Lebesgue Integral Research Articles

The A-Integral and Restricted Riesz Transform

It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds.ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds

Numerical approximation of tempered fractional Sturm‐Liouville problem with application in fractional diffusion equation

SummaryIn this paper, we discuss the numerical approximation to solve regular tempered fractional Sturm‐Liouville problem (TFSLP) using finite difference method. The tempered fractional differential operators considered here are of Caputo type. The numerically obtained eigenvalues are real, and the corresponding eigenfunctions are orthogonal. The obtained eigenfunctions work as basis functions of weighted Lebesgue integrable function space (a,b). Further, the obtained eigenvalues and corresponding eigenfunctions are used to provide weak solution of the tempered fractional diffusion equation. Approximation and error bounds of the solution of the tempered fractional diffusion equation are provided.

The N-Integral

In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon >0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|<epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|<= epsilon}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.

Estimates for the Navier–Stokes equations in the half-space for nonlocalized data

This paper is devoted to the study of the Stokes and Navier-Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely $L^q_{uloc,\sigma}(\R^d_+)$. We prove the analyticity of the Stokes semigroup $e^{-t{\bf A}}$ in $L^q_{uloc,\sigma}(\R^d_+)$ for $1<q\leq\infty$. This follows from the analysis of the Stokes resolvent problem for data in $L^q_{uloc,\sigma}(\R^d_+)$, $1<q\leq\infty$. We then prove bilinear estimates for the Oseen kernel, which enables to prove the existence of mild solutions. The three main original aspects of our contribution are: (i) the proof of Liouville theorems for the resolvent problem and the time dependent Stokes system under weak integrability conditions, (ii) the proof of pressure estimates in the half-space and (iii) the proof of a concentration result for blow-up solutions of the Navier-Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.

Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in [formula omitted

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for a.e. t≥0, and which satisfies a natural assumption. Next, we prove that if f∈M(Bn), then the nonlinear resolvent family {Jr}r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector field is of divergence type, where Jr=Jr[f]=(In+rf)−1, r≥0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1+r)Jr can be embedded as the first element of a normal Loewner chain and the family {(1+r)Jr[f]:0≤r<∞,f∈M(Bn)} is a compact subset of S0(Bn). Also, we deduce that the shearing of (1+r)Jr (r≥0) associated with the family M(B2) is quasi-convex of type A and also starlike of order 4/5. We give a sufficient condition for (1+r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of Cn onto itself. Finally, sharp coefficient bounds for the nonlinear resolvent families of certain subsets of M(Bn), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2, will be obtained.

SoftMax Neural Best Approximation

Neural networks have a great place in approximating nonlinear functions, especially those Lebesgue integrable functions that are approximated by FNNs with one hidden layer and sigmoidal functions. Various operators of neural networks have been defined and achieved to get good rates of approximation depending on the modulus of smoothness. Here we define a new neural network operator with a generalized sigmoidal function (SoftMax) to improve the rate of approximation of a Lebesgue integrable function Lp , with p < 1, to be estimated using modulus of smoothness of order k. The importance of choosing SoftMax function as an activation function is its flexible properties and various applications.

Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

Fubini theorem and generalized Minkowski inequality for the pseudo-integral

This paper is devoted to generalizations of Fubini theorem, Transformation theorem, and generalized Minkowski inequality for the so called pseudo-integral. The approach is based on the relation of double pseudo-integrals and iterated pseudo-integrals. Since the pseudo-integral covers Lebesgue integral, Sugeno's fuzzy integral, and Zhao's (N)fuzzy integral (with a respect to special nonadditive measure: ∨-measure), the obtained results generalize the corresponding previously known results.

Weighted Sp-pseudo S-asymptotic periodicity and applications to Volterra integral equations

This paper is related to the function space formed by weighted Sp-pseudo S-asymptotic periodicity and their applications. Initially, the translation invariance and completeness of the function space are investigated. Additionally, the composition theorem and convolution operator generated by Lebesgue integrable functions are presented. Finally, existence and uniqueness of solutions with weighted Sp-pseudo S-asymptotic periodicity for two classes of Volterra equations are proved by using the results obtained above, and some concrete examples are given. The methods mainly include Minkowski’s inequality, convolution inequality, contraction mapping principle, and especially the generalized Minkowski’s inequality. Our results extend some known results on asymptotic periodicity.

