Some theorems on fractional integrals

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.

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

Fractals ◽

10.1142/s0218348x20502230 ◽

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.

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Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

Abstract and Applied Analysis ◽

10.1155/2012/871912 ◽

2012 ◽

Vol 2012 ◽

pp. 1-24 ◽

Cited By ~ 28

Author(s):

Tatiana Odzijewicz ◽

Agnieszka B. Malinowska ◽

Delfim F. M. Torres

Keyword(s):

Fractional Integral ◽

Necessary Optimality Conditions ◽

Variational Problems ◽

Fractional Integrals ◽

Natural Boundary ◽

Free Boundary Value Problems ◽

Fractional Variational Problems ◽

Generalized Fractional Integral ◽

Fractional Calculus Of Variations ◽

Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

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Analogue of a theorem of Khintchine in fields of formal power series

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100040287 ◽

1966 ◽

Vol 62 (4) ◽

pp. 637-642 ◽

Cited By ~ 2

Author(s):

T. W. Cusick

Keyword(s):

Power Series ◽

Positive Constant ◽

Classical Result ◽

Real Numbers ◽

Linear Forms ◽

Absolute Value ◽

The Difference ◽

Image Position ◽

Formal Power ◽

Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

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An algebraically closed field

Glasgow Mathematical Journal ◽

10.1017/s0017089500000422 ◽

1968 ◽

Vol 9 (2) ◽

pp. 146-151 ◽

Cited By ~ 14

Author(s):

F. J. Rayner

Keyword(s):

Positive Integer ◽

Power Series ◽

Formal Power Series ◽

Characteristic Zero ◽

Closed Field ◽

Algebraically Closed Field ◽

Zero Characteristic ◽

Image Position ◽

Formal Power ◽

Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

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Analytic functions of finite dimensional linear transformations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100033685 ◽

1959 ◽

Vol 55 (1) ◽

pp. 51-61 ◽

Cited By ~ 1

Author(s):

S. N. Afriat

Keyword(s):

Power Series ◽

Integral Formula ◽

Characteristic Root ◽

Interpolation Formula ◽

General Function ◽

Linear Transformations ◽

Multiple Characteristic ◽

Characteristic Roots ◽

Image Position ◽

Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

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Weighted Norm Inequalities for Fractional Integral Operators With Rough Kernel

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-003-1 ◽

1998 ◽

Vol 50 (1) ◽

pp. 29-39 ◽

Cited By ~ 46

Author(s):

Yong Ding ◽

Shanzhen Lu

Keyword(s):

Integral Operator ◽

Fractional Integral ◽

Integral Operators ◽

Rough Kernel ◽

Degree Zero ◽

Weighted Norm ◽

Weighted Norm Inequalities ◽

Norm Inequalities ◽

Fractional Integral Operators ◽

Image Position

AbstractGiven function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

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New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel

AIMS Mathematics ◽

10.3934/math.2021648 ◽

2021 ◽

Vol 6 (10) ◽

pp. 11167-11186

Author(s):

Hari M. Srivastava ◽

◽

Artion Kashuri ◽

Pshtiwan Othman Mohammed ◽

Abdullah M. Alsharif ◽

...

Keyword(s):

Fractional Integral ◽

Integral Operators ◽

Fractional Integrals ◽

Chebyshev Inequality ◽

Fractional Integral Operators ◽

General Family ◽

Chebyshev Functional ◽

Bounded Below

<abstract><p>The main goal of this article is first to introduce a new generalization of the fractional integral operators with a certain modified Mittag-Leffler kernel and then investigate the Chebyshev inequality via this general family of fractional integral operators. We improve our results and we investigate the Chebyshev inequality for more than two functions. We also derive some inequalities of this type for functions whose derivatives are bounded above and bounded below. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. Finally, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view.</p></abstract>

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Объединение классических и динамических неравенств, возникающих при анализе временных масштабов

Вестник КРАУНЦ. Физико-математические науки ◽

10.26117/2079-6641-2020-33-4-26-36 ◽

2020 ◽

pp. 26-36

Author(s):

M.J.S. Sahir

Keyword(s):

Time Scales ◽

Time Scale ◽

Fractional Integral ◽

Fractional Integrals ◽

Liouville Type ◽

Lyapunov’S Inequality ◽

Discrete Analogues ◽

Radon’S Inequality

In this paper, we present an extension of dynamic Renyi’s inequality on time scales by using the time scale Riemann–Liouville type fractional integral. Furthermore, we find generalizations of the well–known Lyapunov’s inequality and Radon’s inequality on time scales by using the time scale Riemann–Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.

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A Spectral Numerical Method for Solving Distributed-Order Fractional Initial Value Problems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4041030 ◽

2018 ◽

Vol 13 (10) ◽

Cited By ~ 9

Author(s):

M. A. Zaky ◽

E. H. Doha ◽

J. A. Tenreiro Machado

Keyword(s):

Integral Equation ◽

Initial Value Problems ◽

Fractional Integral ◽

Recurrence Relations ◽

Fractional Integrals ◽

Initial Value ◽

Fractional Integral Equation ◽

Collocation Scheme ◽

Gauss Quadrature Formula ◽

Distributed Order

In this paper, we construct and analyze a Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems. We first introduce three-term recurrence relations for the fractional integrals of the Legendre polynomial. We then use the properties of the Caputo fractional derivative to reduce the problem into a distributed-order fractional integral equation. We apply the Legendre–Gauss quadrature formula to compute the distributed-order fractional integral and construct the collocation scheme. The convergence of the proposed method is discussed. Numerical results are provided to give insights into the convergence behavior of our method.

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