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
$$ \mathcal{L}_\lambda u := u^{(n)}-\lambda\int_0^l u(s) d_s r(x,s)=f(x), \ \ \ x\in[0,l], \ \ \ B^k(u)=\alpha, $$
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 $\mathcal{L}_\lambda u=0,$ $B^ku=0$ 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.