A note on a third-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third-order multi-point boundary value problem at resonance where f: [0, 1] × R3 R is a continuous function, 0 < ξ1 < ⋅⋅⋅ < ξm < 1, αi ∈ R, i = 1, …, m, m ≥ 1 and 0 < η1 < η2 < ⋅⋅⋅ < ηn < 1, βj ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′′′) is equal to 2. Since all the existence results for third-order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.