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The M-components of level sets of continuous functions in WBV.

Coloma Ballester, Vicent Caselles (2001)

Publicacions Matemàtiques

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and...

The McShane, PU and Henstock integrals of Banach valued functions

Luisa Di Piazza, Valeria Marraffa (2002)

Czechoslovak Mathematical Journal

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...

The Measurability of Complex-Valued Functional Sequences

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions....

The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses

Saïd Abbas, Mouffak Benchohra (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we use the upper and lower solutions method to investigate the existence of solutions of a class of impulsive partial hyperbolic differential inclusions at fixed moments of impulse involving the Caputo fractional derivative. These results are obtained upon suitable fixed point theorems.

The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals

Márcia Federson (2002)

Czechoslovak Mathematical Journal

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, ∫ R d α ( t ) f ( t ) , where R is a compact interval of ℝ n , α and f are functions with values on L ( Z , W ) and Z respectively, and Z and W are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, ∫ R α ( t ) d f ( t ) , as well as to unbounded intervals R .

The non-coincidence of ordinary and Peano derivatives

Zoltán Buczolich, Clifford E. Weil (1999)

Mathematica Bohemica

Let f H ⊂ ℝ → ℝ be k times differentiable in both the usual (iterative) and Peano senses. We investigate when the usual derivatives and the corresponding Peano derivatives are different and the nature of the set where they are different.

The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ( V x ) x ∈ X of X such that d(f(x),f(y)) < ε whenever ( x , y ) ∈ V y × V x . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

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