On the solvability of some multi-point boundary value problems (Q1915629)

The paper deals with the problem of solvability of the multi-point boundary value problem (E), (BC) or (E), (BC)\('\), where \[ x''(t)= f(t,x(t),x'(t))+e(t),\quad t\in (0,1),\tag{E} \] \[ x(0)=\sum^{m-2}_{i=1} c_ix'(\xi_i),\quad x(1)=\sum^{n-2}_{j=1} a_jx(\tau_j),\tag{BC} \] \[ x(0)=\sum^{m-2}_{i=1} c_ix'(\xi_i),\quad x'(1)=\sum^{n-2}_{j=1} a_jx'(\tau_j),\leqno(\text{BC})' \] and the \(c_i\) (resp. \(a_j\)) have the same sign. Using the Leray-Schauder continuation theorem, the authors give conditions on the function \(f\) under which there exists a solution of (E), (BC) or (E), (BC)\('\).