- 1 -
Yang
Yang
Yang
Yang Xiaojun
Xiaojun
Xiaojun
Xiaojun
College of science, China University of Mining and Technology, Xuzhou, Jiangsu,
P.
R.
China
221008
E-mail ： dyangxiaojun@163.com
Abstract
Abstract
Abstract
Abstract
Based on the concept of Jumarie ’ s fractional calculus, local fractional derivative is
modified in detail. Its fundaments of local fractional derivative are proposed.
U niqueness of local fractional derivative is obtained and Rolle ’ s theorem, Cauchy ’ s
generaliz ed mean value theorem and L ’ Hospital ’ s rules are proved. Its fundaments
of the local extreme value of function are discussed . Increasing/ decreasing test and
the -derivative test are proved based on its properties.
2
α
Keyword s:
Keyword s:
Keyword s:
Keyword s:
Rolle ’ s theorem, modified local fractional derivative, uniqueness of
local fractional derivative, local extreme value
1
1
1
1
Introduction
Introduction
Introduction
Introduction
Since kolwankar and Gangal proposed local fractioanl derivative operator throngh renormaliz ation of
Riemann-liouville definition[1-2], it has been developed for recent twenty years. The recent results are that
,
the right-hand and left-hand local fraction al derivative, local fractional Taylor
’
s theorem, the local fractional
increasing/decreasing test, the generaliz ed Rolle
’
s theorem and the property of a local extreme value base
on local fractional derivative are increasingly proposed[3-7].
However, even as derivative of Euclidean space, the problems of local fractional derivative are not enough
to improve and are necessary to modify it. Our work focus on these problems and this paper is organiz ed as
follows.
In section 2, local fractional derivative is modified. Fundamental theorems of local fractional derivative and
uniqueness of local fractional derivative are investigated in Section 3. Fundaments of the local extreme
value of function proposed in Section 4
.
The conclusions are described in section 5.
2
2
2
2
Midified
Midified
Midified
Midified
local
local
local
local
fractional
fractional
fractional
fractional
derivative
derivative
derivative
derivative
2.1Kolwankar
2.1Kolwankar
2.1Kolwankar
2.1Kolwankar and
and
and
and Gangal
Gangal
Gangal
Gangal
’
’
’
’
s
s
s
s
definition
definition
definition
definition
Definition
Definition
Definition
Definition 2.1
2.1
2.1
2.1 Assume that is continuous at a point .Local fractional derivative of
( )
fx
0
x
order
, ,
of on the interval by the expression [1]
α
01
α
<≤
( )
fx
[ ]
,
ab
M odified local fractional derivative and its
fundaments
http://www.paper.edu.cn
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