Nonlinear boundary problem: counterexample

[waytogo]

Maybe someone can come up with any idea about this.

Let us assume nonlinear two-point boundary problem
$\displaystyle x''=f(t,x,x')$
with boundary conditions written as
$\displaystyle L_1(x(a),x(b),x'(a),x'(b))=0,$
$\displaystyle L_2(x(a),x(b),x'(a),x'(b))=0,$
where $\displaystyle f\in C([a,b]\times R^2,R)$ and $\displaystyle L_1,L_2\in C(R^4,R)$.
Further assume that there exists upper and lower boundary functions $\displaystyle \alpha$ and $\displaystyle \beta$,such that
$\displaystyle \alpha(t)\leq\beta(t),t\in[a,b]$ and $\displaystyle L_1(\alpha(a),\alpha(b),\alpha'(a),\alpha'(b))\geq 0 \geq L_2(\beta(a),\beta(b),\beta'(a),\beta'(b))$.
There is a theorem that states:
if $\displaystyle L_1$ is non-decreasing in second and third term, and $\displaystyle L_2$ is non-decreasing in first, but non-increasing in fourth argument, then
there is a solution to this problem such that
$\displaystyle \alpha(t)\leq x(t)\leq \beta(t)$.

Can anyone help with constructing an example where theorem assumptions are not met and showing that there is no soluton for given problem in such case?