折线函数和集值函数的迭代与迭代根
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摘要
迭代是自然界中一个重要的现象。X-射线的透射、流体的渗流、生物体的生长、计算机的运行等过程都包含了迭代现象。在科学计算中，迭代也经常作为有效的工具解决近似问题。而在数学中，一切递推关系，从等差数列、等比数列到微分方程解的Picard逼近都是一个迭代过程。迭代所产生的动力系统刻划了事物运动的主要环节和发展趋势，通过迭代可以预测未来，也就是我们所关心的长期性和终极性状态。另一方面，我们也关心事物运动的全过程，尤其是各环节之间的联系，这就涉及到迭代的逆运算，即迭代根问题。通过这样的方式，我们可以将离散问题的各环节合理地连接起来，还原成完整的连续的衍变过程。
迭代是一种十分复杂的非线性运算。对于映射n次迭代通式的计算，常用的方法有不动点法和共扼相似法，前者的运用需要事先断定映射迭代式的基本代数形式，而后者则需要找一个可逆的桥函数，因此在许多情况下计算迭代是非常困难的。在第二章我们讨论的是区间上一类折线函数的迭代。尽管折线函数是最简单的非线性函数，其迭代的规律十分复杂，函数值在迭代下可能交叉于不同的子区间。我们利用折点的运动轨道变化来探索迭代下折点数不增或有界的条件，并在若干情形下给出其n次迭代表达式。
我们知道，不具有连续性的函数一般来说性质很糟，而具有上半连续及有限集值点的集值函数在一定程度上反映了第一类间断点函数的特性，因此研究集值函数的迭代也是很有意义的。集值分析作为建立非线性数学模型、解决非线性问题的数学理论和有力工具，它已经成为非线性分析的重要组成部分，在控制论和微分对策、数理经济学和决策论、生物数学、物理以及微分包含等众多领域都有着广泛的应用。集值点是集值函数之所以复杂的根本原因。集值点越少，问题相对越简单。如同对折线函数折点的研究一样，本文的第三章将研究一类具有上半连续且单集值点的集值函数的迭代，给出其在迭代下集值点个数不增的条件，并在此条件下给出一般的迭代表达式。
在第三章的基础上，我们讨论这类集值函数的迭代根问题。2004年，W．Jarczyk和张伟年讨论了一类集值函数的2次迭代根，给出若干其迭代根不存在的充分条件。在第四章，我们继续给出一些他们没有给出的2次迭代根不存在的条件。另外，我们还证明了一类集值函数2次迭代根的存在条件。
Iteration is an important phenomenon in nature. The penetration of X-ray, infiltration of liquid, growth of organism and application of computer are examples of iteration. In scientific computation, iteration is as an effective tool to deal with reckon problems. All recurrence relations from equal difference, equal ratios sequences to approximation by Picard are the course of iterating in mathematics. Iteration emerges dynamical system, which describe the main segments and tendency of development. By iterating, we can forecast future, that is long-time behavior and final statements which we were concerned. On the other hand, we also pay attention to its process, especially the relationship between all the segments, which refers to the reversal operation of iteration called iterative root, so that the discrete parts can be connected reasonably.
Iteration is a complicated nonlinear operation. We usually use fixed point and conjugacy method to compute iteration. For fixed point method, we should know its iteration's algebraic form first, and for conjugacy method, a bridge function must be found. Therefore, it is difficult to compute its general iteration in many cases. In chapter 2, we investigate the iteration of polygonal functions on intervals. Although polygonal function is a nonlinear mapping with elementary form, computing its iterates is not easy because the concerned different lines on distinct sub-intervals may interact each other in iteration. We will make use of the dynamics of vertices' orbits to discuss the conditions under which the number of vertices either does not increase or has a bound under iteration. In some cases the explicit expressions of their general iteration are given.
As we know, for a kind of functions without continuity, the property is usually no good enough, but upper semi-continuous multifunctions with finite set-valued points reflect the properties of first set functions which have finite disjoint points, so it is meaningful to consider multifunctions. Multivalued analysis as an effective method in building up and solving nonlinear mathematical models has become an important part in nonlinear analysis. It also has comprehensive applications in control theory, differential decision, economics, bio-mathematics, physics, differential inclusion and so on. The number of set-valued points increasing is the main reason for the complexity of iteration of multifunctions, the problem becomes easier as the reduction of set-valued points. As we discussed for polygonal functions, in chapter 3, we investigate the iteration of upper semi-continuous multifunctions with one set-valued point, giving conditions for the number of set-valued points not to increase under iteration, and the explicit expressions of their general iteration are given in the cases.
