Solvability of a nonlinear Sturm–Liouville boundary-value problem for a second-order integrodifferential equation with one-sided restrictions on the growth of the right side with respect to the first derivative

The following problem is considered: find $u(t)\in C^{(2)}([0,1])$ such that
\begin{equation} u''=F\biggl(t,u,u',\int_0^1K(t,s,u(s))ds\biggr),\quad 0<t<1, \tag{1} \end{equation}

\begin{equation} \begin{gathered} au(0)-bu'(0)=g\varphi\biggl(u(0),u(1),\int_0^1l(s,u(s))\,ds\biggr), \\ cu(1)+du'(1)=h\Psi\biggl(u(0),u(1),\int_0^1m(s,u,(s))\,ds\biggr). \end{gathered} \tag{2} \end{equation}
Both those cases in which there exist both an upper and lower function of problem (1), (2) as well as those cases in which there exist only an upper function, only a lower function, or neither an upper or lower function are considered. The existence of a solution is established under conditions of the type
$$ F(t,u,p,w)\operatorname{sign}u\geqslant-k(u)\omega(|p|)\text{\rm{ for }}A(t)\leqslant u\leqslant B(t), \quad -\infty<p<+\infty, $$
or (for $b>0$, $d>0$)
$$ F(t,u,p,w)\geqslant-k(u)\omega(|p|)\text{\rm{ or }}F(t,u,p,w)\leqslant-k(u)\omega(|p|), $$
or (for $d>0$)
$$ F(t,u,p,w)\operatorname{sign}p\geqslant-k(u)\omega(|p|), $$
or (for $b>0$)
$$ F(t,u,p,w)\operatorname{sign}p\leqslant-k(u)\omega(|p|). $$