# American Institute of Mathematical Sciences

October  2022, 42(10): 4965-4990. doi: 10.3934/dcds.2022082

## Iterative roots of type $\mathcal {T}_2$

 1 Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland 3 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

*Corresponding author: Weinian Zhang

Received  December 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The fourth author is supported by NSFC grant #11831012, #11821001 and #12171336

This paper aims to an open problem on iterative roots of PM functions, a class of non-monotonic functions. The open problem asks: Does a PM function of nonmonotonicity height $\ge 2$ have a continuous iterative root of order $n$ being less than or equal to the number of forts? It was proved that iterative roots of order $n$ being equal to the number of forts (if exist) can be classified into two types: mostly increasing ones (type $\mathcal {T}_1$) and mostly decreasing ones (type $\mathcal {T}_2$) and all roots of type $\mathcal {T}_1$ are found, but the remaining type $\mathcal {T}_2$ is left for more complicated construction. In this paper full description of type $\mathcal {T}_2$ roots is given.

Citation: Liu Liu, Justyna Jarczyk, Witold Jarczyk, Weinian Zhang. Iterative roots of type $\mathcal {T}_2$. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4965-4990. doi: 10.3934/dcds.2022082
##### References:
 [1] Jr. M. K. Fort, The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.  doi: 10.1090/S0002-9939-1955-0080911-2. [2] N. Iannella and L. Kindermann, Finding iterative roots with a spiking neural network, Inform. Process. Lett., 95 (2005), 545-551.  doi: 10.1016/j.ipl.2005.05.022. [3] M. C. Irwin, Smooth Dynamical Systems, Academic Press, 1980. [4] L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. [5] M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publ., Warsaw, 1968. [6] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [7] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Anal., 75 (2012), 286-303.  doi: 10.1016/j.na.2011.08.033. [8] L. Liu, L. Li and W. Zhang, Open question on lower order iterative roots for PM functions, J. Diff. Equ. Appl., 24 (2018), 825-847.  doi: 10.1080/10236198.2018.1437152. [9] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037. [10] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An introduction, Springer, New York, 1982. [11] T. Sun, Iterative roots of anti-N-type functions on intervals, [Chinese], J. Math. Study, 33 (2000), 274-280. [12] T. Sun and H. Xi, Iterative roots of $N$-type of functions on intervals, [Chinese], J. Math. Study, 29 (1996), 40-45. [13] G. Targonski, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. [14] G. Targonski, Progress of iteration theory since 1981, Aequationes Math., 50 (1995), 50-72.  doi: 10.1007/BF01831113. [15] J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, John Wiley & Sons, 1986. [16] G. Zhang, Conjugacy and iterative roots of a class of linear self-mapping (I), [Chinese], Chin. Ann. Math. A, 13 (1992), 33-40. [17] J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta Math. Sinica, 26 (1983), 398-412. [18] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. [19] W. Zhang, A generic property of globally smooth iterative roots, Science in China A, 38 (1995), 267–272. [20] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.  doi: 10.4064/ap-65-2-119-128.

