Box dimensions of Riemann–Liouville fractional integrals of continuous functions of bounded variation

Liang, Yongshun

doi:10.1016/j.na.2010.02.007

Cited by 98 publications

(34 citation statements)

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“…In fact, D −1 f (x) which is Riemann integral of order 1 is a differentiable function on I, then it is of bounded variation on I. By [4], we know (3.5) holds. In Theorem 3.3, there is a condition that orders of Riemann-Liouville integral belong to (0, 1).…”

Section: Proposition 23mentioning

confidence: 91%

“…Furthermore, box dimension of Riemann-Liouville integral of f (x) of order v > 1 − α still is 1. In fact, by [4], Riemann-Liouville integral of a continuous function with bounded variation of any positive order v still is a continuous function with bounded variation. f (x) ∈ H 1 is obviously of bounded variation.…”

Section: Remark 49mentioning

confidence: 99%

“…In [4], Riemann-Liouville fractional integral of a continuous function f (x) of bounded variation on a closed interval has been proved to still be a continuous function with bounded variation. It is obvious that both f (x) and its Riemann-Liouville fractional integral are 1-dimensional.…”

Section: Introductionmentioning

confidence: 99%

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Fractal dimensions of fractional integral of continuous functions

Liang

2016

Acta. Math. Sin.-English Ser.

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In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f (x) of order v(v > 0) which is written as D −v f (x) has been proved to still be continuous and bounded. Furthermore,With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f (x) satisfying Hölder condition, upper box dimension of Riemann-Liouville fractional integral of f (x) seems no more than upper box dimension of f (x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying Hölder condition of order v(v > 0) is strictly less than 2 − v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.

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Section: Proposition 23mentioning

confidence: 91%

Section: Remark 49mentioning

confidence: 99%

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Fractal dimensions of fractional integral of continuous functions

Liang

2016

Acta. Math. Sin.-English Ser.

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“…[8][9][10][11] Fractional calculus such as RiemannLiouville fractional calculus 4,12 which has been used to investigate fractal curves is an important tool in the fractal analysis. Liang 12 carried out research on Riemann-Liouville fractional integral of continuous functions with bounded variation.…”

Section: Introductionmentioning

confidence: 99%

Fractal Dimension of Riemann–liouville Fractional Integral of Certain Unbounded Variational Continuous Function

Xiao

2017

Fractals

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“…Ref. [4] made research on Riemann-Liouville fractional integral of continuous functions with bounded variation. Fractal dimension of Riemann-Liouville fractional integral of any order of such functions has been proved to be 1.…”

Section: Introductionmentioning

confidence: 99%

Upper Bound Estimation of Fractal Dimension of Fractional Calculus of Continuous Functions

Li¹,

Liang²

2017

AAN

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In the present paper, upper bound estimation of upper Box dimension of RiemannLiouville fractional integral of order v of any continuous functions on a closed interval has been proved to be no more than 2 − v when 0 < v < 1. If a continuous function which satisfies α-Hölder condition on a closed interval, upper Box dimension of its Riemann-Liouville fractional integral is no more than 2 − α when 0 < α < 1. Upper bound of upper Box dimension of Riemann-Liouivlle fractional integral of certain type of fractal functions has been proved to be no more than Box dimension of functions themselves.

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Box dimensions of Riemann–Liouville fractional integrals of continuous functions of bounded variation

Cited by 98 publications

References 1 publication

Fractal dimensions of fractional integral of continuous functions

Fractal dimensions of fractional integral of continuous functions

Fractal Dimension of Riemann–liouville Fractional Integral of Certain Unbounded Variational Continuous Function

Upper Bound Estimation of Fractal Dimension of Fractional Calculus of Continuous Functions

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