Implicit Vector Integral Equations Associated with Discontinuous Operators

Abstract

Let $I:=[\mathrm{0,1}]$. We consider the vector integral equation $h(u(t))=f\left(t,{\int }_{I}g(t,z),u(z),dz\right)$ for a.e. $t\in I,$ where $f:I\times J\to \mathbf{R},\hspace{0.17em}g:I\times I\to [0,+\infty [,$ and $h:X\to \mathbf{R}$ are given functions and $X,J$ are suitable subsets of ${\mathbf{R}}^n$. We prove an existence result for solutions $u\in L^s(I, {\mathbf{R}}^n)$, where the continuity of $f$ with respect to the second variable is not assumed. More precisely, $f$ is assumed to be a.e. equal (with respect to second variable) to a function $f^{*}:I\times J\to \mathbf{R}$ which is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a function $f$ can be discontinuous at each point $x\in J$. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case $n=1$.