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The existence and uniqueness of solutions to the Cauchy problem for a first-order quasi-linear partial differential equation is studied in this paper using the Henstock-Kurzweil integral. The classical theory requires certain differentiability and continuity conditions on the coefficients of the derivatives in the equation. It is shown here that in the Henstock-Kurzweil integral setting these conditions can be relaxed and that the resulting solution is differentiable though the derivatives need not be continuous. This sharpens the classical result and provides a bridge between classical and weak solutions in the linear case.

A valid proof that Kubota’s AD-integral is more general than Burkill’s AP integral is given.

There are two different definitions about the approximately continuous Henstock integral or \(AH\) integral, for short. In this paper we prove that they are equivalent, and that the \(AH\) integral is included in the \(AD\) integral defined by Kubota in 1964. Also we give a solution for the problem put forward by Gordon (1994, p. 255).

In this paper, the existence of solutions to a class of retarded functional differential equations is established along with a result concerning the continuous dependence of solutions on a parameter. These results generalize theorems concerning existence and continuous dependence in a Lebesgue integral setting to a Henstock-Kurzweil integral setting.

The author gives a general descriptive definition for integration, denoted by \({\mathcal P}\), which has as special cases the Lebesgue integral for bounded measurable functions, the Lebesgue integral, the Denjoy-Perron integral \({\mathcal D}^*\), the wide Denjoy integral \({\mathcal D}\), the Foran integral, the Iseki integral and the \(S{\mathcal F}\)-integral (\cite{Ene1}). This \({\mathcal P}\)-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying \(f\) with \(g\) whenever \(f = g\) \(a.e.\) on \([a,b]\)). The classical Riesz representation theorem for the linear and continuous functionals on \((C([a,b]),\|\cdot\|_\infty)\) is a consequence of Theorem 2. In addition it is shown that the space of \({\mathcal P}\)-integrable functions is of the first category in itself (see Section 5). Also a characterization of weak convergence on this space is given.

Theorem. If Henstock integrable \(f_n\) converge in measure to a finite \(f\) and their primitives \(F_n\) are equi-\(ACG_*\) and converge pointwise to a continuous \(F\) then \(\int f = \lim F_{n}\).

In this paper the author constructs a \(C^1\) function \(G\) and a Riemann integrable function \(H\), and shows that the composition \(H \circ G\) is not Riemann integrable.

The Banach-Zarecki Theorem states that \(VB \cap (N) = AC\) for continuous functions on a closed set. Hence it is a linear space. In this article we show that \(VB \cap (N)\) is a linear space on any real Borel set. Hence \(VBG \cap (N)\) will also be a real linear space for Borel measurable functions defined on an interval. As a consequence of this result, we show that the \(AK_N\) integral of Gordon (\cite{G14}) is well defined. We also give answers to Gordon’s questions in \cite{G14}.

We define a concept of generalized absolute continuity for additive functions of figures. This makes it possible to give a descriptive definition of the ℱ-integral introduced by Pfeffer in 1993. Finally, we discuss a possible extension to additive functions of sets of bounded variation.

In this paper we show that there are close relationships amongst Cantor bar totality, non-integer integral and Hausdorff integral. If a singular function \(f\) has zero Lipschitz (\(1-\nu\))-numbers on a Cantor set \(C\) with \(H\)-dim~\(C=1-\nu\), \(0\lt \nu\lt 1\), then the \(1-\nu\) order fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Moreover, under strong assumptions on the function \(f\) the \(1-\nu\) order derivative of \(f\) exists everywhere. Consequently, the \(\nu\) order fractional integral of \(f\) equals \(f(0)t^{\nu}\). Using the Concept of th Hausdorff derivative we prove that if a singular function has a Hausdorff derivative on \(C\), then the fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Finally, under some assumptions on \(f\) and \(C\), we establish \begin{align*} (D^{-\nu}f)(t)=&\int_0^t\int_0^uv^{\nu-1}f'_H(u-v)\,dv^{1-\nu}\,du\\ =&\int_0^tu^{1-\nu}J_{\nu}(v^{\nu-1}f'_H(u-v))(u)\,du \end{align*} This identity is similar to the identity \[(D^{-\nu}f)(t)=\frac{1}{B_{\nu}}t^{\nu}(J_{\nu}f)(t)\] which professor R. R. Nigmatullin claimed.

