Geodesic
Non-autonomous vector integral equations with discontinuous right-hand side
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 319-329

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We deal with the integral equation $u(t)=f(t,\int_I g(t,z)u(z)\,dz)$, with $t\in I:=[0,1]$, $f:I\times \Bbb R^n \to \Bbb R^n$ and $g:I\times I\to[0,+\infty[$. We prove an existence theorem for solutions $u\in L^s(I,\Bbb R^n)$, $s\in \,]1,+\infty]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.
Classification : 45G10, 45P05, 47H15, 47J05, 47N20
Keywords: vector integral equations; discontinuity; multifunctions; operator inclusions 
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     author = {Cubiotti, Paolo},
     title = {Non-autonomous vector integral equations with discontinuous right-hand side},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {319--329},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2001},
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     zbl = {1055.45004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2001__42_2_a8/}
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Cubiotti, Paolo. Non-autonomous vector integral equations with discontinuous right-hand side. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 319-329. http://geodesic.mathdoc.fr/item/CMUC_2001__42_2_a8/