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Keywords:
vector integral equations; discontinuity; multifunctions; operator inclusions
vector integral equations; discontinuity; multifunctions; operator inclusions
Summary:
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 $t\in I$.
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