In this paper a Riemann scale of integrals is defined depending on the concept of sifting. Depending on the choice of sifting, the integral defined is equivalent to various classical integrals. The associated concepts of generalized absolute continuity are also defined and used to obtain descriptive definitions of the integrals introduced.

In this paper, we give a necessary and sufficient condition in terms of Lebesgue integrable functions for Henstock integrability in Euclidean space.

In this paper, we extend the \({\mathcal{B}}\)-integral on \(\mathbb{R}\) to one on \(\mathbb{R}^{m}\). After proving a weaker version of the equiintegrability theorem, the equivalence of five generalized convergence theorems will be established, which include the equiintegrability theorem, two versions of the generalized controlled convergence theorem which are based on \(UACG^\nabla\) and \(UACG_{\mathcal{B}}^{\ast\ast}\), respectively, the generalized variational convergence theorem and the uniformly \(MGSRS_{\mathcal{B}}\) (modified \(GSRS\) with respect to \({\mathcal{B}}\)) convergence theorem.

In a former paper ([Ju-No 1]) we introduced an axiomatic approach to the theory of non-absolutely convergent integrals in \(\mathbb{R}^n\). A specialization of this abstract concept leads to the well-behaved \(\nu (S)\)-integral over quite general sets \(A\) which yields the divergence theorem in its presently most general form. (See [Ju-No 3].) While the definition of the \(\nu (S)\)-integral is of descriptive type (i.e. in terms of an additive almost everywhere differentiable set function) we prove in this paper that it can equivalently be defined using Riemann sums. As an application we show that any function being variationally integrable over \(A\) in the sense of [Pf 3] is also \(\nu (S)\)-integrable over \(A\) and both integrals coincide.

A relationship between the \(s\)-dimensional Hausdorff integral and the evolution with losses is established. Professor R. R. Nigmatullin showed that the evolution with loss can be described by a non-integer integral. This paper gives another way to describe the evolution. That is, the evolution can be expressed as Hausdorff integral.

Recently Bartle [B], modifying a convergence theorem of Gordon, has established necessary and sufficient conditions for a Henstock integral convergence theorem. We point out that this result is not specific to the Henstock, or indeed to any definition of, an integral. Rather, it is a result about an arbitrary limit process and can thus be formulated in the setting of abstract convergence theory.

In \cite{L14} (p. 224), P. Y. Lee and T. S. Chew use Corollary 1 of our paper essentially without proof, and without stating it explicitly, claiming that “it is easy to verify”. The same result is also used by P. Y. Lee in \cite{L13} (Theorem 10.2, p. 59). The aim of this article is to prove Corollary 1.

An iterated limits theorem is used to establish two sets of necessary and sufficient conditions for a Henstock integral convergence theorem. The motivation for the iterated limits theorem originates from Bartle’s notion of \(\gamma\)-convergence.

It is proved that the absolute continuity of the variational measure generated by an additive interval function \(F\) implies the differentiability almost everywhere of the function \(F\) and gives a full descriptive characterization of the Denjoy-Perron integral.

A theorem of Marcinkiewicz asserts that the Perron integrability of a function can be deduced from the existence of a single pair of continuous major and minor functions. We show that Perron-type integrals based on various symmetric derivatives do not have this property.

This paper gives a new proof of the equivalence of the McShane and Lebesgue integrals, which is independent of measure theory.

