一類集值映射在迭代下集值點(diǎn)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(yùn)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(jìn)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(yùn)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(diǎn)的折線函數(shù)的迭代,研究其折點(diǎn)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(xiàn)[21]給出了一類單集值點(diǎn)映射在迭代下集值點(diǎn)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(diǎn)稱為集值點(diǎn).進(jìn)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(diǎn)的嚴(yán)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(xiàn)[12]研究了這類集值映射在迭代下集值點(diǎn)個數(shù)不增的條件,并給出該條件下映射迭代的表達(dá)式.將推廣文獻(xiàn)[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(dá)式.為方便起見,令V(F)表示函數(shù)F的集值點(diǎn)個數(shù),l(F)為F的集值區(qū)間.

1 F1、F2為嚴(yán)格遞增的連續(xù)函數(shù)

在F1和F2嚴(yán)格遞增的情形下,注意到集值點(diǎn)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(xiàn)[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(yán)格遞減的連續(xù)函數(shù)

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