Based on chapter 3, we continue to discuss the problem of its iterative roots. In 2004, W. Jarczyk and Weinian Zhang investigate the 2-th order iterative roots of a kind multifunctions and give some conditions of nonexistence. In chapter 4, we continue to find more possibilities of nonexistence. In addition, we also present the condition of existence for a kind of multifunctions.
迭代是一种十分复杂的非线性运算。对于映射n次迭代通式的计算，常用的方法有不动点法和共扼相似法，前者的运用需要事先断定映射迭代式的基本代数形式，而后者则需要找一个可逆的桥函数，因此在许多情况下计算迭代是非常困难的。在第二章我们讨论的是区间上一类折线函数的迭代。尽管折线函数是最简单的非线性函数，其迭代的规律十分复杂，函数值在迭代下可能交叉于不同的子区间。我们利用折点的运动轨道变化来探索迭代下折点数不增或有界的条件，并在若干情形下给出其n次迭代表达式。
我们知道，不具有连续性的函数一般来说性质很糟，而具有上半连续及有限集值点的集值函数在一定程度上反映了第一类间断点函数的特性，因此研究集值函数的迭代也是很有意义的。集值分析作为建立非线性数学模型、解决非线性问题的数学理论和有力工具，它已经成为非线性分析的重要组成部分，在控制论和微分对策、数理经济学和决策论、生物数学、物理以及微分包含等众多领域都有着广泛的应用。集值点是集值函数之所以复杂的根本原因。集值点越少，问题相对越简单。如同对折线函数折点的研究一样，本文的第三章将研究一类具有上半连续且单集值点的集值函数的迭代，给出其在迭代下集值点个数不增的条件，并在此条件下给出一般的迭代表达式。
在第三章的基础上，我们讨论这类集值函数的迭代根问题。2004年，W．Jarczyk和张伟年讨论了一类集值函数的2次迭代根，给出若干其迭代根不存在的充分条件。在第四章，我们继续给出一些他们没有给出的2次迭代根不存在的条件。另外，我们还证明了一类集值函数2次迭代根的存在条件。
Iteration is an important phenomenon in nature. The penetration of X-ray, infiltration of liquid, growth of organism and application of computer are examples of iteration. In scientific computation, iteration is as an effective tool to deal with reckon problems. All recurrence relations from equal difference, equal ratios sequences to approximation by Picard are the course of iterating in mathematics. Iteration emerges dynamical system, which describe the main segments and tendency of development. By iterating, we can forecast future, that is long-time behavior and final statements which we were concerned. On the other hand, we also pay attention to its process, especially the relationship between all the segments, which refers to the reversal operation of iteration called iterative root, so that the discrete parts can be connected reasonably.
Iteration is a complicated nonlinear operation. We usually use fixed point and conjugacy method to compute iteration. For fixed point method, we should know its iteration's algebraic form first, and for conjugacy method, a bridge function must be found. Therefore, it is difficult to compute its general iteration in many cases. In chapter 2, we investigate the iteration of polygonal functions on intervals. Although polygonal function is a nonlinear mapping with elementary form, computing its iterates is not easy because the concerned different lines on distinct sub-intervals may interact each other in iteration. We will make use of the dynamics of vertices' orbits to discuss the conditions under which the number of vertices either does not increase or has a bound under iteration. In some cases the explicit expressions of their general iteration are given.
As we know, for a kind of functions without continuity, the property is usually no good enough, but upper semi-continuous multifunctions with finite set-valued points reflect the properties of first set functions which have finite disjoint points, so it is meaningful to consider multifunctions. Multivalued analysis as an effective method in building up and solving nonlinear mathematical models has become an important part in nonlinear analysis. It also has comprehensive applications in control theory, differential decision, economics, bio-mathematics, physics, differential inclusion and so on. The number of set-valued points increasing is the main reason for the complexity of iteration of multifunctions, the problem becomes easier as the reduction of set-valued points. As we discussed for polygonal functions, in chapter 3, we investigate the iteration of upper semi-continuous multifunctions with one set-valued point, giving conditions for the number of set-valued points not to increase under iteration, and the explicit expressions of their general iteration are given in the cases.