Let $f$ be a distribution (generalised function) on the real line. If there is a continuous function $F$ with real limits at infinity such that $F'=f$ (distributional derivative) then the distributional integral of $f$ is defined as $\int_{-\infty}^\infty f = F(\infty) - F(-\infty)$. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock--Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, H\"older inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed., Comment: To appear in Real Analysis Exchange

In the present paper, some Olech and Visintin-type results are obtained in Henstock-Kurzweil-Pettis integration. More precisely, under extreme or denting point condition, one can pass from weak convergence (i.e. convergence with respect to the topology induced by the tensor product of the space of real functions of bounded variation and the topological dual of the initial Banach space) or from the convergence of integrals to strong convergence (i.e. in the topology of Alexiewicz norm or, even more, of Pettis norm). Our results extend the results already known in the Bochner and Pettis integrability setting.

We define two interval functions $U_\delta$ and $V_\delta$ using Riemann sums of Henstock integrable functions, as major and minor functions. Then we formulate two dominated convergence theorems for the Henstock integral in the $n$-dimensional space.

It is proved that for a wide class of bases in ${\mathbb R}^m$ a function generates a $\sigma$-finite absolutely continuous variational measure if and only if this function belongs to $ACG_\delta$-class. It is also shown that under some additional assumptions on a basis, $\sigma$-finiteness follows from the absolute continuity of the variational measure.

WWe prove the following extension of the Mean Value Theorem. Let E be a Banach space and let $F:[a,b]\rightarrow E$and $\varphi :[a,b]\rightarrow\mathbb{R}$ be two functions for which there exists a subset $A\subset\lbrack a,b]$ such that: ¶ i) $F$ and $\varphi$ have negligible variation on $A$, ¶ ii) $F$ and $\varphi$ are differentiable on $[a,b]\setminus A$ and $\left\Vert F^{\prime}\right\Vert \leq\varphi^{\prime}$ on $[a,b]\setminus A$. ¶ Then $\left\Vert F(b)-F(a)\right\Vert \leq\varphi(b)-\varphi(a).$ ¶ Several applications are included.

Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in $(a,b)$, $g_n\to g$ in measure or in the $L^1$ norm. The same conditions are necessary and sufficient for $\|f(g_n-g)\|\to 0$ for all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e. then convergence $\|fg_n\|\to\|fg\|$ for all Henstock--Kurzweil integrable functions $f$ is equivalent to $\|f(g_n-g)\|\to 0$. This extends a theorem due to Lee Peng-Yee., Comment: See http://www.math.ualberta.ca/~etalvila/research.html. Real. Anal. Exchange (to appear)

The McShane integral of Banach space-valued functions $f:I\to X$ defined on an $m$-dimensional interval $I$ is considered in this paper. We show that a McShane integrable function is integrable over measurable sets contained in $I$ (Theorem 9). A certain type of absolute continuity of the indefinite McShane integral with respect to Lebesgue measure is derived (Theorem 11) and we show that the indefinite McShane integral is countably additive (Theorem 16). Allowing more general partitions using measurable sets instead of intervals another McShane type integral is defined and we show that it is equivalent to the original McShane integral (Theorem 21)

The purpose of this paper is to give a simple and completely elementary proof that the derivative of the indefinite Riemann-complete integral equals the integrand almost everywhere. Elementary in this case means using the least amount of measure theory possible.

In this paper, give a new, full characterization of the primitive of a Kurzweil-Henstock integrable function in multidimensional space.

In this paper, we give an example to show that a theorem of Nakanishi for the Henstock integral does not hold for the general Denjoy integral.

The present paper concerns with the introduction of a new type of generalized Stieltjes integral with an integrator function which depends on multiple points in a division and cannot be induced by a measure. Some properties of this integral were studied.

The classical Black-Scholes-Merton method for pricing European call options uses the It\^{o} calculus to model the processes involved. We show how to model stochastic process using Henstock integrands instead of It\^{o} differentials (or stochastic integrals), and we show how to derive the Black-Scholes partial differential equation and pricing formulae using elementary methods.

