Two point boundary value problems for the equation \(({\Psi{}}(T,X'))'=F(T,X,X')\) (Q1191768)

The author proves some existence theorems for the boundary value problem \(x'=\varphi(t,y)\), \(y'=g(t,x,y)\), \(x\in E\), where \(g:[0,1]\times\mathbb{R}^ 2\to\mathbb{R}\) and \(\varphi(t,y)y\geq 0\), \(g\) satisfies a Bernstein-Nagumo type condition, and \(E\) is a closed linear subspace of \(C^ 1([0,1];\mathbb{R})\) of codimension two, such that for each \(u\in E\) there exists \(r_ 0=r_ 0(u)\in[0,1]\) such that \(| u(t)|\leq| u(t_ 0)|\) and \(u'(t_ 0)=0\). The results are then applied to boundary value problems for the equation \((\psi(t,x'))'=f(t,x,x')\), \(x\in E\).