J. Math. Anal. Appl. 306 (2005) 730–739

www.elsevier.com/locate/jmaa

A family of the Cauchy type mean-value theorems

Josip E. Pecariˇcacute; ^a, Ivan Pericacute; ^b, H.M. Srivastava ^c,

1. Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, HR-10000 Zagreb, Croatia ^b Faculty of Chemical Engineering and Technology, University of Zagreb, Maruliacute;cev trg. 19,
2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada

Received 6 October 2004

Available online 4 February 2005

Submitted by William F. Ames

Abstract

The Cauchy type mean-value theorems for the Riemann–Liouville fractional derivative are de-duced here from known mean-value theorems of the Lagrange type. A general method for deducing these Cauchy type formulas is extracted. Two Cauchy type formulas are then deduced without a priori knowledge about the Lagrange type mean-value theorems.

2004 Elsevier Inc. All rights reserved.

Keywords: Cauchy mean-value theorem; Lagrange mean-value theorem; Riemann–Liouville fractional integral and fractional derivative; Newton–Cotes quadrature formulas; Jensenrsquo;s inequality; Trapezoidal rule

1. Introduction

Mean-value theorems are of great importance in mathematical analysis. In particular, the Lagrange type and the Cauchy type mean-value theorems are most frequently used. The usual approach is to prove first the Lagrange type mean-value theorems and then deduce from them the Cauchy type mean-value theorems. As a typical example of this method,

• Corresponding author.

E-mail addresses: pecaric@hazu.hr (J.E. Pecariˇc),acute; iperic@pbf.hr (I. Peric),acute; hmsri@uvvm.uvic.ca,

harimsri@math.uvic.ca (H.M. Srivastava).

0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2004.10.018

J.E. Peˇcariacute; et al. / J. Math. Anal. Appl. 306 (2005) 730–739

731

in Section 2, we first show how this approach works for the Riemann–Liouville fractional derivative. Then, in Section 3, we extract a general abstract method which contains the crucial step in this procedure. Finally, in Section 4, we make use of the perfect symmetry of the Cauchy type mean-value theorems in order to show that, in many cases, one can easily guess the form of the Cauchy type mean-value theorem and then deduce from it the exact form of the Lagrange type mean-value theorem.

1. Generalized Cauchy type formulas for the Riemann–Liouville fractional derivative

Let us first consider the Riemann–Liouville fractional integral of order								minus;alpha;, that is,
Dalpha; f (x)	=		I minus;alpha; f (x).
a				a
Here the Riemann–Liouville fractional integral operator Iabeta; is defined as follows:
						x	(x minus; t )beta;minus;1f (t ) dt
Iabeta; f (x) := Gamma; (beta;) a
				1
(x gt; a; a isin; R; beta; isin; R )								(1)
and
Dalpha; f (x)	=		DnI nminus;alpha; f (x)
a					a
D :=			d		; n minus; 1 alpha; lt; n; n isin; N := {1, 2, 3, . . .}			(2)

d x

with, of course,

I_a⁰f (x) = f (x).

The sequential fractional derivative is denoted by (see, for example, [5, p. 86 et seq.])

Danalpha; := Daalpha; Da(nminus;1)alpha;n isin; N0 := N cup; {0} .

(3)

Let Ω be a real interval and alpha; isin; [0, 1]. Let F (Ω ) denote the space of Lebesgue mea-surable functions with domain in Ω and suppose that x₀ isin; Ω . Then a function f is called

alpha;-continuous at x₀ if there exists lambda; isin; [0, 1 minus; alpha;) for which the function g given by

g(x) = |x minus; x₀|^lambda;f (x)

is continuous at x₀. Thus, in the present terminology, the function f is called 1-continuous at x₀ if it is continuous at x<su

剩余内容已隐藏，支付完成后下载完整资料</su

---

译文

柯西型中值定理的一个族

JosiP.E.佩克·阿里奇A，佩里克B，H.M. Srivastava C，A.

