THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050052
Author(s):  
JUNRU WU
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Closed Interval ◽  
Open Interval ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Positive Order ◽  
Lipschitz Continuous ◽  
Hölder Continuous Functions

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050005
Author(s):  
JIA YAO ◽  
YING CHEN ◽  
JUNQIAO LI ◽  
BIN WANG
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order ◽  
One Dimensional ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050123
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractal Dimension ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Condition ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Liouville Fractional Derivative

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

RELATIONSHIP BETWEEN UPPER BOX DIMENSION OF CONTINUOUS FUNCTIONS AND ORDERS OF WEYL FRACTIONAL INTEGRAL

Fractals ◽  
2021 ◽  
Author(s):  
H. B. GAO ◽  
Y. S. LIANG ◽  
W. XIAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Dimension One ◽  
Weyl Fractional Integral

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on [Formula: see text], a better estimation of fractal dimension can be obtained. When a function is continuous on [Formula: see text] and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

ESTIMATION OF FRACTAL DIMENSION OF WEYL FRACTIONAL INTEGRAL OF CERTAIN CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050030 ◽  
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Condition ◽  
Positive Order ◽  
Weyl Fractional Integral

In this work, we consider fractal dimension such as Box dimension, of Weyl fractional integral of certain continuous functions. Upper Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition of positive order [Formula: see text] is no more than both [Formula: see text] and [Formula: see text]. Furthermore, it is no more than [Formula: see text] which means strictly less than [Formula: see text]. Meanwhile, [Formula: see text], Box dimension of Weyl fractional integral of continuous functions satisfying [Formula: see text]-order Hölder condition must be one.

Invariant subsets of expanding mappings of the circle

1987 ◽  
Vol 7 (4) ◽  
pp. 627-645 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Compact Interval ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Piecewise Monotone ◽  
Hölder Continuous Functions ◽  
Expanding Mapping

AbstractThe continuity of Hausdorff dimension of closed invariant subsetsKof aC2-expanding mappinggof the circle is investigated. Ifg/Ksatisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that iffis a piecewise monotone continuous mapping of a compact interval and φ a continuous function withP(f,φ)> sup(φ), then the pressureP(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.

HÖLDER CONTINUITY AND BOX DIMENSION FOR THE WEYL FRACTIONAL INTEGRAL

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050032 ◽  
Author(s):  
LONG TIAN
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Hölder Continuity ◽  
Box Dimension ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Weyl Fractional Integral

In this paper, we investigate the Hölder continuity and the estimate for the box dimension of the Weyl fractional integral of some continuous function [Formula: see text], denoted by [Formula: see text]. We obtain that if [Formula: see text] is [Formula: see text]-order Hölder continuous, then [Formula: see text] is [Formula: see text]-order Hölder continuous. Moreover, if [Formula: see text] belongs to [Formula: see text], then [Formula: see text] is [Formula: see text]-order Hölder continuous with [Formula: see text].

Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions

2018 ◽  
Vol 21 (6) ◽  
pp. 1651-1658 ◽  
Author(s):  
Yong Shun Liang
Keyword(s):  
Fractal Dimension ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order

Abstract The present paper investigates fractal dimension of fractional integral of continuous functions whose fractal dimension is 1 on [0, 1]. For any continuous functions whose Box dimension is 1 on [0, 1], Riemann-Liouville fractional integral of these functions of any positive order has been proved to still be 1-dimensional continuous functions on [0, 1].

On a quantitative theory of limits: Estimating the speed of convergence

2020 ◽  
Vol 23 (4) ◽  
pp. 1013-1024
Author(s):  
Renato Spigler
Keyword(s):  
Modulus Of Continuity ◽  
Fractional Derivatives ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Speed Of Convergence ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Quantitative Theory ◽  
Hölder Continuous Functions ◽  
Definition Of

AbstractThe classical “ε-δ” definition of limits is of little use to quantitative purposes, as is needed, for instance, for computational and applied mathematics. Things change whenever a realistic and computable estimate of the function δ(ε) is available. This may be the case for Lipschitz continuous and Hölder continuous functions, or more generally for functions admitting of a modulus of continuity. This, provided that estimates for first derivatives, fractional derivatives, or the modulus of continuity might be obtained. Some examples are given.

On the nonexistence of strictly monotonic Hölder continuous functions

2006 ◽  
Vol 2 (1-2) ◽  
pp. 89-91
Author(s):  
Sergio Amat ◽  
Sonia Busquier ◽  
Antonio Escudero
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Simple Analysis ◽  
Hölder Continuous ◽  
Holder Space ◽  
Hölder Space ◽  
Analysis Tools ◽  
Hölder Continuous Functions

This note is devoted to the study of the monotony of the Hölder continuous functions. We prove the nonexistence of a strictly monotonic (increasing or decreasing) hölder continuous function with exponent s ϵ (0, 1) such that it does not belongs for anypoint in a Hölder space with exponent s + ε ε > 0. We use simple analysis’ tools.

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