Here, the existence of solutions to the following nontrivial initial value problem is established: \[ x'(t) = f(t,x(t)) \text{ a.e. } t \in [0,1], \quad x(0) + \sum_{k=1}^m a_kx(t_k) = 0, \] with \(\sum_{k=1}^m a_k \neq 1\). The novelty in this paper is the fact that the growth conditions imposed on \(f\) are split in two: one on the interval \([0,t_m]\), and a second one on \([t_m,1]\). The cases where \(f\) is continuous and where \(f\) satisfies a Lipschitz condition with respect to the second variable are treated. The proofs rely on the Leray-Schauder alternative and on the Banach contraction principle according to the cases considered.