Nonlinear boundary problem: counterexample

waytogo

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Jan 22, 2012
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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?
 
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