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On the Lipschitz perturbation of monotonic functions
被引：0
作者：
Zita Makó
Zsolt Páles
机构：
[1] Institute of Mathematics,
[2] University of Debrecen,undefined
[3] Institute of Mathematics,undefined
[4] University of Debrecen,undefined
来源：
Acta Mathematica Hungarica
|
2006年
/
113卷
关键词：
Lipschitz function;
Lipschitz perturbation;
monotonicity;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
A real valued function <InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"26"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"27"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"28"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"29"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"30"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"31"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"32"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"33"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"34"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>f$ defined on a real interval $I$ is called \emph{$d$-Lipschitz} if it satisfies $|\ell(x)- \ell(y)| \le d(x,y)$ for $x,y\in I$. In this paper, we investigate when a function $p\: I \to \bR$ can be decomposed in the form $p=q+ \ell$, where $q$ is increasing and $\ell$ is $d$-Lipschitz. In the general case when $d\: I^{2} \to \bR$ is an arbitrary semimetric, a function $p\: I \to \bR$ can be written in the form $p=q+ \ell$ if and only if \vspace{-4pt} <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{i=1}^{n}{\big(p(s_{i})-p(t_{i})-d(t_{i},s_{i}) \big)^{+}} \le \sum_{j=1}^{m}{\big(p(v_{j})-p(u_{j})+d(u_{j},v_{j}) \big)} \vspace{-4pt} $$ is fulfilled for all real numbers $t_{1}<s_{1}, \dots, t_{n}<s_{n}$ and $u_{1}<v_{1}, \dots, u_{m}<v_{m}$ in $I$ satisfying the condition \vspace{-4pt} $$ \sum_{i=1}^{n} 1_{\left]t_i,s_i\right]}= \sum_{j=1}^{m} 1_{\left]u_j,v_j\right]}, \vspace{-4pt} $$ where $1_{\left]a,b\right]}$ denotes the characteristic function of the interval $\left]a,b\right]$. In the particular case when $d\: I^{2} \to R$ is a so-called concave semimetric, a function $p\: I \to \bR$ is of the form $p=q+ \ell$ if and only if \vspace{-4pt} $$ 0 \le \sum_{k=1}^{n}{d(x_{2k-1},x_{2k})} + d(x_0,x_{2n+1}) + \sum_{k=0}^{n}{\big(p(x_{2k+1})-p(x_{2k})\big)} \vspace{-4pt} $$ holds for all $x_0\le x_1\ki \cdots\ki x_{2n}\le x_{2n+1}$ in $I$.
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页码：1 / 18
页数：17
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