Consider the boundary value problem \[ x'(t)= f(t, x(t), x(\alpha(t))\quad\text{for }[0,T]\setminus \{t_1,t_2,\dots, t_m\}, \] \[ \Delta x(t_k)= I_k(x(t_k))\quad\text{for }k= 1,\dots, m,\tag{\(*\)} \] \[ 0= g(x(0), x(T)), \] where \(f\), \(\alpha\), \(g\) and \(I_k\) \((1\leq k\leq m)\) are continuous functions, \(0\leq t\leq\alpha(t)\leq T\), \(\Delta x(t_k)= x(t^+_k)- x(t^-_k)\). Using the method of lower and upper solutions in reversed order and the method of coupled lower and upper solutions, the author establishes conditions guaranteeing the existence of a solution and of a quasi-solution to \((*)\), respectively.