Existence of solutions for second-order \(m\)-point boundary value problems at resonance (Q2780906)

The authors study the existence of solutions to the nonlinear second-order \(m\)-point boundary value problem NEWLINE\[NEWLINE(\rho(t) x')'=f(t, x(t), x'(t)),\quad 0< t< 1,\quad x(0)= 0,\quad x(1)= \sum^{m-2}_{i=1} \alpha_i x(\eta_i),NEWLINE\]NEWLINE where \(\rho:[0,1]\to (0,\infty)\) is a continuous and differentiable function, \(f: [0,1]\times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous, \(\eta_i\), \(i= 1,2,\dots, m-2\), are given constants satisfying \(0< \eta_1< \eta_2<\cdots< \eta_{m-2}< 1\), and \(\alpha_i\), \(i= 1,2,\dots, m-2\), are nonnegative constants with \(\sum^{m-2}_{i=1} \alpha_i= 1\). The proof is based upon the coincidence degree continuation theorem and the concept of autonomous curvature bound set related to the \(m\)-point boundary value problem.