Let I ⊆ R be a compact interval. In this paper we prove an existence result for solutions u ∈ Lp (I, Rn), with p ∈]1, +∞], of the implicit functional-integral equation (formula presented) for almost every t ∈ I, I where f: I × S × Rn → R, φ: I → I and ξ: I × I → [0, +∞[ are given functions, and S ⊆ Rn is a suitable closed connected and locally connected set. The main peculiarity of our result is the regularity assumption on f with respect to the third variable, considerably weaker than the us...
Let $I=[0,1]$. We deal with the existence of solutions $u\in L^p(I)$ of the implicit functional-integral
equation
$$
\psi(u(t))=f\Big(t,\int_I k(t,s)\,u(\varphi(s))\,ds\Big)\quad\hbox{for a.e.}\quad
t\in I,
$$
where $Y\subseteq{\bf R}$ is a closed interval, $p\in]1,+\infty[\,$, and
$\psi:Y\to{\bf R}$,
$f:I\times {\bf R}\to{\bf R}$, $k:I\times I\to [0,+\infty[$ and
$\varphi:I\to I$ are given functions.
We prove an existence result where the function $f$ can be discontinuous, ...
In this paper we deal with the functional-integral equation
\[
u(t)=f\Big(t,\int_I k(t,s)\,g(s,u(\varphi(s)))\,ds\Big)\quad\hbox{for a.e.}\quad
t\in I,
\]
where $I:=[0,1]$, and $f:I\times {\bf R}_0^+\to{\bf R}$, $g:I\times{\bf R}\to{\bf R}_0^+$,
$k:I\times I\to {\bf R}_0^+$, and $\varphi:I\to I$ are given functions.
We prove an existence result for solutions $u\in L^p(I)$, where
the usual Carathéodory conditions on the function $f$ are meaningfully weakened. In particular, a funct...
Let $n\in{\bf N}$, with $n\ge2$, and let $\Omega\subseteq{\bf R}^n$ be a bounded domain with smooth boundary $\partial\Omega$.
Let $p\in ]1,+\infty[$, let $Y\subseteq{\bf R}^n$, and let $f:\Omega\times{\bf R}^h\to{\bf R}$ and $\psi:Y\to{\bf R}$ be two given functions.
We deal with the existence of solutions $u=(u_1,\ldots,u_h)\in W^{1,p}_0(\Omega,{\bf R}^h)$ of the implicit equation
$\psi(-\Delta_p u)=f(x,u)$, where $\Delta_p u=(\Delta_p u_1,\Delta_p u_2,
\ldots,\Delta_p u_h)$.
We pro...
Let $I$ be a real compact interval. We deal with the problem of the existence of solutions $u\in L^p(I)$ of the implicit functional-integral equation
\begin{equation*}
f\Big(t,u(t),\int_Ik(t,s)\,u(\varphi(s))\,ds\Big)=0\quad\hbox{for a.e.}\quad
t\in I,
\end{equation*}
where $Y$ is a closed interval, and
$f:I\times Y \times {\bf R}\to{\bf R}$, $k:I\times I\to
[0,+\infty[$ and $\varphi:I\to I$ are given functions. Such an equation includes, as
special cases, many integral equation...