On the existence of \(m\)-point boundary value problem at resonance for higher order differential equation. (Q1417985)

The paper considers an \(m\)-point boundary value problem for a higher-order differential equation of the form \[ x^{(k)}(t)=f(t,x(t),x'(t),\dots,x^{(k-1)}(t))+e(t), \quad t\in (0,1), \] \[ x'(0)=0,\;x''(0)=0, \dots, x^{(k-1)}(0)=0, \quad x(1)=\sum_{i=1}^{m-2} a_ix(\xi_i), \] where \(f:[0,1]\times \mathbb{R}^k \to \mathbb{R}\) and \(e:[0,1]\to \mathbb{R}\) are continuous functions. Further, \(m\geq 3\), \(k\geq 2\) are two integers, \(a_i\in \mathbb{R}\), \(\xi_i \in (0,1)\), \(i=1,2,\dots, m-2\), \(\xi_1 <\xi_2<\dots <\xi_{m-2}\). The authors study the problem at resonance because they assume that \[ \sum_{i=1}^{m-2}a_i=1. \] Moreover, they do not need that all \(a_i\)'s, \(1\leq i\leq m-2\), have the same sign. The authors prove a new existence result under certain sign and growth conditions imposed on \(f\). The growth of \(f\) in some variables can be superlinear. The proofs are based on Mawhin's continuation theorem. Some examples illustrate the obtained result.