Paolo Cubiotti
Non-autonomous vector integral equations with discontinuous right-hand side

Comment.Math.Univ.Carolinae 42,2 (2001) 319-329.

Abstract: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 $t\in I$.

Keywords: vector integral equations, discontinuity, multifunctions, operator inclusions
AMS Subject Classification: 45P05, 47H15