For a piecewise monotone function F of height 1,an open question was raised:Does F have an iterative root f of order n≤N(F)+1 if the‘characteristic endpoints condition’is not satisfied?This question was answered pa...For a piecewise monotone function F of height 1,an open question was raised:Does F have an iterative root f of order n≤N(F)+1 if the‘characteristic endpoints condition’is not satisfied?This question was answered partly in the case that F is strictly increasing on its characteristic interval K(F)but f is strictly decreasing on K(F).In this paper we discuss the question for F increasing on K(F)in some remaining cases,giving the necessary and sufcient conditions for the existence of continuous iterative roots f decreasing on K(F)of order n=N(F)>2 with H(f)=n-1.展开更多
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 func...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.展开更多
基金Supported by the Natural Science Foundation of Sichuan(Grant No.2022NSFSC1812)。
文摘For a piecewise monotone function F of height 1,an open question was raised:Does F have an iterative root f of order n≤N(F)+1 if the‘characteristic endpoints condition’is not satisfied?This question was answered partly in the case that F is strictly increasing on its characteristic interval K(F)but f is strictly decreasing on K(F).In this paper we discuss the question for F increasing on K(F)in some remaining cases,giving the necessary and sufcient conditions for the existence of continuous iterative roots f decreasing on K(F)of order n=N(F)>2 with H(f)=n-1.
基金supported by the Natural Science Foundation of Shandong Province(ZR2017MA019)the Scientific Research Fund of Binzhou University(BZXYL1802)+2 种基金supported by the National Science Foundation of China(11501394)the Science Research Fund of Sichuan Provincial Education Department(15ZB0041)funding of School of Mathematical Sciences and V.C.&V.R.Key Lab of Sichuan Province。
文摘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.
