On the product of derivatives
In this paper we explain the relationship between Stieltjes type integrals of Young, Dushnik and Kurzweil for functions with values in Banach spaces. To this aim also several new convergence theorems will be stated and proved.
The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions....
In this paper we prove that each differentiation basis associated with a -adic path system defined by a bounded sequence satisfies the Ward Theorem.
The -finiteness of a variational measure, generated by a real valued function, is proved whenever it is -finite on all Borel sets that are negligible with respect to a -finite variational measure generated by a continuous function.
In this paper a full totalization is presented of the Kurzweil-Henstock integral in the multidimensional space. A residual function of the total Kurzweil-Henstock primitive is defined.
We discuss variations of functions that provide conceptually similar descriptive definitions of the Lebesgue and Denjoy-Perron integrals.
The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.
In this paper we give a representation theorem for the orthogonally additive functionals on the space in terms of a non-linear integral of the Henstock-Kurzweil-Stieltjes type.
In this paper we show that a path-wise solution to the following integral equationYt = ∫0t f(Yt) dXt, Y0 = a ∈ Rd,exists under the assumption that Xt is a Lévy process of finite p-variation for some p ≥ 1 and that f is an α-Lipschitz function for some α > p. We examine two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric, the other forward. The geometric solution is obtained by adding fictitious time and solving an associated...