Sergey Labovskiy
abstract:
The question of the positivity of the Green function $G(x,s,\lambda)$ of the
two-point boundary value problem
$$(-1)^k u^{(n)}-\lambda \int\limits_0^l u(s)\,d_sr(x,s)=f(x), \;\; x\in[0,l],
\;\; B_k u=0,$$
where
$$B_mu:=\big(u(0),u'(0),\dots,u^{(n-m-1)}(0),u(l),-u'(l),u''(l),\dots,(-1)^{m-1}u^{(m-1)}(l)\big),$$
with non-decreasing $r(x,\,\cdot\,)$ for almost all $x\in[0,l]$ is reduced to
estimating the eigenvalues of auxiliary boundary value problems. The Green
function $G(x,s,\lambda)$ is positive if and only if
$$ -\min\{\lambda_{k-1},\lambda_{k+1}\}\le \lambda < \lambda_k $$
(there are also small clarifying details that are not needed for the ordinary
differential equation $(-1)^{k+1}u^{(n)}-\lambda p(x) u = f(x))$. Here, $\lambda_m$
is the smallest positive eigenvalue of the boundary value problem
$$ (-1)^{m}u^{(n)} - \lambda \int\limits_0^l u(s)\,d_sr(x,s) =0, \;\; B_m u=0 $$
$(m\in\{0,\dots,n\})$.
Mathematics Subject Classification: 34B05, 34B27, 34K10, 34K12
Key words and phrases: Boundary value problem, Green's function, positive solutions