It is shown that if is of bounded variation in the sense of Hardy-Krause on , then is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.
The paper is concerned with integrability of the Fourier sine transform function when , where is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of to be integrable in the Henstock-Kurzweil sense, it is necessary that . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.
The least concave majorant, , of a continuous function on a closed interval, , is defined by We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on . Given any function , it can be well-approximated on by a clamped cubic spline . We show that is then a good approximation to . We give two examples, one to illustrate, the other to apply our algorithm.
This paper considers a Volterra's population system of fractional order and describes a bi-parametric homotopy analysis method for solving this system. The homotopy method offers a possibility to increase the convergence region of the series solution. Two examples are presented to illustrate the convergence and accuracy of the method to the solution. Further, we define the averaged residual error to show that the obtained results have reasonable accuracy.
We consider the wave equation damped with a boundary nonlinear velocity feedback p(u'). Under some geometrical conditions, we prove that the energy of the system decays to zero with an explicit decay rate estimate even if the function ρ has not a polynomial behavior in zero. This work extends some results of Nakao, Haraux, Zuazua and Komornik, who studied the case where the feedback has a polynomial behavior in zero and completes a result of Lasiecka and Tataru. The proof is based on the construction...
It is shown that there exist a continuous function and a regulated function defined on the interval such that vanishes everywhere except for a countable set, and the -integral of with respect to does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.
We establish a Banach-Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of L¹(μ) × L¹(μ) and c₀ × c₀. As another consequence, we get a Banach-Mazurkiewicz type theorem on some residual subset of C[0,1] involving Kharazishvili's notion of Φ-derivative.