Geodesic
The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 279-289

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In this paper we prove an existence theorem for the Cauchy problem \[ x^{\prime }(t) = f(t, x(t)), \quad x(0) = x_0, \quad t \in I_{\alpha } = [0, \alpha ] \] 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.
Classification : 28B05, 34G20
Keywords: pseudo-solution; Pettis integral; Henstock-Kurzweil integral; Cauchy problem 
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     title = {The {Henstock-Kurzweil-Pettis} integrals and existence theorems for the {Cauchy} problem},
     journal = {Czechoslovak Mathematical Journal},
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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, Tome 54 (2004) no. 2, pp. 279-289. http://geodesic.mathdoc.fr/item/CMJ_2004__54_2_a1/