Based on chapter 3, we continue to discuss the problem of its iterative roots. In 2004, W. Jarczyk and Weinian Zhang investigate the 2-th order iterative roots of a kind multifunctions and give some conditions of nonexistence. In chapter 4, we continue to find more possibilities of nonexistence. In addition, we also present the condition of existence for a kind of multifunctions.
引文
[1] Abel N. H., Oeuvres completes, Christiana, 1881, Ⅱ: 36-39.
[2] Agarwal R. P., O'Regan D., Set valued mappings with applications in nonlinear analysis, Series in mathematical analysis and application 4, Taylor & Francis, London, 2002.
[3] Antoniou I., Qiao B., Spectral decomposition of piecewise linear monotonic maps, Chaos Solitons & Fractals, 1996, 7: 1895-1911.
[4] Aubin J. P., Fracnkowska H., Set-Valued Analysis, Birkhazuser, Boston, 1990.
[5] Babbage Ch., Essay towards the calculus of functions, Philosoph. Transact., 1815, 389-423.
[6] Blokh A. M., Coven E., Misiurewicz M., Nitecki Z., Roots of continuous piecewise monotone maps of an interval, Acta Math. Univ. Comenian. (N. S.), 1991, 60: 3-10.
[7] Bogatyi S., On the nonexistence of iterative roots, Topology Appl., 1997, 76: 97-123.
[8] Choi C. S., Nam D. G., Interpolation for partly hidden diffusion processes, Stochastic Process. Appl., 2004, 113: 199-216.
[9] Dombi J., Gera Z., Dombi J., Gera Z., The approximation of piecewise linear membership functions and Lukasiewicz operators, Fuzzy Sets and Systems, 2005: 154: 275-286.
[10] Dubbey J. M., The Mathematical Work of Charles Babbage, Cambridge University Press, Cambridge, 1978.
[11] El-Sayed Ahmed M. A., Ibrahim A. -G., Set-valued integral equations of fractional-orders, Appl. Math. Comput., 2001, 118: 113-121.
[12] Guo C. -M., Zhang D. -L., On set-valued fuzzy measures, Information Science, 2004, 160: 13-25.
[13] He L. -X., The monotone iterative roots of a class of self-mappings on the interval, J. Math. Res. & Exp., 2000, 20: 483-490.
[14] 何连法，牛东晓，一类S~1上自同胚的迭代根，数学研究与评论，1991，11：305-310．
[15] Hu S., Papageorgiou N. S., Handbook of Multivalued Analysis, Kluwer Academic, Dordrecht, 1997.
[16] Ichiishi T., Idzik D., Equitable allocation of divisible goods, Journal of Mathematical Economics, 1999, 32: 389-400.
[17] Isaacs R., Iterates of fractional order, Canad. J. Math., 1950, 2: 409-416.
[18] Jarczyk W., Powierza T., On the smallest set-valued iterative roots of bijections, Dynamical Systems and Functional Equations (Murcia, 2000), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2003, 13: 1889-1893.
[19] Jarczyk W., Zhang W., Also multifunctions do not like iterative roots, Elemente Math., 2006, in press.
[20] Johansson K. H., Rantzer A., Astrom K. J., Fast switches in relay feedback system, Automatica, 1999, 35: 539-552.
[21] Kobza J., Iterative functional equation x(x(t)) = f(t) with f(t) piecewise linear, J. Comput. Appl. Math., 2002, 115: 331-347.
[22] Kuczma M., Functional equation in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warszawa, 1968.
[23] Kuczma M., Choczewski B., Ger R., Iterative functional equations, Encyclopedia of Mathematics and Its Applications 32, Cambridge University Press, Cambridge, 1990.
[24] Lawry J., A framework for linguistic modelling, Artificial Intelligence, 2004, 155: 1-39.
[25] Lee J., Wilson D., Polyhedral methods for piecewise linear functions: the lambda method, Discrete Appl. Math., 2001, 108: 269-285.
[26] Levinshteim M. E., Ivanov P. A., Agarwal A. K., Palmour J. W., Fast switchesoff of high voltage 4H-SiC npn bipolar junction transistor from deep saturation regime, Solid-State Electronics, 2004, 48: 491-493.
