THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

Fractals ◽  
2021 ◽  
Author(s):  
XIA TING ◽  
CHEN LEI ◽  
LUO LING ◽  
WANG YONG
Keyword(s):  
Linear Relationship ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Hölder Condition ◽  
Weierstrass Function ◽  
Variation Function ◽  
Weyl Fractional Integral

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

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.

scholarly journals Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation

2020 ◽  
Vol 12 (1) ◽  
pp. 30-53
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Fractional Integrals ◽  
Functions Of Bounded Variation ◽  
Hölder Continuous ◽  
Hölder Continuous Functions ◽  
Hadamard Fractional Integrals

AbstractIn this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.

Some theorems on fractional integrals

1959 ◽  
Vol 55 (1) ◽  
pp. 31-50 ◽  
Author(s):  
T. M. Flett
Keyword(s):  
Power Series ◽  
Fractional Integral ◽  
Real Variable ◽  
Fractional Integrals ◽  
Image Position ◽  
Weyl Fractional Integral

1. Let φ(z) be regular in │z│ < 1, letand for any real α letwhere (in)−α = n−α exp(— ½απi). The function φα is an adaptation to power series of the Weyl fractional integral of order a of α function of a real variable.† It exists and is regular in │z│ < 1 for all α. Let also

scholarly journals Ostrowski and Trapezoid Type Inequalities for the Generalized k-g-Fractional Integrals of Functions with Bounded Variation

2017 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Fractional Integral ◽  
Integrable Function ◽  
Fractional Integrals ◽  
Continuous Derivative ◽  
Functions Of Bounded Variation ◽  
Lebesgue Integrable Function ◽  
Increasing Function

Let g be a strictly increasing function on a , b , having a continuous derivative g&prime; on a , b . For the Lebesgue integrable function f : a , b &rarr; C , we define the k-g-left-sided fractional integral of f by S k , g , a + f x = &int; a x k g x - g t g &prime; t f t d t , x &isin; a , b and the k-g-right-sided fractional integral of f by S k , g , b - f x = &int; x b k g t - g x g &prime; t f t d t , x &isin; [ a , b ) , where the kernel k is defined either on 0 , &infin; or on 0 , &infin; with complex values and integrable on any finite subinterval. In this paper we establish some Ostrowski and trapezoid type inequalities for the k-g-fractional integrals of functions of bounded variation. Applications for mid-point and trapezoid inequalities are provided as well. Some examples for a general exponential fractional integral are also given.

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