Optimal existence theorems for positive solutions of second order multi-point boundary value problems (Q2912435)

The paper concerns the second-order ordinary differential equation NEWLINE\[NEWLINEy''(t)+a(t)f(y),\quad 0\leq t\leq 1NEWLINE\]NEWLINE subject to either Dirichlet or Neumann multi-point boundary conditions which may be nonhomogeneous, for instance, \(y(0)=\sum_{i=1}^{m}\beta_iy(\xi_i)+b\) with some non-negative real constant \(b\). The coefficient \(a\) is continuous and the nonlinearity \(f\in C([0,1]\times\mathbb R_+,\mathbb R_+)\). Criteria for the existence of a positive solution are given using a comparison of the behavior of the ratio \(f(t,x)/x\) at \(0\) and positive infinity with the smallest eigenvalue of an associated linear problem, the eigenvalues being given by the Krein-Rutman theorem. The proof uses the integral formulation of the problem and relies on the Guo-Lakshmikantham computation of the fixed point index on cones in Banach spaces. The obtained results improve recent ones. In this respect, the paper ends with a very nice discussion of recent results and open questions.