A quasisum is a function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ F: I_1 \times \dots \times I_n \longrightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>:</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>I</mi> <mi>n</mi> </msub> <mo stretchy="false">⟶</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> of the form <Equation ID="Equ8"> <EquationSource Format="TEX">$\begin{aligned} F \left( x_1 \,, \dots , x_n \right) = g \bigl ( f_1(x_1) + \dots + f_n(x_n) \bigr ) \hspace{10mm} \left( x_1 \in I_1 \,, \dots , x_n \in I_n \right) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>F</mi> <mfenced close=")" open="("> <msub> <mi>x</mi> <mn>1</mn> </msub> <mspace width="0.166667em" /> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> </mfenced> <mo>=</mo> <mi>g</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mspace width="28.45274pt" /> <mfenced close=")" open="("> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>∈</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mspace width="0.166667em" /> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>∈</mo> <msub> <mi>I</mi> <mi>n</mi> </msub> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ n \ge 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is an integer and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ f_k: I_k \longrightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <mo>:</mo> <msub> <mi>I</mi> <mi>k</mi> </msub> <mo stretchy="false">⟶</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a continuous, strictly monotone function defined on a nonempty open interval of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> (for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ k = 1 , \dots , n $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>), moreover <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ g: f_1(I_1) + \dots + f_n(I_n) \longrightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟶</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is also continuous, strictly monotone. In this paper we will show that if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ p \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and the quasisum F is p-times continuously differentiable then each of the generator functions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ g , f_1 , \dots , f_n $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>,</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are p-times continuously differentiable as well. We present applications of our results for p-times continuously differentiable semigroup operations and additively separable utility functions as well.

Regularity preservation for quasisums

期刊论文
出版模式：

开放获取
发表日期：

2025年8月25日

查看全文

Péter Tóth

摘要

A quasisum is a function $ F: I_1 \times \dots \times I_n \longrightarrow \mathbb {R}$ $F : I_{1} \times \dots \times I_{n} ⟶ R$ of the form $\begin{aligned} F \left( x_1 \,, \dots , x_n \right) = g \bigl ( f_1(x_1) + \dots + f_n(x_n) \bigr ) \hspace{10mm} \left( x_1 \in I_1 \,, \dots , x_n \in I_n \right) \end{aligned}$ $\begin{matrix} F (x_{1}, \dots, x_{n}) = g (f_{1} (x_{1}) + \dots + f_{n} (x_{n})) (x_{1} \in I_{1}, \dots, x_{n} \in I_{n}) \end{matrix}$ where $ n \ge 2 $ $n \geq 2$ is an integer and $ f_k: I_k \longrightarrow \mathbb {R}$ $f_{k} : I_{k} ⟶ R$ is a continuous, strictly monotone function defined on a nonempty open interval of $ \mathbb {R}$ $R$ (for $ k = 1 , \dots , n $ $k = 1, \dots, n$ ), moreover $ g: f_1(I_1) + \dots + f_n(I_n) \longrightarrow \mathbb {R}$ $g : f_{1} (I_{1}) + \dots + f_{n} (I_{n}) ⟶ R$ is also continuous, strictly monotone. In this paper we will show that if $ p \in \mathbb {N}$ $p \in N$ and the quasisum F is p-times continuously differentiable then each of the generator functions $ g , f_1 , \dots , f_n $ $g, f_{1}, \dots, f_{n}$ are p-times continuously differentiable as well. We present applications of our results for p-times continuously differentiable semigroup operations and additively separable utility functions as well.