Bounds for the Generalized (&amp;#934;, f)-Mean Difference

In this paper we establish some bounds for the \( (\Phi;f) \)-mean difference introduced in the general settings of measurable spaces and Lebesgue integral, which is a two functions generalization of Gini mean difference that has been widely used by economists and sociologists to measure economic inequality.

HÖLDER CONDITIONS IN THE DIVERGENCE THEOREM

In the framework of Lebesgue integration and bounded sets of finite perime- ter, we present a straightforward proof of the divergence theorem for bounded vector fields satisfying Holder conditions on sets of appropriate Hausdorff measures. Given a compact BV set A ⊂ R n , the divergence theorem holds for every vector field v : A → R n that is continuous outside an H n−1 negligible set, and pointwise Lipschitz outside an H n−1 σ-finite set (9, Theorem 2.9). Since continuous and pointwise Lipschitz are extreme points of the scale represented by Holder conditions, it is natural to ask whether the divergence theorem remains valid under the following assumptions: for 0 <s< 1, the vector field v is pointwise Lipschitz outside an H n−1+s σ-finite (negligible) set E ⊂ A, and the s-Holder constant of v is zero (finite) at each x ∈ E. We use results of W.B. Jurkat (see Remark 2.5 below) to prove the divergence theorem which takes into account Holder conditions for all 0 ≤ s ≤ 1 simultaneously. One of the consequences, obtained by restricting the number s to 0 and 1, is a new and simpler proof of the divergence theorem cited above. Our starting point is the well-known divergence theorem for bounded BV sets and con- tinuously differentiable vector fields, which is assumed without proof. Modulo a few techni- calities resulting from the use of BV sets, the exposition is elementary. The crux of the proof involves only dyadic cubes. As an application, we present two theorems about integration by parts. Both are easy corollaries of the main result.

RALPH HENSTOCK'S INFLUENCE ON THE INTEGRATION THEORY

Professor Ralph Henstock, a distinguished analyst who devoted most of his research to integration theory, died on January 7, 2007. It was my good fortune to know Ralph for almost 30 years, and to derive much inspiration from his pioneering work. Rather than commenting briefly on a variety of Ralph’s achievements, I will elaborate on that which has been the most influential: a Riemann type definition of the Denjoy-Perron integral . An early recognition that the powerful Lebesgue integral does not integrate the derivatives of all differentiable functions led to the development of the Denjoy-Perron integral. Three equivalent definitions of the Denjoy-Perron integral were available at the beginning of the last century: descriptive and constructive definitions presented by Denjoy [16, Chapter 8], and a definition based on approximations by majorants and minorants due to Perron [16, Chapter 6, Section 6]. These definitions differ widely, and establishing their equivalence is not easy — for a comprehensive treatment see [4, Chapter 11]. Neither definition is simple, and attempts to generalize any of them to higher dimensions were not successful. Indeed, none of the multidimensional integrals based on these definitions integrates partial derivatives of all differentiable functions. It was a major independent accomplishment of Henstock [5] and Kurzweil [10] to observe that a minor but ingenious change in the classical definition of the Riemann integral produces the Denjoy-Perron integral. The striking simplicity of their definition revitalized the efforts toward finding a multidimensional analog of the Denjoy-Perron integral. The initial impetus was further promoted by Henstock’s diligent work on the general properties of Riemann type integrals, summarized in monographs [6, 7, 8]. After introducing the Henstock-Kurzweil result without proof, I will discuss two separate but related topics:

Rough integration via fractional calculus

On the basis of fractional calculus, the author's previous study [9] introduced an approach to the integral of controlled paths against Holder rough paths. The integral in [9] is defined by the Lebesgue integrals for fractional derivatives without using any arguments based on discrete approximation. In this paper, we revisit the approach of [9] and show that, for a suitable class of Holder rough paths including geometric Holder rough paths, the integral in [9] is consistent with that obtained by the usual integration theory of rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.

A stochastic Fubini theorem for [formula omitted]-stable process

In this paper, we give an elementary proof of a stochastic Fubini theorem, which says that one can interchange a Lebesgue integral and a stochastic integral with respect to an α-stable (1<α<2) Lévy process. As an application, the equivalence between weak and mild solutions for a class of semi-linear SPDEs with α-stable (1<α<2) noise is established.

