Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, 2021
http://actams.apm.ac.cn
A GENERALIZED RESULT ON THE POLYNOMIAL-LIKE ITERATIVE EQUATION∗
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Pingping ZHANG (
College of Science, Binzhou University, Binzhou 256603, China E-mail : [email protected]
QCC)
†
Yingying ZENG (
School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu 610068, China E-mail : [email protected] Abstract Most results on the polynomial-like iterative equation are given under the condition that the given function is monotone, while a work by L. Liu and X. Gong gets nonmonotone PM solutions with height 1 when the given function is of the same case. Removing the condition on height for the given function, we first give a method to assert the nonexistence of C 0 solutions, then present equivalent conditions for the existence of PM solutions with finite height. Finally, as an application of the equivalent conditions, we construct the PM solutions in the case that the given function has one fort. Key words
height; characteristic interval; fort; PM function
2010 MR Subject Classification
1
39B12; 37E05
Introduction
Let I := [a, b] ⊆ R and C(I, I) consist of all C 0 self-maps ϕ on I. For each nonnegative integer n, the n-th iterate of ϕ ∈ C(I, I) is defined by ϕn = ϕ ◦ ϕn−1 and ϕ0 = id, recursively. A functional equation involving an iterate as its main operation is an iterative equation. The main difficulty here comes from the nonlinearity of the iterate operator. The iterative roots problem ([1, 2]), connecting discrete and continuous dynamical systems, solves a special iterative equation f n (x) = F (x),
x ∈ I,
where F : I → I is a given self-map. ∗ Received October 29, 2019; revised September 11, 2020. The first author is supported by the Natural Science Foundation of Shandong Province (ZR2017MA019) and the Scientific Research Fund of Binzhou University (BZXYL1802); the second author is supported by the National Science Foundation of China (11501394), the Science Research Fund of Sichuan Provincial Education Department (15ZB0041) and funding of School of Mathematical Sciences and V.C. & V.R. Key Lab of Sichuan Province. † Corresponding author: Yingying ZENG.
178
ACTA MATHEMATICA SCIENTIA
Vol.41 Ser.B
In recent years, as a generalization of iterative roots, the polynomial-like iterative equation f n (x) + λn−1 f n−1 (x) + · · · + λ1 f (x) = F (x), x ∈ I
(1.1)
has attracted a lot of attention ([3, 5, 7, 8, 14]). Here λi s (i = 1, 2, · · · , n − 1) are real constants and the known function F may not be a self-map. For strictly monotone function F , different kinds of solutions of eq. (1.1), such as continuous ([9, 19]), smooth ([4]), analytic ([12]), convex ([6]) and equivariant ([17, 18]) solutions, have been investigated widely. For non-monotone function F , the results (e.g. periodic solution ([11])) are relatively few. Recently, for a given PM function F with height H(F ) = 1, the PM solutions f of eq. (1.1) are investigat
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