1. Introduction
Fractals are extremely non-smooth sets. It is very difficult to define derivatives on fractals. Kigami [1] said “Since fractals like the Sierpinski gasket and Koch curve do not have any structures, to define differential operator like the Laplacian is not possible from the classical viewpoint of analysis. To overcome such difficulty is a new challenge in mathematics”.
In fact, Kigami made it open that the problem of finding the derivative of a fractal was a very difficult one. In this work, we will focus on the everywhere continuous and nowhere differentiable Koch curve in our series of studies of the “derivative” of fractals. It is known that one characteristic of continuous but not differentiable functions
is for some
, the Hölder equality is satisfied
(1)
For example, Hardy proved [2] that for Weierstrass function
(1) is satisfied for
. Yang et al. [3] [4] [5] also proved (1) for the Koch curve, the Levy curve, and the Kiesswetter functions respectively with different α values.
Traditional differentiability can be almost considered as a result of (1) for
, i.e. the Lipschitz inequality holds. Naturally, one guesses that the α-Hölder derivative of fractals should be
Besicovitch was one of the mathematicians who were interested in this [6] . However, there has not been any example of fractal that possesses such Hölder derivative. Thomson introduced [7] the concept of upper α-Hölder derivative, proposed as
There are many generalizations of traditional derivatives, but none of which is helpful for our problem. Based on the dyadic derivatives [8] [9] introduced by Kahane and Pecguement respectively, Fu [10] proved the Cantor function on the Cantor set had ternary Hölder derivative with
:
Even though the Cantor set is a special case with zero Lebesque measure, this result provides some helpful ideas.
There have not been many activities in the area of differentialbility of fractals. We have published some results not long ago [5] . Recently, Prodanov obtain differentiability of a continuous function in terms of fractals [11] and Scott analyzed differentiable Iterated Function Systems [12] with computer technology and Arif [13] analyzed a fractal-fractional derivative of a stress fluid through a porous medium. In this paper, we introduce an α-Hölder p-adic derivative that can be applied to fractal curves with different Hölder exponent α. As for the Koch curve, it should satisfy the Hölder condition with exponent
and has a 4-adic arithmetic-analytic representation. In fact, in this article, we can prove that the Koch curve has an exact order of
-Hölder 4-adic derivative. The contents of this paper are organized as follows: in Section 2, the α-Hölder p-adic derivatives of fractals are defined; in Section 3, analytic properties of the Koch curve are analyzed and, in Section 4, the α-Hölder of the Koch curve are proven.
2. α-Hölder p-Adic Derivatives
In the p-adic system, the expansion of
is
where p is a positive integer and
is the n-th digit of real number t in the decimal system. All the p-adic rational numbers
make up of a countable set D, which is a subset of
. So, a sequence of nested intervals
exists in the form of
such that
.
Definition 2.1 Let
be a function defined on
,
and
, if
exists, then
is called α-Hölder p-adic differentiable at point t or simply
differentiable, and the derivative is denoted by
, called Hölder-adic derivative.
We can further define the upper and lower Hölder adic derivative as
It is easy to see that the α-Hölder p-adic derivatives have some simple properties:
1) If
is α-Hölder p-adic differentiable at t,
is a real constant, then
2) If
and
are
differentiable at t, then
3) If
is
differentiable at t,
, then
If
is
differentiable at t,
, then
In particular, if
is differentiable at t, then
is
differentiable at t, and
If
is
differentiable at t and
, then
However, if
is
differentiable at t,
does not necessarily exist.
Definition 2.2 Let
be a p-adic rational number with
,
, if
exists, then the limit is called the right (resp. left) α-Hölder p-adic derivative of
at t and denote it by
(resp.
).
The left and right α-Hölder p-adic derivatives of
at t also have the similar properties as the α-Hölder p-adic derivatives of
at t.
3. The Analytic Properties of the Koch Curve
It is well known that the Koch curve is a typical example of fractal curves. Discussing its analytic properties is obviously important. Von Koch initially constructed the curve with a recursive way of using pure geometric descriptions [14] . Recently, works of analytic representation of the curve have made some progresses. An arithmetic-analytic representation based on 4-adic expansion is obtained [3] , which will be used in our investigation in this paper.
Due to the geometric properties of Koch curve [15] , the 4-adic expansion
is appropriate, where
takes the value of 0, 1, 2 or 3. (6) can also be represented by the 2-adic form
where
takes values 0 or 1. Both
and
satisfy the relations
Lemma 3.1 [4] Suppose the parametric equation of the Koch curve with argument
is
Then, the arithmetic-analytic representation of Koch curve from the Iterated Function System (IFS) [16] is expressed as
(2)
(3)
where
,
.