Вариационная постановка одной обратной задачи для параболического уравнения с интегральными условиями

Coefficient inverse problems for partial differential equations can be posed as optimal control problems, i. e. in variation form. In such formulations, the sought-for coefficients of the state equations play the role of control functions, and the objective functionals are compiled on the basis of additional conditions. The paper discusses a variational formulation of the inverse problem of determining the lower coefficient of a multidimensional parabolic equation with an integral boundary condition and an additional integral condition. In this case, the role of the control function is played by the lower coefficient of the parabolic equation and is an element of the space of Lebesgue integrable functions with a finite summability index. The solution to the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The objective functional is based on an additional integral condition. The existence of a solution to the problem is proved and the necessary optimality condition is obtained.

Multivariate weighted kantorovich operators

Recently, some weighted Durrmeyer type operators were used as research tools in Learning Theory. In this paper we introduce a class of multidimensional weighted Kantorovich operators \begin{document}$ K_n $\end{document} on \begin{document}$ C(Q_d) $\end{document} where \begin{document}$ Q_d $\end{document} is the \begin{document}$ d $\end{document} -dimensional hypercube \begin{document}$ [0,1]^d $\end{document} . We show that each \begin{document}$ K_n $\end{document} has a unique invariant probability measure and determine this measure. Then, using results from approximation theory and the theory of ergodic operators, we find the limit of the iterates of \begin{document}$ K_n $\end{document} and give rates of convergence of the iterates toward the limit. Finally, we show that some Kantorovich type operators previously investigated in literature fall into the class of operators introduced in our paper. Other properties and applications, involving Learning Theory, will be presented in a forthcoming paper, where we will consider also operators on spaces of Lebesgue integrable functions on the hypercube \begin{document}$ Q_d $\end{document} .

Observability of Fractional-Order Impulsive Control Integro-Differential System with Nonlocal Initial Condition

The article describes a new concept for initial and exactly observability of nonlocal fractional-order impulsive control integro-differential system. This is based on the concepts of the abstract Cauchy problem, which depended on some necessary and sufficient conditions. These conditions established on the semigroup theory of bounded operators as a dynamical operator system, which generated by bounded linear operators. Moreover, invertible operators play a primary role, and we presented a necessary condition for some nonlinear multi variables functions. Thus, all these operators were treated in nonlinear functional analysis to guaranty the initial observable and exactly observability. Therefore, from the mild solution of the system and exactly homogenous part, we proved the equivalent concepts between the initial observability and exactly the observability. Thus, our approach in this article is to prove the uniqueness of initial nonlocal values with admissible control, which belongs to the second-order Lebesgue integrable. The interest of observability results in this article lies by proving a unique fixed point, which is nonlocal initial values that are described in the proposal formula by using Banach’s fixed point theory. The processing observability for complexly systems (such as this system) with all components and properties was established and can be used for many control system applications.

The Lebesgue integral of trapezoid rule based on derivative

The Lebesgue integral of trapezoid rule based on derivative

Familiarizing Students with Definition of Lebesgue Outer Measure Using Mathematica: Some Examples of Calculation Directly from Its Definition

In this paper we present some examples of calculation the Lebesgue outer measure of some subsets of mathbb {R}^2 directly from definition 1. We will consider the following subsets of mathbb {R}^2: displaystyle {(x, y)in mathbb {R}^2: 0le y le x^2, xin [0, 1]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le exp (-x), xge 0}, big { (x,y) in mathbb {R}^2: ln x le y le 0, xin (0, 1]big }, big { (x,y) in mathbb {R}^2: 0le y le 1/x, xge 1big }, displaystyle {(x, y)in mathbb {R}^2: 0le y le sin x, xin [0, pi /2]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le exp (x), xin [0, 1]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le ln (1-2r cos x+r^2), x in [0, pi ]}, r>1 and some others. We could not find any analogical examples in available literature (except for rectangle and countable sets), so this paper is an attempt to fill this gap. We calculate sums, limits and plot graphs and dynamic plots of needed sets and unions of rectangles sums of which volumes approximate Lebesgue outer measure of the sets, using Mathematica. We also show how to calculate the needed sums and limits by hand (without CAS). The title of this paper is very similar to the title of author’s article (Wojas and Krupa in Math Comput Sci 11:363–381, 2017) which deals with definition of Lebesgue integral but this paper deals with definition of Lebesgue outer measure instead.

English

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrodinger equation $-y''+qy=f$, where $q$ and $f$ are Henstock-Kurzweil integrable functions on $[a,b]$. Results presented in this article are generalizations of the classical results for the Lebesgue integral.