[27] 陆青，对线性分式函数n次迭代的一个补充，数学通讯，2006，39．
[28] 李林，林淑容，一类集值映射在迭代下集值点个数不增的条件，四川大学学报(自然科学版)，2007，已接受．
[29] 麦结华，圆周上自同胚有Ⅳ阶迭代根的条件，数学学报，1987，30：280-283．
[30] Nikodem K., Zhang W., On a multivalued iterative equation, Publ. Math. Debrecen, 2004, 64: 427-435.
[31] Powierza T., Set-valued iterative square roots of bijections, Bull. Pol. Ac.: Math., 1999, 47: 377-383.
[32] Powierza T., Higher order set-valued iterative roots of bijections, Publ. Math. Debrecen, 2002, 61: 315-324.
[33] Rice R. E., Iterative square roots of (?)eby(?)ev polynomials, Stochastica, 1979, 3: 1-14.
[34] Rice R. E., Schweizer B., Sklar A., When is f(f(z)) = az~2+bz+c?, Amer. Math. Monthly, 1980, 87: 252-263.
[35] SchrSder E., Uber iterierte funktionen, Math. Ann., 1871, 3: 296-322.
[36] Schwaiger J., Roots of formal power series in one variable, Aequationes Math., 1985, 29: 40-43.
[37] 孙道椿，拟二次多项式的迭代，数学杂志，2004，24：237-240．
[38] 孙太祥，关于区间上单峰函数的迭代根，广西民族学院学报，1996，2：62-64．
[39] 孙太祥，区间上反N型函数的迭代根，数学研究，2000，33：274-280．
[40] 孙太祥，蒋运然，区间上平顶单峰自映射的迭代根，广西科学，2000，7：111-114．
[41] 孙太祥，席鸿建，区间上N型函数的迭代根，数学研究，1996，29：40-45．
[42] 孙太祥，席鸿建，区间上k段单调连续自映射的k阶迭代根，数学研究与评论，1998，18：575-579．
[43] 孙太祥，席鸿建，区间上平顶双峰连续自映射的迭代根，系统科学与数学，2001．21：348-361．
[44] 孙太祥，区间上满的扩张Markov自映射的迭代根，纯粹数学与应用数学，2001，17：99-116．
[45] 孙太祥，圆周上扩张自映射的迭代根，广西科学，1996，3：13-14．
[46] 唐建国，分形迭代a_(k+1)=a_k+a_k~2／n第n次的估计，数学实践与认识，2006，36(9)：309-316．
[47] Targonski Gy., Topics in iteration theory, Vandenhoeck and Ruprecht, G(?)ttingen, 1981.
[48] 吴昭君，孙道椿，拟多项式的迭代，数学物理学报，2006，26A：493-497．
[49] 肖柳青，周石鹏，非线性离散动力系统理论与应用，上海交通大学出版社，上海，2000．
[50] 许璐，许绍元，关于线性分式函数的n次迭代及其应用，数学实践与认识，2006，36(6)：225-228．
[51] Zdun M. C., On iterative roots of homeomorphisms of the circle, Bull. Pol. Acad. Math., 2000, 48: 203-213.
[52] 张广远，一类线段自映射的共轭与迭代根(Ⅰ)，数学年刊，1992，13A：33-44．
[53] 张广远，一类线段自映射的共轭与迭代根(Ⅱ)，数学年刊，1992，13A：473-478．
[54] 张景中，杨路，论逐段单调连续函数的迭代根，数学学报，1983，26：398-412．
[55] 张景中，杨路，张伟年，迭代方程与嵌入流，上海科技教育出版社，上海，1998．
[56] 张景中，熊金城，函数迭代与一维动力系统，四川教育出版社，成都，1992．
[57] Zhang J., Yang L., Zhang W., Some advances on functional equations, Adv. Math. Chin., 1995, 24: 385-405.
[58] Zhang M. -R., Li W. -G., A note on iterative maps, Adv. Math. (China), 1994, 23: 25-30.
[59] Zhang W., A generic property of globally smooth iterative roots, Sci. China Set A, 1995, 38: 267-272.
[60] Zhang W., Iteration, iterative roots and iterative equations, Trends in Math., 2001, 4: 151-160.
[61] Zhang W., PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 1997, 65: 119-128.
[62] 张伟年，动力系统基础，高等教育出版社，北京，2002．
[63] 张伟年，杨地莲，邓圣福，函数方程，四川教育出版社，成都，2002．
[64] Zhang W. -X., Zhang W., Computing iterative roots of polygonal functions, J. Comput. Appl. Math., 2006, In press.
