一類集值映射在迭代下集值點和集值區(qū)間的變化

章靜靜, 李 林

(嘉興學(xué)院數(shù)理與信息工程學(xué)院,浙江嘉興314001)

對于一個非空集合X和一個正整數(shù)n,映射f:X→X的n次迭代可定義為:fn(x)=f(fn-1(x)),?x∈X.特別地,記f0(x)≡x.近幾十年來由于關(guān)于周期性的Sharkovsky序、關(guān)于分岔的Feigenbaum現(xiàn)象、關(guān)于運動復(fù)雜性的Smale馬蹄等重大發(fā)現(xiàn)的不斷涌現(xiàn),動力系統(tǒng)的新成就促進了迭代函數(shù)方程的發(fā)展.關(guān)于映射迭代的研究,至少可追溯到一百多年以前 E.Schr?der[1]、N.H.Abel[2]、C.Babbage[3]等數(shù)學(xué)家的工作.由于迭代工作與代數(shù)運算的迥然不同,研究工作艱難曲折[4-12].對于一些具體函數(shù)的迭代研究目前主要是關(guān)于多項式函數(shù)、折線函數(shù)[13-16]等一些特殊的非單調(diào)函數(shù).例如金蕾等[17]對高次多項式這類非線性映射通過共軛相似法給出了一般的n次迭代計算結(jié)果,并且討論了f(x)=1/(a+bxr)1/r這類非多項式型映射的迭代,給出了二維映射F:(x,y)→(u(x,y),v(x,y))在u(x,y)和v(x,y)均為線性函數(shù)時的n次迭代結(jié)果.L.Li[18]在2007年研究了區(qū)間上單折點的折線函數(shù)的迭代,研究其折點的個數(shù)不會增加或者有界的條件.孫太祥等又討論了區(qū)間I=[0,1]上所有的平頂單峰和雙峰自映射的迭代問題[19-20].最近,文獻[21]給出了一類單集值點映射在迭代下集值點個數(shù)不增的條件.

令2X為X的所有子集構(gòu)成的族,則稱映射F:X→2X為X上的一個集值映射,而X中取到集值的點稱為集值點.進一步,對于X中的任意子集Y?X,其像F(Y)定義為,那么F的n次迭代Fn定義為,其中F0(x):={x},x∈X.

本文討論的是一類定義在單位區(qū)間I=[0,1]上具有單個集值點的嚴格單調(diào)映射的迭代.這類集值映射可定義為

其中A?I為F的集值區(qū)間,而F1和F2分別是定義在[0,c)和(c,1]上的線性函數(shù),并滿足以下條件之一:

顯然,F為定義在I=[0,1]上的上半連續(xù)函數(shù).文獻[12]研究了這類集值映射在迭代下集值點個數(shù)不增的條件,并給出該條件下映射迭代的表達式.將推廣文獻[9,12]中的結(jié)論,研究該函數(shù)在迭代下的集值區(qū)間的變化,并給出一般的迭代表達式.為方便起見,令V(F)表示函數(shù)F的集值點個數(shù),l(F)為F的集值區(qū)間.

1 F1、F2為嚴格遞增的連續(xù)函數(shù)

在F1和F2嚴格遞增的情形下,注意到集值點個數(shù)V(Fn)取決于函數(shù)值與c的關(guān)系.為行文方便,稱單位區(qū)間[0,1]上的一個遞增(或遞減)的數(shù)列為m次跨越c∈(0,1),如果存在正整數(shù)m≥2有xic)并且xm≥c(xm≤c),其中i=1,2,…,m-1(見文獻[7]).根據(jù)F的單調(diào)性,容易得到為關(guān)于n的遞增序列;為關(guān)于n的遞減序列.接下來,將根據(jù)以下幾種情況分別討論.

2 F1、F2為嚴格遞減的連續(xù)函數(shù)

[1]Schr?der E.ber Iterate funktionen[J].Math Ann,1871,3:295-322.

[2]Abel N H.Oeuvres completes[J].Christiana,1881,II:36-39.

[3]Dubbey J M.The Mathematical Work of Charles Babbage[M].New York:Cambridge University Press,1978.

[4]Chen J M,Zhang W N.Leading coefficient problem for polynomial-like iterative equations[J].J Math Anal Appl,2009,349:413-419.

[5]Li L,Yang D L,Zhang W N.A note on iterative roots of PM functions[J].J Math Anal Appl,2008,341:1482-1486.

[6]Li L,Zhang W M.Continuously decreasing solutions for polynomial-like iterative equations[J].Sci China,2013,A56:1051-1058.

[7]Li L,Zhang W N.Construction of usc solutions for a multivalued iterative equation of ordern[J].Results Math,2012,62:203-216.

[8]Liu L,Jarczyk W,Li L,et al.Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2[J].Nonlinear Anal,2012,75:286-303.

[9]Liu L,Zhang W N.Non-monotonic iterative roots extended from characteristic intervals[J].J Math Anal Appl,2011,378:359-373.

[10]Shi Y G,Li L,Lesniak Z.On conjugacy ofr-modal interval maps with nonmonotonicity height equal to 1[J].J Difference Equ Appl,2013,19:573-584.

[11]Xu B,Zhang W.Construction of continuous solutions and stability for the polynomial-like iterative equation[J].J Math Anal Appl,2007,325:1160-1170.

[12]Zhang W M,Zhang W N.Continuity of iteration and approximation of iterative roots[J].J Comput Appl Math,2011,235:1232-1244.

[13]Yang L L,Yang L,Yu Z H,et al.Real polynomial iterative roots in the case of nonmonotonicity height≥2[J].Sci China,2012,A55:2433-2446.

[14]Yu Z H,Yang L,Zhang W N.Discussion on polynomials having polynomial iterative roots[J].J Symb Comput,2012,47(10):1154-1162.

[15]李林.折線函數(shù)與集值函數(shù)的迭代與迭代根[D].成都:四川大學(xué),2007.

[16]Zhang W X,Zhang W N.Computing iterative roots of polygonal functions[J].J Comput Appl Math,2007,205:497-508.

[17]金蕾,周喆,劉旭.迭代的計算與估計[J].數(shù)學(xué)的實踐與認識,2004,34(4):170-175.

[18]Li L.Number of vertices for polygonal functions under iteration[J].Korea Soc Math Educ,2007,14:99-109.

[19]孫太祥,蔣運然.區(qū)間上平頂單峰自映射的迭代根[J].廣西科學(xué),2000,7(2):111-114.

[20]孫太祥,席鴻建.區(qū)間上平頂雙峰連續(xù)自映射的迭代根[J].系統(tǒng)科學(xué)與數(shù)學(xué),2001,21(3):348-361.

[21]李林,林淑容,李春曄.一個集值映射在迭代下集值點個數(shù)不增的條件[J].四川大學(xué)學(xué)報:自然科學(xué)版,2010,47:17-20.

[22]Aubin J P,Frankowska H.Set-Valued Analysis[M].Boston,Basel,Berlin:Birkhauser,1990.

[23]Babbage C.Essay towards the calculus of functions[J].Philosoph Transact,1815:389-423.

[24]Baron K,Jarczyk W.Recent results on functional equations in a single variable,perspectives and open problems[J].Aequationes Math,2001,61:1-48.

[25]Kuczma M.Functional equation in a single variable[C]//Monografie Mat.Warszawa:Polish Scientific Publishers,1968,46.

[26]Kuczma M,Choczewski B,Ger R.Iterative functional equations[C]//Encyclopedia of Mathematics and Its Applications.Cambridge:Cambridge University Press,1990,32.