Let $I:=[0,1]$. In this paper we consider the implicit functional-integral equation \begin{equation*} \psi(t,u(t))=g(t)+f(t,\int_Ik(t,s)\,u(\varphi(s))\,ds)\quad\hbox{for a.e.}\quad t\in I, \end{equation*} where $\psi:I\times{\bf R}\to{\bf R}$, $g:I\to{\bf R}$, $k:I\times I\to[0,+\infty[$, $\varphi:I\to I$ and $f:I\times{\bf R}\to{\bf R}$ are given functions. We prove an existence result for solutions $u\in L^p(I)$ (with $p\in]1,+\infty]$), where the regularity assumptions on $f$ are considerably weaker than the usual Carath\'eodory condition required in the literature. In fact, a function $f$ satisfying the assumptions of our main result can be discontinuous, with respect to the second variable, even at all points $x\in{\bf R}$.

An existence result for functional-integral equations associated with discontinuous functions

Paolo Cubiotti;
2023-01-01

Abstract

Let $I:=[0,1]$. In this paper we consider the implicit functional-integral equation \begin{equation*} \psi(t,u(t))=g(t)+f(t,\int_Ik(t,s)\,u(\varphi(s))\,ds)\quad\hbox{for a.e.}\quad t\in I, \end{equation*} where $\psi:I\times{\bf R}\to{\bf R}$, $g:I\to{\bf R}$, $k:I\times I\to[0,+\infty[$, $\varphi:I\to I$ and $f:I\times{\bf R}\to{\bf R}$ are given functions. We prove an existence result for solutions $u\in L^p(I)$ (with $p\in]1,+\infty]$), where the regularity assumptions on $f$ are considerably weaker than the usual Carath\'eodory condition required in the literature. In fact, a function $f$ satisfying the assumptions of our main result can be discontinuous, with respect to the second variable, even at all points $x\in{\bf R}$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/3248133
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