[65] 张筑生，微分动力系统原理，科学出版社，北京，1987．
[66] Zimmermann G., (?)ber die Existenz iterativer Würzeln yon Abbildungen, Doctoral Dissertation, Marburg/Lahn, 1978.
[2] Agarwal R. P., O'Regan D., Set valued mappings with applications in nonlinear analysis, Series in mathematical analysis and application 4, Taylor & Francis, London, 2002.
[3] Antoniou I., Qiao B., Spectral decomposition of piecewise linear monotonic maps, Chaos Solitons & Fractals, 1996, 7: 1895-1911.
[4] Aubin J. P., Fracnkowska H., Set-Valued Analysis, Birkhazuser, Boston, 1990.
[5] Babbage Ch., Essay towards the calculus of functions, Philosoph. Transact., 1815, 389-423.
[6] Blokh A. M., Coven E., Misiurewicz M., Nitecki Z., Roots of continuous piecewise monotone maps of an interval, Acta Math. Univ. Comenian. (N. S.), 1991, 60: 3-10.
[7] Bogatyi S., On the nonexistence of iterative roots, Topology Appl., 1997, 76: 97-123.
[8] Choi C. S., Nam D. G., Interpolation for partly hidden diffusion processes, Stochastic Process. Appl., 2004, 113: 199-216.
[9] Dombi J., Gera Z., Dombi J., Gera Z., The approximation of piecewise linear membership functions and Lukasiewicz operators, Fuzzy Sets and Systems, 2005: 154: 275-286.
[10] Dubbey J. M., The Mathematical Work of Charles Babbage, Cambridge University Press, Cambridge, 1978.
[11] El-Sayed Ahmed M. A., Ibrahim A. -G., Set-valued integral equations of fractional-orders, Appl. Math. Comput., 2001, 118: 113-121.
[12] Guo C. -M., Zhang D. -L., On set-valued fuzzy measures, Information Science, 2004, 160: 13-25.
[13] He L. -X., The monotone iterative roots of a class of self-mappings on the interval, J. Math. Res. & Exp., 2000, 20: 483-490.
[14] 何连法，牛东晓，一类S~1上自同胚的迭代根，数学研究与评论，1991，11：305-310．
[15] Hu S., Papageorgiou N. S., Handbook of Multivalued Analysis, Kluwer Academic, Dordrecht, 1997.
[16] Ichiishi T., Idzik D., Equitable allocation of divisible goods, Journal of Mathematical Economics, 1999, 32: 389-400.
[17] Isaacs R., Iterates of fractional order, Canad. J. Math., 1950, 2: 409-416.
[18] Jarczyk W., Powierza T., On the smallest set-valued iterative roots of bijections, Dynamical Systems and Functional Equations (Murcia, 2000), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2003, 13: 1889-1893.
[19] Jarczyk W., Zhang W., Also multifunctions do not like iterative roots, Elemente Math., 2006, in press.
[20] Johansson K. H., Rantzer A., Astrom K. J., Fast switches in relay feedback system, Automatica, 1999, 35: 539-552.
[21] Kobza J., Iterative functional equation x(x(t)) = f(t) with f(t) piecewise linear, J. Comput. Appl. Math., 2002, 115: 331-347.
[22] Kuczma M., Functional equation in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warszawa, 1968.
[23] Kuczma M., Choczewski B., Ger R., Iterative functional equations, Encyclopedia of Mathematics and Its Applications 32, Cambridge University Press, Cambridge, 1990.
[24] Lawry J., A framework for linguistic modelling, Artificial Intelligence, 2004, 155: 1-39.
[25] Lee J., Wilson D., Polyhedral methods for piecewise linear functions: the lambda method, Discrete Appl. Math., 2001, 108: 269-285.
[26] Levinshteim M. E., Ivanov P. A., Agarwal A. K., Palmour J. W., Fast switchesoff of high voltage 4H-SiC npn bipolar junction transistor from deep saturation regime, Solid-State Electronics, 2004, 48: 491-493.
[27] 陆青，对线性分式函数n次迭代的一个补充，数学通讯，2006，39．
[28] 李林，林淑容，一类集值映射在迭代下集值点个数不增的条件，四川大学学报(自然科学版)，2007，已接受．
[29] 麦结华，圆周上自同胚有Ⅳ阶迭代根的条件，数学学报，1987，30：280-283．
[30] Nikodem K., Zhang W., On a multivalued iterative equation, Publ. Math. Debrecen, 2004, 64: 427-435.
