On a necessary and sufficient condition for the negativeness of the Green’s function of a two-point boundary value problem for a functional differential equation

Conditions of negativity for the Green's function of a two-point boundary value problem
\[
    \Lc_\lambda u := u^{(n)}-\lambda\int_0^l u(s) d_s r(x,s)=f(x), \ \ \ x\in[0,l], \ \ \ B^k(u)=0,
\]
where $B^k(u)=(u(0),\ldots,u^{(n-k-1)}(0),u(l),-u'(l),\ldots,(-1)^{(k-1)}u^{(k-1)}(0)),$
$n\ge3,$ $0\!<\!k\!<\!n,$ $k$ is odd,  are considered. The function $r(x,s)$ is assumed to be non-decreasing in the second argument.
A necessary and sufficient condition for the nonnegativity of the solution of this boundary value problem on the set $E$ of functions satisfying the conditions
\[
    u(0)=\cdots=u^{(n-k-2)}(0)=0, \ \ \ u(l)=\cdots=u^{(k-2)}(l)=0,
\]
$u^{(n-k-1)}(0)\ge0,$ $u^{(k-1)}(l)\ge0,$ $f(x)\le 0$ is obtained.
This condition lies in the subcriticality of boundary value problems with vector functionals $B^{k-1}$ and $B^{k+1}.$ Let $k$ be even and $\lambda^k$ be the smallest positive value of $\lambda$ for which the problem $\Lc_\lambda u = 0,$ $B^ku = 0$ has a nontrivial solution.
Then the pair of conditions $\lambda <\lambda^{k-1}$ and $\lambda <\lambda^{k+1}$ is necessary and sufficient for positivity of the solution of the problem.