In this work, we present a new constructive technique which is based on Green's functional concept. According to this technique, a linear completely nonhomogeneous nonlocal problem for a second-order ordinary differential equation is reduced to one and only one integral equation in order to identify the Green's solution. The coefficients of the equation are assumed to be generally variable nonsmooth functions satisfying some general properties such as p-integrability and boundedness. A system of three integro-algebraic equations called the special adjoint system is obtained for this problem. A solution of this special adjoint system is Green's functional which enables us to determine the Green's function and the Green's solution for the problem. Some illustrative applications and comparisons are provided with some known results.