We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, 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 the measure of weak noncompactness.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1087,
author = {A. Sikorska-Nowak},
title = {Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {27},
year = {2007},
pages = {315-327},
zbl = {1149.34053},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1087}
}
A. Sikorska-Nowak. Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 27 (2007) pp. 315-327. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1087/
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