[31] Powierza T., Set-valued iterative square roots of bijections, Bull. Pol. Ac.: Math., 1999, 47: 377-383.
[32] Powierza T., Higher order set-valued iterative roots of bijections, Publ. Math. Debrecen, 2002, 61: 315-324.
[33] Rice R. E., Iterative square roots of (?)eby(?)ev polynomials, Stochastica, 1979, 3: 1-14.
[34] Rice R. E., Schweizer B., Sklar A., When is f(f(z)) = az~2+bz+c?, Amer. Math. Monthly, 1980, 87: 252-263.
[35] SchrSder E., Uber iterierte funktionen, Math. Ann., 1871, 3: 296-322.
[36] Schwaiger J., Roots of formal power series in one variable, Aequationes Math., 1985, 29: 40-43.
[37] 孙道椿，拟二次多项式的迭代，数学杂志，2004，24：237-240．
[38] 孙太祥，关于区间上单峰函数的迭代根，广西民族学院学报，1996，2：62-64．
[39] 孙太祥，区间上反N型函数的迭代根，数学研究，2000，33：274-280．
[40] 孙太祥，蒋运然，区间上平顶单峰自映射的迭代根，广西科学，2000，7：111-114．
[41] 孙太祥，席鸿建，区间上N型函数的迭代根，数学研究，1996，29：40-45．
[42] 孙太祥，席鸿建，区间上k段单调连续自映射的k阶迭代根，数学研究与评论，1998，18：575-579．
[43] 孙太祥，席鸿建，区间上平顶双峰连续自映射的迭代根，系统科学与数学，2001．21：348-361．
[44] 孙太祥，区间上满的扩张Markov自映射的迭代根，纯粹数学与应用数学，2001，17：99-116．
[45] 孙太祥，圆周上扩张自映射的迭代根，广西科学，1996，3：13-14．
[46] 唐建国，分形迭代a_(k+1)=a_k+a_k~2／n第n次的估计，数学实践与认识，2006，36(9)：309-316．
[47] Targonski Gy., Topics in iteration theory, Vandenhoeck and Ruprecht, G(?)ttingen, 1981.
[48] 吴昭君，孙道椿，拟多项式的迭代，数学物理学报，2006，26A：493-497．
[49] 肖柳青，周石鹏，非线性离散动力系统理论与应用，上海交通大学出版社，上海，2000．
[50] 许璐，许绍元，关于线性分式函数的n次迭代及其应用，数学实践与认识，2006，36(6)：225-228．
[51] Zdun M. C., On iterative roots of homeomorphisms of the circle, Bull. Pol. Acad. Math., 2000, 48: 203-213.
[52] 张广远，一类线段自映射的共轭与迭代根(Ⅰ)，数学年刊，1992，13A：33-44．
[53] 张广远，一类线段自映射的共轭与迭代根(Ⅱ)，数学年刊，1992，13A：473-478．
[54] 张景中，杨路，论逐段单调连续函数的迭代根，数学学报，1983，26：398-412．
[55] 张景中，杨路，张伟年，迭代方程与嵌入流，上海科技教育出版社，上海，1998．
[56] 张景中，熊金城，函数迭代与一维动力系统，四川教育出版社，成都，1992．
[57] Zhang J., Yang L., Zhang W., Some advances on functional equations, Adv. Math. Chin., 1995, 24: 385-405.
[58] Zhang M. -R., Li W. -G., A note on iterative maps, Adv. Math. (China), 1994, 23: 25-30.
[59] Zhang W., A generic property of globally smooth iterative roots, Sci. China Set A, 1995, 38: 267-272.
[60] Zhang W., Iteration, iterative roots and iterative equations, Trends in Math., 2001, 4: 151-160.
[61] Zhang W., PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 1997, 65: 119-128.
[62] 张伟年，动力系统基础，高等教育出版社，北京，2002．
[63] 张伟年，杨地莲，邓圣福，函数方程，四川教育出版社，成都，2002．
[64] Zhang W. -X., Zhang W., Computing iterative roots of polygonal functions, J. Comput. Appl. Math., 2006, In press.
[65] 张筑生，微分动力系统原理，科学出版社，北京，1987．
[66] Zimmermann G., (?)ber die Existenz iterativer Würzeln yon Abbildungen, Doctoral Dissertation, Marburg/Lahn, 1978.