The authors study the following resonant second-order boundary value problem \(u^{\prime \prime }=f\left( t,u,u^{\prime }\right)\) with periodic boundary conditions \(u\left( 0\right) =u\left(T\right)\), \(u^{\prime }\left( 0\right) =u^{\prime }\left( T\right)\). They modify the problem at resonance and consider an equivalent nonresonant boundary value problems \(u^{\prime \prime }+\beta u^{2}=f\left(t,u,u^{\prime }\right) +\beta u^{2}\) and \(u^{\prime \prime }-\beta u^{2}=f\left( t,u,u^{\prime }\right) -\beta u^{2}\). Then, they study the properties of the corresponding Green's function for both modified problems. Next, the authors obtain some sufficient conditions for the existence of solutions of the modified boundary value problem, using fixed-point theory. Consequently, these conditions are sufficient for the existence of solutions of the original boundary value problem. At the end of the paper, the authors demonstrate the applicability of the established results through some examples.