Representation of bounded and compact linear operators in the Banach space of regulated functions is given in terms of Perron-Stieltjes integral.

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of...

In 1990, Hönig proved that the linear Volterra integral equation $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)ds=f\left(t\right),t\in \left[a,b\right],$$ where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[a,b\right]$ of the real line $ℝ$, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)dg\left(s\right)=f\left(t\right),t\in \left[a,b\right],$$ in a real-valued context.

Riemann sums based on $\delta$-fine partitions are illustrated with a Maple procedure.

The McShane integral of functions $fI\to ℝ$ defined on an $m$-dimensional interval $I$ is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.

We use an elementary method to prove that each $BV$ function is a multiplier for the $C$-integral.

Assume that for any $t$ from an interval $[a,b]$ a real number $u\left(t\right)$ is given. Summarizing all these numbers $u\left(t\right)$ is no problem in case of an absolutely convergent series ${\sum }_{t\in [a,b]}u\left(t\right)$. The paper gives a rule how to summarize a series of this type which is not absolutely convergent, using a theory of generalized Perron (or Kurzweil) integral.

We study a scale of integrals on the real line motivated by the $M{C}_{\alpha }$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

We investigate some properties of the normed space of almost periodic functions which are defined via the Denjoy-Perron (or equivalently, Henstock-Kurzweil) integral. In particular, we prove that this space is barrelled while it is not complete. We also prove that a linear differential equation with the non-homogenous term being an almost periodic function of such type, possesses a solution in the class under consideration.

The Henstock-Kurzweil approach, also known as the generalized Riemann approach, has been successful in giving an alternative definition to the classical Itô integral. The Riemann approach is well-known for its directness in defining integrals. In this note we will prove the Fundamental Theorem for the Henstock-Kurzweil-Itô integral, thereby providing a characterization of Henstock-Kurzweil-Itô integrable stochastic processes in terms of their primitive processes.

In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.

The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function $f:[a,b]\to c_0$ which is not Pettis integrable on any subinterval in [a,b], while ${ʃ}_{J}f$ belongs to $c_0$ for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord...