An existence theorem for a class of BVP without restrictions of the Bernstein-Nagumo type (Q2367762)

If \(k\geq 2\), \(p\) and \(q\) are continuous, \(q\) has two zeros of opposite sign and \(E\) is a closed linear subspace of \(C^ k([0,1],\mathbb{R})\) of codimension \(k\) such that the closure of \(E\) in \(C^{k-1}([0,1],\mathbb{R})\) does not contain polynomials of degree less than \(k-1\), then there is a solution of the equation \(x^{(k)}=q(x^{(k-1)}) p(t,x,x',\dots,x^{(k-1)})\), \(0\leq x\leq 1\), which belongs to \(E\).