Lebesgue Integral Research Articles

Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces

In this paper we study set-valued stochastic differential equations in M-type 2 Banach spaces. Their drift terms and diffusion terms are assumed to be set-valued and single-valued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define set-valued stochastic Lebesgue integral in a Banach space. This integral is a set-valued random variable. We state its properties such as additivity with respect to the interval of integration, continuity as a function of the upper limit of integration, integrable boundedness. The existence and uniqueness of solution to set-valued differential equations in M-type 2 Banach space is obtained by a method of successive approximations. We show that the approximations are uniformly bounded and converge to the unique solution. A distance between $n$th approximation and exact solution is estimated and a continuous dependence of solution with respect to the data of the equation is proved. Finally, we construct a fuzzy stochastic Lebesgue integral in a Banach space and examine fuzzy stochastic differential equations in M-type 2 Banach spaces. We investigate properties like those in set-valued cases. All the results are achieved without assumption on separability of underlying sigma-algebra.

On a Class of Quadratic Urysohn–Hammerstein Integral Equations of Mixed Type and Initial Value Problem of Fractional Order

In this paper, we study the existence of monotonic solution of a general nonlinear functional quadratic Urysohn–Hammerstein integral equations of mixed type in the class of Lebesgue integrable functions on the half-axis (\({{\mathbb{R}}^+ = [0, \infty))}\). As an application of our results, we prove the existence of solution for the initial value problem of fractional order. The main tools used are Darbo fixed point theorem associated with the measure of noncompactness and fractional calculus.

Almost everywhere approximation capabilities of double Mellin approximate identity neural networks

The research works in approximation theory of artificial neural networks is still far from completion. To fill a gap in this issue, this study focuses on the almost everywhere approximation capabilities of single-hidden-layer feedforward double Mellin approximate identity neural networks. First, the notion of double Mellin approximate identity is introduced. Using this notion, an auxiliary theorem is proved. The auxiliary theorem provides a connection between a class of double Mellin convolution linear operators and the notion of almost everywhere convergence. This theorem is applied to prove a main theorem. The proof of the main theorem is based on the notion of epsilon-net. The main theorem shows almost everywhere approximation capability of single-hidden-layer feedforward double Mellin approximate identity neural networks in the space of almost everywhere continuous bivariate functions on $$ \mathbb {R}_{+}^{2} $$R+2. Moreover, similar results are obtained in the spaces of almost everywhere Lebesgue integrable bivariate functions on $$ \mathbb {R}_{+}^{2} $$R+2.

Properties of the McShane Integrability

In 1990 Gordon[12] introduced the concepts of the McShane integral of Banach-valued functions. This integral is a generalized Riemann integral of functions which have values in a Banach space. For real-valued functions the McShane integral and the Lebesgue integral are equivalent. Gordon[12] and Fremlin and Mendoza[3] have developed the properties of this integral. In this paper, we will investigate some properties and relations among the McShane integral, the Dunford integral, and the Pettis integral. Let a given separable  contain no copy of C 0 and T be a weakly compact operator. Then we will show that a Dunford integrable function f:[a,b] →X is intrinsically-separable valued when f is McShane integrable. Also, we will show that if there exists a sequence (f n ) of McShane integrable functions from [a,b] to X such that for each x ? ∈X ? ,x ? f n → x ? f a·e·, then f is McShane integrable. Finally, we will show that if f :[a,b] →X is McShane integrable, then {x ? f x ? ∈Bx ? } is uniformly integrable in L¹(μ) where X contains no copy of C 0 .

On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the $$\sigma $$ -Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

Some properties of Choquet integral based probability functions

The Choquet integral permits us to integrate a function with respect to a non-additive measure. When the measure is additive it corresponds to the Lebesgue integral. This integral was used recently to define families of probability-density functions. They are the exponential family of Choquet integral (CI) based class-conditional probability-density functions, and the exponential family of Choquet– Mahalanobis integral (CMI) based class-conditional probability-density functions. The latter being a generalization of the former, and also a generalization of the normal distribution.In this paper we study some properties of these distributions, and study the application of a few normality tests.

An existence result for a class of nonlinear functional integral equations

We consider a general nonlinear functional integral equation, and we prove the existence of solutions of this equation in the space of Lebesgue integrable functions on $\R^+$. Our analysis uses a recent version of Krasnosel'skii's fixed point theorem (Theorem \ref{2t1}) and the concept of the measure of weak noncompactness. In the appendix, we give an extension of Theorem~\ref{2t1} to expansive mappings.

Existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on [formula omitted

Let Lp(R+) denote the space of Lebesgue integrable functions on R+ with the standard norm∥x∥p=(∫0∞|x(t)|pdt)1p.First, we define a new measure of noncompactness on the spaces Lp(R+) (1 ≤ p < ∞). In addition, we study the existence of entire solutions for a class of nonlinear functional integral equations of convolution type using Darbo’s fixed point theorem, which is associated with the new measure of noncompactness. We provide some examples to demonstrate that our results are applicable whereas the previous results are not.

Absolute Continuity of a Function and Uniform Integrability of Its Divided Differences

We describe a proof of the fundamental theorem of calculus for the Lebesgue integral, based on the following result: A real-valued function f on a compact interval is absolutely continuous if and only if its family of divided difference functions, {x ↦ [f(x + h) − f(x)]/h}0<h≤1, is uniformly integrable. The Vitali convergence theorem states that, for a sequence of uniformly integrable functions, pointwise convergence implies integrability of the limit function and convergence of the integrals. Our characterization of absolute continuity, together with Vitali's theorem, shows that the fundamental theorem for absolutely continuous functions follows simply by passing to the limit in its discrete formulation.

An indirect pseudospectral method for the solution of linear-quadratic optimal control problems with infinite horizon

We consider a class of linear-quadratic infinite horizon optimal control problems in Lagrange form involving the Lebesgue integral in the objective. The key idea is to introduce weighted Sobolev spaces as state spaces and weighted Lebesgue spaces as control spaces into the problem setting. Then, the problem becomes an optimization problem in Hilbert spaces. We use the weight functions in our consideration. This problem setting gives us the possibility to extend the admissible set and simultaneously to be sure that the adjoint variable belongs to a Hilbert space too. For the class of problems proposed, existence results as well as a Pontryagin-type Maximum Principle, as necessary and sufficient optimality condition, can be shown. Based on this principle we develop a Galerkin method, coupled with the Gauss–Laguerre quadrature formulas as discretization scheme, to solve the problem numerically. Results are presented for the introduced model and different weight functions.

ON HENSTROCK INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

In this paper we introduce the interval-valued Hen- stock integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was flrst deflned by Henstock (2) in 1963. The Henstock integral is more powerful and simpler than the Lebesgue, Wiener and Feynman integrals. The Henstock delta integral on time scales was introduced by Allan Peterson and Bevan Thompson (5) in 2006. In 2000, Congxin Wu and Zengtai Gong introduced the concept of the Henstock integral of interval-valued functions (6). In this paper we introduce the concept of the Henstock delta integral of interval-valued function on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real numberR with the subspace topology inherited from the standard topology ofR. For t 2 T we deflne the forward jump operator ae(t) = inffs 2 T : s > tg where inf ` = supfTg, while the backward jump operator ‰(t) = supfs 2 T : s t, we say that t is right-scattered, while if ‰(t) < t, we say that t is left-scattered. If ae(t) = t, we say that t is right-dense, while if ‰(t) = t, we say that t is left-dense. The forward graininess function (t) of t 2 T is deflned by (t) = ae(t) i t, whlie the backward graininess function (t) of t 2 T is deflned by (t) = t i ‰(t). For a;b 2 T we denote the closed interval (a;b)T = ft 2 T : atbg.

A Characterization Theorem for Aumann Integrals

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be written as the Aumann integral with respect to some $\sigma$-finite measure. These conditions include the analog of the conlinearity and monotone convergence properties of the classical Daniell-Stone theorem for the Lebesgue integral, and three geometric properties that are peculiar to the set-valued case as they are redundant in the one-dimensional setting.

Foundational aspects of singular integrals

We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-L1 element in the class, and the Hadamard finite part integral (p.f.) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs.The Lebesgue integral as well as p.f., starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if f=g on (0,a), then their integrals from 0 to x coincide for any x∈(0,a).We find that integrals from zero with the essential properties of p.f., plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on (0,1] which are L1((ε,1]) for all ε>0. However, this existence proof does not provide a satisfactory construction. Without some regularity at 0, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-L1 element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator.Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze p.f. on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.

Approximation in different smoothness spaces with the RAFU method

The RAFU method is an original and unknown approximation procedure we can use in Approximation Theory. We know that the RAFU method provides a linear space uniformly dense in C[a,b] by using some separation conditions. In this work, we will show we can employ the RAFU method to approximate functions of C0(R) and C00(R), Riemann integrable functions, Lebesgue integrable functions, functions of Lp[a,b] and Lp(R), 1≤p&lt;¥ and measurable functions. Moreover, Riemann integrals can be approximated by the integrals of the functions that the RAFU method provides.

On a functional-integral equation with deviating arguments

We study the solvability of a functional-integral equation with deviating arguments, where our investigations take place in the space of Lebesgue integrable functions on an unbounded interval. In this space, we show that our functional-integral equation has at least one nonnegative and nonincreasing solution. The proof of our main result is based on a suitable combination of the technique associated with measures of noncompactness (in both the weak and the strong sense) and the Darbo fixed point. In the end, we conclude an example to illustrate our abstract results.

Filtering for stochastic systems driven by Poisson processes

This paper investigates the filtering problem for stochastic systems driven by Poisson processes. By utilising the martingale theory such as the predictable projection operator and the dual predictable projection operator, this paper transforms the expectation of stochastic integral with respect to the Poisson process into the expectation of Lebesgue integral. Then, based on this, this paper designs an filter such that the filtering error system is mean-square asymptotically stable and satisfies a prescribed performance level. Finally, a simulation example is given to illustrate the effectiveness of the proposed filtering scheme.

On the Kluvánek construction of the Lebesgue integral with respect to a vector measure

Abstract The Kluvánek construction of the Lebesgue integral is extended in two directions. First, instead of a compact interval [a, b] in the real line an abstract non-empty set X is considered, instead of the ring generated by subintervals of [a, b] an arbitrary ring A of subsets of X. Secondly, instead of the length of intervals (λ([c, d]) = d−c) any vector measure λ: A→V is considered, where V is a Riesz space.

Global Synchronization of Complex Networks Perturbed by Brown Noises and Poisson Noises

Stochastic disturbances, such as Brown noises and Poisson noises, are ubiquitous in the real world. It is necessary to consider these noises' effect for dynamical behaviors of complex networks. This paper addresses the synchronization problem for complex networks perturbed by Brown noises and Poisson noises. By utilizing the martingale theory such as the predictable projection operator and the dual predictable projection operator, we transform the expectation of the stochastic integral with respect to the Poisson process into the expectation of Lebesgue integral. Then, based on this, this paper presents a global synchronization criterion for complex networks perturbed by Brown noises and Poisson noises. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Evaluating Lebesgue Integrals Efficiently with the FTC

This note addresses evaluation of Lebesgue integrals on the real line using the Fundamental Theorem of Calculus, without having to verify that the primitive is absolutely continuous.