Displaying similar documents to “Equations containing locally Henstock-Kurzweil integrable functions”

The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem

Mieczysław Cichoń, Ireneusz Kubiaczyk, Sikorska-Nowak, Aneta Sikorska-Nowak, Aneta (2004)

Czechoslovak Mathematical Journal

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In this paper we prove an existence theorem for the Cauchy problem x ' ( t ) = f ( t , x ( t ) ) , x ( 0 ) = x 0 , t ∈ I α = [ 0 , α ] using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of measures of weak noncompactness.

Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral

Márcia Federson, Ricardo Bianconi (2001)

Archivum Mathematicum

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In 1990, Hönig proved that the linear Volterra integral equation x t - ( K ) ∫ a , t α t , s x s d s = f t , t ∈ a , b , where the functions are Banach space-valued and f is a Kurzweil integrable function defined on a compact interval a , b 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 t - ( K ) ∫ a , t α t , s x s d g s = f t , t ∈ a , b , in a real-valued context.