show all references

##### References:
 [1] Jr. M. K. Fort, The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6 (1955), 960-967.  doi: 10.1090/S0002-9939-1955-0080911-2. [2] N. Iannella and L. Kindermann, Finding iterative roots with a spiking neural network, Inform. Process. Lett., 95 (2005), 545-551.  doi: 10.1016/j.ipl.2005.05.022. [3] M. C. Irwin, Smooth Dynamical Systems, Academic Press, 1980. [4] L. Kindermann, Computing iterative roots with neural networks, Proc. Fifth Conf. Neural Info. Processing, 2 (1998), 713-715. [5] M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publ., Warsaw, 1968. [6] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [7] L. Liu, W. Jarczyk, L. Li and W. Zhang, Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2, Nonlinear Anal., 75 (2012), 286-303.  doi: 10.1016/j.na.2011.08.033. [8] L. Liu, L. Li and W. Zhang, Open question on lower order iterative roots for PM functions, J. Diff. Equ. Appl., 24 (2018), 825-847.  doi: 10.1080/10236198.2018.1437152. [9] L. Liu and W. Zhang, Non-monotonic iterative roots extended from characteristic intervals, J. Math. Anal. Appl., 378 (2011), 359-373.  doi: 10.1016/j.jmaa.2011.01.037. [10] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An introduction, Springer, New York, 1982. [11] T. Sun, Iterative roots of anti-N-type functions on intervals, [Chinese], J. Math. Study, 33 (2000), 274-280. [12] T. Sun and H. Xi, Iterative roots of $N$-type of functions on intervals, [Chinese], J. Math. Study, 29 (1996), 40-45. [13] G. Targonski, Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981. [14] G. Targonski, Progress of iteration theory since 1981, Aequationes Math., 50 (1995), 50-72.  doi: 10.1007/BF01831113. [15] J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, John Wiley & Sons, 1986. [16] G. Zhang, Conjugacy and iterative roots of a class of linear self-mapping (I), [Chinese], Chin. Ann. Math. A, 13 (1992), 33-40. [17] J. Zhang and L. Yang, Iterative roots of a piecewise monotone continuous self-mapping, Acta Math. Sinica, 26 (1983), 398-412. [18] J. Zhang, L. Yang and W. Zhang, Some advances on functional equations, Adv. Math. Chin., 24 (1995), 385-405. [19] W. Zhang, A generic property of globally smooth iterative roots, Science in China A, 38 (1995), 267–272. [20] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.  doi: 10.4064/ap-65-2-119-128.
type $\mathcal {T}_1$, lower
type $\mathcal {T}_1$, upper
type $\mathcal {T}_2$, upper
type $\mathcal {T}_2$, lower
$S(f) = \{c_1\}$
$S(f) = \{c_n\}$
$F$ is in (i-1)
$F$ is in (i-2)
$F$ is in (i-3)
$F$ is in (i-3)
$F$ is in (ii-1)
$F$ is in (ii-2)
$F$ is in (iii-2)
$F$ is in (iv-2)
$F_1$ with $N(F_1) = 3$, $f\in \mathcal {T}_2$ with $S(f) = \frac{1}{4}$
$F_2$ with $N(F_2) = 4$, $f\in \mathcal {T}_2$ with $S(f) = \frac{7}{8}$
 [1] Liu Liu, Weinian Zhang. Genetics of iterative roots for PM functions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2391-2409. doi: 10.3934/dcds.2020369 [2] Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003 [3] Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871 [4] Antonis Papapantoleon, Robert Wardenga. Continuous tenor extension of affine LIBOR models with multiple curves and applications to XVA. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 1-. doi: 10.1186/s41546-017-0025-4 [5] Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180 [6] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895 [7] David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499 [8] Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks and Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191 [9] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [10] Xiaoming Zheng, Gou Young Koh, Trachette Jackson. A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, platelet-derived growth factor-B, and pericytes. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1109-1154. doi: 10.3934/dcdsb.2013.18.1109 [11] Andrew L. Nevai, Richard R. Vance. The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 101-124. doi: 10.3934/mbe.2008.5.101 [12] Santanu Sarkar, Subhamoy Maitra. Some applications of lattice based root finding techniques. Advances in Mathematics of Communications, 2010, 4 (4) : 519-531. doi: 10.3934/amc.2010.4.519 [13] Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011 [14] Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 67-84. doi: 10.3934/dcdss.2020004 [15] Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109 [16] Jacek Serafin. A faithful symbolic extension. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051 [17] Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187 [18] Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2039-2058. doi: 10.3934/cpaa.2014.13.2039 [19] Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409 [20] Yan Liu, Minjia Shi, Hai Q. Dinh, Songsak Sriboonchitta. Repeated-root constacyclic codes of length $3\ell^mp^s$. Advances in Mathematics of Communications, 2020, 14 (2) : 359-378. doi: 10.3934/amc.2020025

2021 Impact Factor: 1.588