It is worthy noticing that the arguments of sinusoidal and cosine function are standard symbolic sequences determined by the expansion coefficient
or
of binary system. The sharp-angled vertices without derivative in almost everywhere can be completely described by the symbolic sequences. So instead of being directly determined by the value of t, (3.1), (3.2) are a special type of parametric functions defined by the binary system. The following propositions present some of their properties that will be used for later discussions.
Lemma 3.2 [4] The Koch curve
satisfies the Hölder condition with exponent
, i.e.
(4)
(5)
Lemma 3.3 The arithmetic-analytic representation of the Koch curve (2), (3) is uniquely determined at a 4-adic rational point
,
,
.
Proof. 1) If
i.e.
, then from (2),
where
. So
If
takes the other form:
(6)
where
,
,
,
,
, then
And from
we have:
Fractal
2) If
i.e.
, then from (2),
If t takes the form of
where
,
,
,
, then
3) If
i.e.
, then from (2),
If t takes the form of (6), where
,
,
,
,
,
, then
Similarly, it can be proved that function
is also uniquely determined at a 4-adic rational point
.
Next, by using (2), (3) the values of
and
at certain points can be calculated. In order to discuss them, we will proceed with various cases of
. Partition the interval
into 4 congruent segments and then partition each sub-segments similarly. After repeating these steps for m times, a family of subintervals is obtained:
Let
, where
,
, and denote the interval
as
where
. Then
So, any point
is the intersection of one of the above nest of intervals
All points
fall into three groups:
1)
is a 4-adic rational point. Altogether they form the countable set D. Its 4-adic decimal expansion contains finite terms (or an infinite cyclic decimal).
2)
. That is to say the 4-adic decimal expansion of t is infinite acylic, but from the
decimal place,
for
.
3)
where
for
.
Using the arithmetic-analytic representation of the Koch curve of 4-adic rational points (see Lemma 3.3), we can prove the following two lemmas:
Lemma 3.4. If
then
In particular, if
, where
, then
Lemma 3.5. If
then
where
when
,
when
.
4. The
-Hölder 4-Adic Derivatives of the Koch Curve
The Koch curve with 4-adic decimal expansion satisfies the Hölder condition (4), (5), so it is reasonable to consider dyadic derivatives of
and
, which are defined in Section 3. Now we shall prove that for the Koch curve the exact
-Hölder 4-adic derivative exists which is the main result of this paper.
Theorem 4.1 If
and
(7)
is point type 2), then
where
.
Proof. By (4.1), it can be seen that t is contained in the following sequence of intervals
and
Therefore
. According to Lemma 3.4,
So, by Definition 2.1,
Similarly
Theorem 4.2 If
is point type 3), then
Proof. By (4.1) t is contained in the following nested intervals
where
, and
Therefore
According to Lemma 4,
By Definition 2.1,
Similarly
The conclusions of Theorem 4.1 and Theorem 4.2 can be written as: for
, then
Note that when t is the point of type 2),
.
We have shown that the Koch curve
has
-Hölder 4-adic derivatives for every
. Next, we consider the case of a 4-adic rational point on D.
Theorem 4.3 If
is 4-adic rational point on
Then, there exists
-Hölder 4-adic left and right derivatives. The right derivatives are
Fractal
The left derivatives are
where
when
,
when
.
Proof. By Lemma 4,
Then, by Definition 2.2,
Similarly
According to Lemma 5,
where
when
,
when
. So, by Definition 2.2,
Similarly
Theorems 4.1 - 4.3 then have established the
-Hölder 4-adic derivatives of the Koch curve on [0, 1]. As He pointed out recently [17] that fractal derivative/calculus has very important applications in many applied fields including mathematics, engineering and fluid dynamics, and researchers have tried to define various derivatives of fractals. Our results will not only allow us to further investigate the differentiability of other fractals [4] [18] , but also provide a new type of derivative for researchers in other fields to conduct their investigations.
Some observations: It is obvious that the right and left
-Hölder 4-adic derivatives of the Koch curve at the 4-adic rational point are not equal, as these points should be at the sharp-angled vertices. Furthermore, 1) There exists unequal left and right
-Hölder 4-adic derivatives for the Koch curve at countable points set (i.e. the 4-adic rational points) on
. This indicates that the knot point quality of a non-differentiable function seems not to be eliminated no matter how the derivative is defined. 2) For 4-adic irrational points, the
-Hölder 4-adic derivatives are determined at the second type of points, which are
and
. 3) For the points of the third type, although the set of the
-Hölder 4-adic derivatives also contains
and
, it is not definite. This reflects the oscillatory quality of non-differentiable function, i.e. it is also the case of the knot points without one-sided derivatives [19] . Of course, in this case, it might be better to consider the upper α-Hölder derivative.