Advances in Pure Mathematics, 2015, 5, 27-30 Published Online January 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.51003

How to cite this paper: Liang, Y.S. (2015) Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Conti-nuous Functions. Advances in Pure Mathematics, 5, 27-30. http://dx.doi.org/10.4236/apm.2015.51003

Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions Yongshun Liang Faculty of Science, Nanjing University of Science and Technology, Nanjing, China Email: liangyongshun@gmail.com Received 1 December 2014; revised 15 December 2014; accepted 1 January 2015

Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional inte- gral of any continuous functions on a closed interval is no more than 2 − v.

Keywords Box Dimension, Riemann-Liouville Fractional Calculus, Fractal Function

1. Introduction In [1], fractional integral of a continuous function of bounded variation on a closed interval has been proved to still be a continuous function of bounded variation. The upper bound of Box dimension of the Weyl-Marchaud fractional derivative of self-affine curves has given in [2]. Previous discussion about fractal dimensions of frac-tional calculus of certain special functions can be found in [3] [4].

In the present paper, we discuss fractional integral of fractal dimension of any continuous functions on a closed interval.

If U is any non-empty subset of n-dimensional Euclidean space, nR , the diameter of U is defined as { }sup : ,U x y x y U= − ∈ , i.e. the greatest distance apart of any pair of points in U. If { }iU is a countable

collection of sets of diameter at most δ that cover F, i.e. 1 iiF U∝

=⊂

with 0 iU δ< ≤ for each i, we say that { }iU is a δ-cover of F.

Suppose that F is a subset of nR and s is a non-negative number. For any positive number define.

( ) { }1

inf : is a -cover of ssi i

iF U U Fδ δ

∞

=

=

∑

Y. S. Liang

28

Write

( ) ( )0

lims sF Fδδ→=

( )s Fδ is called s-dimensional Hausdorff measure of F. Hausdorff dimension is defined as follows: Definition 1.1 [5] Let F be a subset of nR and s is a non-negative number. Hausdorff dimension of F is

( ) ( ){ } ( ){ }dim inf : 0 sup :s sH F s F s F= = = = ∞

If ( )dimHs F= , then ( )s F may be zero or infinite, or may satisfy

( )0 s F< < ∞

A Borel set satisfying this last condition is called an s-set. Box dimension is given as follows: Definition 1.2 [5] Let F be any non-empty bounded subset of nR and let ( )N Fδ be the smallest number of

sets of diameter at most δ which can cover F. Lower and upper Box dimensions of F respectively are defined as

( ) ( )0

logdim lim

logB

N FF δ

δ δ→=−

(1.1)

and

( ) ( )0

logdim lim

logB

N FF δ

δδ

→=−

(1.2)

If (1.1) and (1.2) are equal, we refer to the common value as Box dimension of F

( ) ( )0

logdim lim

logB

N FF δ

δ δ→=−

(1.3)

Definition 1.3 [6] Let ( ) [ ]0,1f x C∈ and 0v > . For [ ]0,1t∈ we call

( ) ( ) ( ) ( )1

0

1 dx vvD f x x t f t t

v−− = −

Γ ∫

Riemann-Liouville integral of ( )f x of order v.

2. Riemann-Liouville Fractional Integral of 1-Dimensional Fractal Function Let ( )f x be a 1-dimensional fractal function on I. We will prove that Riemann-Liouville fractional integral of ( )f x is bounded on I. Box dimension of Riemann-Liouville fractional integral of ( )f x will be estimated.

2.1. Riemann-Liouville Fractional Integral of ( )f x Theorem 2.1 Let ( )vD f x− be Riemann-Liouville integral of ( )f x of order v. Then, ( )vD f x− is bounded.

Proof. Since ( )f x is continuous on a closed interval I, there exists a positive constant M such that

( ) f x M x I≤ ∀ ∈

From Definition 1.3, we know

( ) ( ) ( ) ( )1

0

1 d 0 1x vvD f x x t f t t v

v−− = − < <

Γ ∫

For any x I∈ , it holds

Y. S. Liang

29

( ) ( ) ( ) 0 1

1v vM MD f x x v

v v v− ≤ ≤ < <

Γ Γ +

( )vD f x− is a bounded function on I.

