Solvability of multi-point boundary value problems for \(2n\)th order ordinary differential equations at resonance. II (Q2488024)

The authors investigate the existence of solutions and positive solutions of the multipoint boundary value problem for 2nth-order differential equations \[ (-1)^{n-1}x^{(2n)}=f(t,x(t),x'(t),\cdots,x^{(2n-1)}(t)), \;t\in(0,1),\tag{1} \] subject to one of the following boundary conditions \[ x^{(2i-1)}(0)=0, \;i=1,\cdots,n, \] \[ x^{(2i-1)}(1)=0, \;i=1,\cdots,n-1,\tag{2} \] \[ x(1)=\sum_{i=1}^{m}\beta_ix(\xi_i), \] and \[ x^{(2i-1)}(0)=0, \;i=1,\cdots,n, \] \[ x^{(2i)}(1)=0, \;i=1,\cdots,n-1,\tag{3} \] \[ x(0)=\sum_{i=1}^{m}\beta_ix(\xi_i), \] where \(f:[0,1]\times \mathbb{R}^{2n}\rightarrow \mathbb{R}\) is a continuous function, \(n\geq 1\) is an integer, \(0<\xi_1<\cdots <\xi_m<1\) and \(\beta_i\in \mathbb{R}\) for \(i=1,\dots,m.\) By applying the coincidence degree theory and the Schauder fixed point theorem, some sufficient conditions for the existence of solutions of the boundary value problem (1) and (2) and of the boundary value problem (1) and (3) at resonance and at nonresonance are obtained.