萨格勒布大学纺织技术系，皮埃奥蒂耶娃6，克罗地亚萨格勒布H-10000

萨格勒布大学化学工程与技术学院，Marulic EV Trg。19，

克罗地亚萨格勒布H-10000

维多利亚大学数学与统计系，Victoria，不列颠哥伦比亚，V8W 3P4，加拿大

2004年10月6日收到

网上2005年2月4日由William F. Ames提交

摘要

本文从拉格朗日型的已知中值定理出发，导出了黎曼-刘维尔分数阶导数的柯西型中值定理。导出了推导这些柯西型公式的一般方法。在没有拉格朗日型中值定理先验知识的前提下，推导出两个柯西型公式。

2004爱思唯尔公司版权所有。

关键词：柯西中值定理；拉格朗日中值定理；黎曼-刘维尔分数积分和分数导数；牛顿-科特斯求积公式；延森不等式；Trapezoidal规则

1、引言

中值定理在数学分析中具有重要的意义。特别地，最常用的是拉格朗日型和柯西型中值定理。通常的方法是首先证明拉格朗日型中值定理，然后从中推导出柯西型中值定理。作为这种方法的一个典型例子，

对应作者。

(电子邮件地址：PICARICH@ HAZUHR（J.E.PEC·ARIC），IpICIC@ PBF.HR（I. Pric），HMSRIIUVVM UVIC.CA，HARIMSRIH-MATH.UVIC.CA（H.M. Srivastava）。

022-247X/$-见前面事项2004爱思唯尔公司保留所有权利。DOI:101016/J.JMAA.2014.10018)

在第2节中，我们首先展示了这种方法对于黎曼-刘维尔分数阶导数的工作原理。然后，在第3节中，我们提取了一个通用的抽象方法，其中包含了这个过程中的关键步骤。最后，在第4节中，我们利用柯西型中值定理的完全对称性，说明在许多情况下，人们可以很容易地猜出柯西型中值定理的形式，然后从中推导出拉格朗日型中值定理的精确形式。

2、黎曼- Liouville分数阶导数的广义柯西型公式

让我们先考虑阶alpha;黎曼—Liouville分数次积分，即D^alpha;f (x) = I ^minus;alpha;f (x).

a

a

这里，黎曼- Liouville分数阶积分算子Ibeta;定义如下：

D = n minus; 1 le; alpha;lt; n; n isin; N := {1, 2, 3,.. .}

当然了 f (x) = f (x).顺序分数导数由（参见，例如，[ 5，p 86 et eq）]表示。

(n isin; N₀ := N cup; {0}).

设为实区间alpha;isin;〔0，1〕。设F（alpha;）表示勒贝格域上的可积函数的空间，并假设X0。函数f在x0处称为alpha;-连续，如果存在由函数G给出的alpha;[ 0, 1，alpha;]。

g(x) = |x minus; x₀|lambda;f (x)（2）

在x0处是连续的。因此，在目前的术语中，函数f在x0处被称为1-连续，如果它在x0处是连续的。此外，如果函数X是alpha;-连续的，则函数f称为alpha;-连续的。为了方便起见，我们现在在alpha;上表示alpha;-连续函数类

对于alpha;，函数f称为的alpha;奇异。

对于alpha;，函数f称为alpha;alpha;的alpha;奇异。

让alpha;isin;isin;R ，和Ωsub;，这样，Ω对于任意 Xisin;Ele;x则我们写

a I_alpha;(E) = f f isin; F (Ω) 和I^alpha;f (x) lt; infin; (forall;x isin; E) , (4)

a

其中，f，（f）表示域中勒贝格可测函数的空间。最近，Trujillo等人。〔6〕证明了以下结论。

定理1（广义中值定理）。设alpha;alpha;〔0, 1〕和fomega;c（a，b]

Dalpha;F[C]〔A，B〕。然后

（5）

一个具有小于plusmn;x的x〔a，b〕

定理2（广义泰勒公式）。设alpha;〔0, 1〕和nn n为f满足（a，b]满足下列条件的一个连续函数：

一）djalpha;fisin;（a，b）和c aialpha;djalpha;fisin;〔a，b〕for j＝1，hellip;，n。

（二）d（n 1）的连续在线alpha;f is〔a，b〕。

（三）如果alpha;＜1／2，然后，for each jisin;{1，hellip;，n} such that（J 1）D（alpha;le;1，j＋1）（x）是alpha;f

连续gamma;-gamma;（x =（1minus;a for some（J 1）gamma;le;alpha;le;1）或alpha;-阶奇异协会。

然后，对每个xisin;（a，b），

在我们的调查中，我们提出了一些相关的结果，通过使用在[4 ]中给出的我

们首先陈述我们的第一个结果如下。

定理3。设alpha;alpha;〔0, 1〕和f f，Gεc（a，b]是这样的

f

for every x isin; [a, b].

然后，对于每一个x*（a，b]，存在一个ZEA（Aomega;chi;times;x）。

比如

证明。设x[a，b]是固定的。由K1和K2表示以下函数：

K₁ = f (x) minus;[(x minus; a)^{1minus;alpha;}f (x)](a )(x minus; a)^{alpha;minus;1}

和K₂ = g(x) minus; [(x minus; a)^{1minus;alpha;}g(x)](a )(x minus; a)^{alpha;minus;1}.

