The authors consider the boundary value problem
\[ \begin{aligned} &y''(t) = f(t,y(t)),\quad \text{a.e.}\;t \in [0,1], t \neq t_i,\\ &\Delta y|_{t=t_i} = I_i(y(t_i-)), \quad i = 1,\dots,m\\ &\Delta y|_{t=t_i} = \bar I_i(y(t_i-)), \quad i = 1,\dots,m\\ &y(0) - k_1y'(0) = \int_0^1 h_1(s,y(s))\;\text{d}s,\\ &y(1) + k_2y'(1) = \int_0^1 h_2(s,y(s))\;\text{d}s, \end{aligned} \]
where \(0 < t_1 < \dots < t_m < 1\), \(f, h_1, h_2 : [0,1]\times {\mathbb R} \to {\mathbb R}\), \(I_i\), \(\bar I_i : {\mathbb R} \to {\mathbb R}\), \(k_1\), \(k_2 \geq 0\). Using Banach’s fixed-point theorem, an existence and uniqueness result for the BVP is obtained. Using nonlinear alternative of Leray-Schauder type, an obtained existence result is obtained.