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On the strong McShane integral of functions with values in a Banach space

Štefan Schwabik, Ye Guoju (2001)

Czechoslovak Mathematical Journal

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....

On the σ -finiteness of a variational measure

Diana Caponetti (2003)

Mathematica Bohemica

The σ -finiteness of a variational measure, generated by a real valued function, is proved whenever it is σ -finite on all Borel sets that are negligible with respect to a σ -finite variational measure generated by a continuous function.

On variations of functions of one real variable

Washek Frank Pfeffer (1997)

Commentationes Mathematicae Universitatis Carolinae

We discuss variations of functions that provide conceptually similar descriptive definitions of the Lebesgue and Denjoy-Perron integrals.

Operator-valued functions of bounded semivariation and convolutions

Štefan Schwabik (2001)

Mathematica Bohemica

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.

Orthogonally additive functionals on B V

Khaing Aye Khaing, Peng Yee Lee (2004)

Mathematica Bohemica

In this paper we give a representation theorem for the orthogonally additive functionals on the space B V in terms of a non-linear integral of the Henstock-Kurzweil-Stieltjes type.

Path-wise solutions of stochastic differential equations driven by Lévy processes.

David R. E. Williams (2001)

Revista Matemática Iberoamericana

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...

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