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In this study, the basic theory of the sequential Henstock-Kurzweil delta integral on time scales will be discussed. First, we give the notion and the elementary properties of this integral; then we show the equivalence of the Henstock-Kurzweil delta integral and the sequential Henstock-Kurzweil delta integral on time scales. In addition, we consider the Cauchy criterion and the Fundamental Theorems of Calculus. Finally, we prove Henstock’s lemma and give some convergence theorems. As an application, we consider the existence theorem of a kind of functional dynamic equations.

We obtain recovery formulas for coefficients of orthonormal spline series by means of its sum, if the partial sums of an orthonormal spline series converge in measure to a function and the majorant of partial sums satisfies some necessary condition, provided that the spline system corresponds to a "regular" sequence. Additionally, it is proved that the regularity of the sequence is essential. [ABSTRACT FROM AUTHOR]

Given arbitrary r ≥ 1 , we construct an HK r -integrable function which is not P 1 -integrable. This is an extension of a recently published construction [Musial, P., Skvortsov, V., Tulone, F.: The HK r -integral is not contained in the P r -integral. Proceedings of the American Mathematical Society 150(5), 2107–2114 (2022)]. [ABSTRACT FROM AUTHOR]

We construct a continuous function possessing an infinite derivative that cannot be integrated using the approximately continuous extension of the Denjoy-Khinchin integral introduced by Kubota. [ABSTRACT FROM AUTHOR]

In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent. [ABSTRACT FROM AUTHOR]

It is known that the controlled convergence theorem is one of the convergence theorems for $\mathcal{HK}$-integrable functions, which deals with nonabsolute $\mathcal{HK}$-integrable functions. This convergence theorem uses the concept of $U AC_\delta^{\star\star}(X)$, and the proof of the convergence theorem in the $n$-dimensional space is rather involved. In this paper, we shall give a necessary and sufficient condition for $UAC_\delta^{\star\star}(X)$, which is in terms of Lebesgue integrable functions on $X$ and primitive functions, see Theorem 9. Furthermore, we shall give three convergence theorems, Theorems 5, 7 and 10, for $\mathcal{HK}$-integrable functions in the $n$-dimensional space, which may be easier to apply. [ABSTRACT FROM AUTHOR]

In this article we develop the theory of H-Orlicz space generated by generalised Young function. Modular convergence of H-Orlicz space for the case of vector-valued functions and norm convergence in H θ (X , μ ¯) where X is any Banach space are discussed. Relationships of modular convergence and norm convergence of H-Orlicz spaces are discussed. [ABSTRACT FROM AUTHOR]

We improve known results on the existence and uniqueness of solutions for two types of scalar linear Stieltjes integral equations, which are adjoint to each other. The main results are explicit solution formulas for homogeneous as well as nonhomogeneous problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence and uniqueness of solutions of the following boundary value problem concerning generalized ODEs dx d τ = D [ A (t) x + F (t) ] , ∫ a b d [ K (s) ] x (s) = r , for operators taking values in general Banach spaces. Up to now, this problem was only tackled in the case of finite dimensional space-valued functions. We establish necessary and sufficient conditions not only for the existence of at least one solution, but also for the uniqueness of a solution. Another important result describes the solution in terms of a Green function and the fundamental operator of the corresponding homogeneous problem. We also explore the problem dx d τ = D [ A (t) x + F (t) ] and we give necessary and sufficient conditions for the existence of a (θ , T) -periodic solution, where θ ∈ R , with θ ≠ 0 , and T > 0 , recalling that the notion of (θ , T) -periodic solution generalizes the notions of periodic, anti-periodic, almost periodic, and quasi-periodic solution. Examples are given to illustrate the main results. In addition, we apply the main theorems to abstract ODEs, as well as to a Volterra-Stieltjes-type integral equation, and we include examples of these as well. [ABSTRACT FROM AUTHOR]

We fill up some gaps in the existing literature on the Henstock-Kurzweil integration on metric measure spaces. The most important one is the choice of suitable candidates for 'intervals' in metric spaces for which the conclusion of Cousin's lemma holds. We also provide some characterizations of compactness and completeness in terms of Cousin's lemma, along with some alternative proofs of a few related results. Then we improve the result regarding differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces, and a few consequences. We propose shorter proofs of measure theoretic characterizations of the HK-integral in terms of the variational measure $V_F$ as well as the $ACG^\Delta$ functions. Finally, we present alternative proofs of some generalizations of two results on Lebesgue integration. The first one bypasses Vitali covering lemma for a result on absolute continuity, while the second one is a version of the fundamental theorem of calculus in our setting. [ABSTRACT FROM AUTHOR]

Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation. [ABSTRACT FROM AUTHOR]

It is well-known that the generalized Riemann approach using non-uniform mesh has given rise to integrals which are more general than the Riemann integral. In this note, motivated by earlier studies on stochastic integrals, we consider special interval-point pair in defining the Riemann sum, where the point (or the tag) is the left-hand point of the interval. We show that this approach in fact is equivalent to the Lebesgue integral. Hence this simplifies McShane's construction of interval-point pair. Moreover, by restricting to the tag as the left hand point of the interval, the integration-by-substitution and by-parts formulae become easy consequences of the definition. [ABSTRACT FROM AUTHOR]

We define the L r -variational integral and we prove that it is equivalent to the H K r -integral defined in 2004 by P. Musial and Y. Sagher in the Studia Mathematica paper The L r -Henstock–Kurzweil integral. We prove also the continuity of L r -variation function. [ABSTRACT FROM AUTHOR]

The Denjoy integral is an integral that extends the Lebesgue integral and canintegrate any derivative. In this paper, it is shown that the graph of theindefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borelrelation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is thePolish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$is the Polish space of real-valued continuous functions on $[a,b]$. Using thesame methods, it is also shown that the class of indefinite Denjoy integrals,called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space$C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basicmodel theory of the associated spaces of integrable functions is also studied.Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with theindeterminate $X$ being interpreted as the indefinite integral, the space ofcontinuous functions on the interval $[a,b]$ is elementarily equivalent to theLebesgue-integrable and Denjoy-integrable functions on this interval, and eachis stable but not superstable, and that they all have a common decidable theorywhen viewed as $\mathbb{Q}[X]$-modules.

The space ℋ K of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Fréchet spaces ℋ K (X). In this paper, on each Fréchet space ℋ K (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the ℋ K (X) space. It is known that every control-convergent sequence in the ℋ K space always belongs to a ℋ K (X) space for some X. We illustrate how to apply results for Fréchet spaces ℋ K (X) to control-convergent sequences in the ℋ K space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved. [ABSTRACT FROM AUTHOR]

During the last years several fractional integrals were investigated. Having this idea in mind, in the present article, some new generalized fractional integral inequalities of the trapezoidal type for λφ–preinvex functions, which are differentiable and twice differentiable, are established. Then, by employing those results, we explore the new estimates on trapezoidal type inequalities for classical integral and Riemann–Liouville fractional integrals, respectively. Finally, we apply our new inequalities to construct inequalities involving moments of a continuous random variable.

We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil–Henstock-type integral related to this basis. We prove a version of Hake's theorem in terms of a variational measure. [ABSTRACT FROM AUTHOR]

In this paper, we find the completion of the space of Henstock (respectively, Kurzweil) integrable vector-valued functions. Also we establish some properties of the vector distributional Henstock–Kurzweil integral. Fundamental theorem of calculus, integration by parts formula, monotone and dominated convergence theorems are given. [ABSTRACT FROM AUTHOR]

We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by f = ∑ n = 1 ∞ x n χ E n where x n ∈ X and the sets E n are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of ∑ n = 1 ∞ x n m (E n) is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane (SM) - integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and M : SM → SM is a bounded linear operator, then there exists g ∈ L ∞ ([ 0 , 1 ] , X) such that M (f) = f g for all f ∈ SM . Some results on multipliers of McShane (M) - integrable functions are also derived. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger 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. [ABSTRACT FROM AUTHOR]

Based on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems). [ABSTRACT FROM AUTHOR]

For functions which are Henstock–Kurzweil integrable but not Lebesgue-integrable we prove a theorem which resembles the Riemann theorem on the rearrangement of conditionally convergent series. [ABSTRACT FROM AUTHOR]

We study the Cauchy representation formula for analytic functions on the unit disc whose pointwise boundary value function is distributionally integrable. We prove that the formula holds when the distributional boundary values exist, and give examples that show that it may not be true when that is not the case. We also prove a maximum principle for pointwise boundary values valid for functions with distributional boundary values. [ABSTRACT FROM AUTHOR]