2.2. Fractal Dimensions of Riemann-Liouville Fractional Integral of ( )f x Theorem 2.2 Let ( )vD f x− be Riemann-Liouville integral of ( )f x of order v. Then,

( ) ( )1 dim , dim , 2 , 0 1v vBH D f I D f I v v− −≤ Γ ≤ Γ ≤ − < <

Proof. Let 0 1 2δ< < , and m is the least integer greater than or equal to 1 δ . If 1 10 a b δ≤ < ≤ , we have

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )

1 1

1 1

1

1 11 1 1 10 0

1 1 11 1 10

d d

d d .

b av vv v

a bv v v

a

v D f b D f a b t f t t a t f t t

b t a t f t t b t f t t

− −− −

− − −

Γ − = − − −

= − − − + −

∫ ∫

∫ ∫

For 1 i m≤ ≤ , let ( ) ( )1 ,maxi x i iM f xδ δ∈ − = , ( ) ( )1 ,mini x i im f xδ δ∈ −

= ( )max x IM f x∈= If

( ) ( )1 1 0v vD f b D f a− −− ≥ , it holds

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 11 11 vvv v vv D f b D f a b a M m b a m− − Γ + − ≤ − − + −

If ( ) ( )1 1 0v vD f b D f a− −− < , it holds

( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 vv vv D f b D f a b a M m− −Γ + − ≤ − −

We have

( ) ( ) ( ) ( ) ( )1 1 1 1 1 11

1vv v vD f b D f a b a M m M

vδ− −− ≤ − − +

Γ +

Let 1 1n m≤ ≤ − . If ( )1 1 1n nn a b nδ δ+ +≤ ≤ ≤ + , we have

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 11 11 1 1 10 0

1 11 10

d d

d

n nn b n av vv vn n n n

n v vn n

v D f b D f a n b t f t t n a t f t t

n b t n a t f t t

δ δ

δ

δ δ

δ δ

+ ++ +− −− −+ + + +

− −+ +

Γ − = + − − + −

= + − − + −

∫ ∫

∫( ) ( ) ( )

( ) ( )

1

1

1

1 11 1

11

d

d .

n

n

n

n a v vn nn

n b vnn a

n b t n a t f t t

n b t f t t

δ

δ

δ

δ

δ δ

δ

+

+

+

+ − −+ +

+ −++

+ + − − + −

+ + −

∫

∫

If ( ) ( )1 1 0v vn nD f b D f a− −+ +− ≥ , it holds

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1 1 11 vv v v vn n n n n n n n nv D f b D f a b a M m b a m− −+ + + + + + + + + Γ + − ≤ − − + −

If ( ) ( )1 1 0v vn nD f b D f a− −+ +− < , it holds

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 2v vv vn n n n n nv D f b D f a b a M m Mδ− −+ + + + + +Γ + − ≤ − − +

We get

( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 2

1vv v v

n n n n n nD f b D f a b a M m Mv

δ− −+ + + + + +− ≤ − − +

Γ +

There exists a positive constant C, such that

Y. S. Liang

30

( ), 1 1 1, vv

D fR i i C i mδ δ δ− + ≤ ≤ ≤ −

If ( )vN D fδ− is the number of squares of the δ mesh that intersects ( ),vD f I−Γ , by Proposition 11.1 of

[1], we have

( ) ( )1

1 2

02 , 1v

mv v

D fi

N D f m R i i Cδ δ δ δ δ−

−− − −

=

≤ + + ≤ ∑

From (1.2) of Definition 1.2, we know

( ) ( )0

logdim , lim 2 , 0 1

log

vv

BN D f

D f I v vδδ

δ

−−

→Γ = ≤ − < <−

With Definition 1.1, we get the conclusion of Theorem 2.2. This is the first time to give estimation of fractal dimensions of fractional integral of any continuous function

on a closed interval.

Acknowledgements Research is supported by NSFA 11201230 and Natural Science Foundation of Jiangsu Province BK2012398.

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Curves. Analysis Theory and Application, 26, 222-227. http://dx.doi.org/10.1007/s10496-010-0222-9 [3] Liang, Y.S. and Su, W.Y. (2011) The Von Koch Curve and Its Fractional Calculus. Acta Mathematic Sinica, Chinese

Series [in Chinese], 54, 1-14. [4] Zhang, Q. and Liang, Y.S. (2012) The Weyl-Marchaud Fractional Derivative of a Type of Self-Affine Functions. Ap-

plied Mathematics and Computation, 218, 8695-8701. http://dx.doi.org/10.1016/j.amc.2012.01.077 [5] Falconer, J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York. [6] Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.