In the linear space$\mathcal{DP}[a,b]$ of all Denjoy-Perron integrable functions on an interval $[a,b]$ one wishes to introduce the most natural topology. Herein are some considerations that suggest what topology might be the most natural.

An integral is defined using the Moore-Smith limit and this new integral is compared to the Henstock integral.

The exact relations between the approximately continuous Perron and Denjoy integrals of Burkill [1] and Kubota [5, 6] are re-established by rectifying the faulty proofs of Kubota, and the related questions of Gordon [3] are resolved completely.

We prove that the \(n\)-dimensional Perron integral with respect to the full interval basis, without any regularity condition, defined by continuous major and minor functions is equivalent to the one defined by major and minor functions which are not supposed to be continuous.

We prove that if \(T\) is a continuous linear functional on the space \({\mathcal D}\) of Henstock-Kurzweil integrable functions on \([a_1,b_1] \times \cdots \times [a_m,b_m]\), then there exists a function \(g\) of strong bounded variation on \([a_1,b_1] \times \cdots \times [a_m,b_m]\) such that \[ T(f) = (HK) {\int \ldots \int \atop \scriptstyle{[a_1,b_1] \times \cdots \times [a_m,b_m]}} f(x_1,\ldots,x_m) g(x_1,\ldots,x_m)\, dx_1 \ldots dx_m\, . \]

Our Theorem 4.1 provides a new and direct proof of the measurability of Perron integrable functions, and hence of Kurzweil-Henstock integrable functions since both integrals coincide (cf. Theorem 11.5 and Theorem 11.6 in \cite{Go}, or the original paper of Kurzweil \cite{K}\h).

The change of variables formula for the Riemann integral is discussed and a theorem is proved which perhaps compares favorably with its counterpart in Lebesgue theory.

The paper deals with the space of all Denjoy-Perron integrable functions on a fixed interval endowed with the Alexiewicz norm and the completion of this space. The relatively weakly compact subsets of each space are characterized.

We define a generalized Riemann-Stieltjes integral with respect to charges (i.e., additive continuous functions on bounded \(BV\) sets), and study the linear space of all functions that are integrable with respect to every charge.

It is proved that if a function is continuous on an interval, then its variations on a set taken relative to some covering relations are also continuous as interval-functions. This result is applied to a direct construction of continuous Perron major and minor functions of Henstock integrable function. The Henstock Integral with respect to a derivation base is known to be equivalent to the Perron Integral defined with respect to the same base. (See [9] for an abstract setting.) But the Perron Integral is understood in this theory to be defined using major and minor functions which are not necessarily continuous with respect to the base. The question of whether we get an equivalent definition by restricting the major and minor functions to be continuous is still an open one in a general case. (See [1, p. 202] and [2].) This question is closely related to the continuity property of a variation which arises in the theory of Henstock integration. The classical Perron Integral on an interval in the real line R is defined by means of continuous major and minor functions. The well-known proof (see [11]) of the fact that this definition is equivalent to the one without the continuity condition is based on the descriptive theory of the Denjoy Integral and we are not aware of any satisfactory proof of this equivalence having been presented which uses just elementary properties of the Henstock integral. (We shall indicate below a gap in the proof of related theorems in [4] and [8]). A

Let \(\{\epsilon_n\}\) be a sequence of positive numbers converging to \(0\). For each \(n\) divide the unit interval \([0, 1]\) into sub-intervals of the type \([k \epsilon_n, (k+1) \epsilon_n)\), \(k = 0, 1, 2, \dots\) and define a function \(f^n\) as \(1, -1, 1, -1, \dots\) successively on these intervals. If \(\{f^{n_k}\}\) is any subsequence of \(\{f^n\}\) then the set of points at which \(\{f^{n_k}\}\) converges has Lebesgue measure zero. This is a generalization of the well known analogous result for Rademacher functions.