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

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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.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614, Poznań, Poland

    M. Cichoń, I. Kubiaczyk & A. Sikorska

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  1. M. Cichoń
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  2. I. Kubiaczyk
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  3. A. Sikorska
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Cichoń, M., Kubiaczyk, I. & Sikorska, A. The Henstock-Kurzweil-Pettis Integrals and Existence Theorems for the Cauchy Problem. Czechoslovak Mathematical Journal 54, 279–289 (2004). https://doi.org/10.1023/B:CMAJ.0000042368.51882.ab

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