Solvability of a multi-point boundary value problem at resonance (Q1904438)

The Leray-Schauder continuation method is used to find sufficient conditions which imply the existence of at least one solution for multi-point boundary value problems of the form \[ x'' (t) = f \bigl( t,x(t), x'(t) \bigr) + e(t),\;x(0) = 0,\;x(1) = {1 \over k} \sum^k_{i = 1} a_i x(\eta_i). \] The assumptions involve growth and sign conditions upon \(f\).