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The Henstock-Kurzweil-Pettis Integrals and Existence Theorems for the Cauchy Problem

Authors:

M Cichoń I Kubiaczyk A Sikorska

Institution:

(1) Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Pozna nacute

, Poland

Abstract:

In this paper we prove an existence theorem for the Cauchy problem

$x'\left( t \right) = f\left( {t,x\left( t \right)} \right),x\left( 0 \right) = x_0 ,t \in I_\alpha = \left {0,\alpha } \right]$

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.

Keywords:

pseudo-solution Pettis integral Henstock-Kurzweil integral Cauchy problem

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