我们考虑函数f（t）由F (t ) = K₂f (t ) minus; K₁g(t) t isin; [a, b]

由于F和G满足定理1中的条件，所以对于F是相同的，所以我们有

对于一些ZEA（Aomega;ztimes;x）。这给了我们

其中定理3的断言（9）很容易跟随。

推论1。设alpha;alpha;〔0, 1〕和f f，Gεc（a，b]是这样的

当 对于每一个x[a，b]。

然后，对于每一个x*（a，b]，存在一个ZEA（A）

证明。在（x（a））alpha;-1f（x）和g（x）（（x，a））alpha;1g（x）替换f（x）时，定理3很容易得到推论1。

定理4。假设函数f和g满足定理2中的条件，其中

D^{(n 1)alpha;}g(x) /= 0 for every x isin; [a, b].

a

然后，对于每一个x*（a，b]，存在一个ZEA（Aomega;zomega;z），使得

证明。设X（A，B）固定。根据（7）定义的RN，我们用1和2表示以下函数：K₁ = R_n(g; x, a) and K₂ = R_n(f ; x, a),和并考虑函数f所定义的

F (t ) = K₁f (t ) minus; K₂g(t) t isin; [a, b]

利用由（7）定义的RN的线性性质，定理4的其余部分与定理3的证明一样。

定理4的一个简单结果由下面的推论给出。

外文原文

A family of the Cauchy type mean-value theorems

Josip E. Pecˇaricacute; a, Ivan Pericacute; b, H.M. Srivastava c,lowast;

a Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, HR-10000 Zagreb, Croatia

b Faculty of Chemical Engineering and Technology, University of Zagreb, Marulicacute;ev trg. 19,

HR-10000 Zagreb, Croatia

c Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada

Received 6 October 2004

Available online 4 February 2005 Submitted by William F. Ames

Abstract

The Cauchy type mean-value theorems for the Riemann–Liouville fractional derivative are de- duced here from known mean-value theorems of the Lagrange type. A general method for deducing these Cauchy type formulas is extracted. Two Cauchy type formulas are then deduced without a priori knowledge about the Lagrange type mean-value theorems.

 2004 Elsevier Inc. All rights reserved.

Keywords: Cauchy mean-value theorem; Lagrange mean-value theorem; Riemann–Liouville fractional integral and fractional derivative; Newton–Cotes quadrature formulas; Jensenrsquo;s inequality; Trapezoidal rule

Introduction

Mean-value theorems are of great importance in mathematical analysis. In particular, the Lagrange type and the Cauchy type mean-value theorems are most frequently used. The usual approach is to prove first the Lagrange type mean-value theorems and then deduce from them the Cauchy type mean-value theorems. As a typical example of this method,

*Corresponding author.

E-mail addresses: pecaric@hazu.hr (J.E. Pecˇaricacute;), iperic@pbf.hr (I. Pericacute;), hmsri@uvvm.uvic.ca, harimsri@math.uvic.ca (H.M. Srivastava).

0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.10.018

in Section 2, we first show how this approach works for the Riemann–Liouville fractional derivative. Then, in Section 3, we extract a general abstract method which contains the crucial step in this procedure. Finally, in Section 4, we make use of the perfect symmetry of the Cauchy type mean-value theorems in order to show that, in many cases, one can easily guess the form of the Cauchy type mean-value theorem and then deduce from it the exact form of the Lagrange type mean-value the

剩余内容已隐藏，支付完成后下载完整资料

---

资料编号：[462806]，资料为PDF文档或Word文档，PDF文档可免费转换为Word

您可能感兴趣的文章

• 政策放松后地摊经济将何去何从？——基于杭州放松的经济政策执行困境研究外文翻译资料
• 特征值、特征向量及对角化在环境科学中的应用外文翻译资料
• 多重逻辑回归外文翻译资料
• K-means聚类分析外文翻译资料
• 求解线性方程组的LA = U分解法外文翻译资料
• 401(K)计划策略的蒙特卡罗研究：美元成本平均、价值平均和比例再平衡外文翻译资料
• 第二章：比特币如何去中心化外文翻译资料
• 通过深度强化学习实现人类水平的控制外文翻译资料
• 通过可编程开关加速大数据排序外文翻译资料
• 会计师兼RAMANUJAN的数学朋友——NARAYANA AIYAR外文翻译资料