Adjoint systems and Green functionals for second-order linear integro-differential equations with nonlocal conditions

In this work, we generalize so called Green's functional concept in literature to second-order linear integro-differential equation with nonlocal conditions. According to this technique, a linear completely nonhomogeneous nonlocal problem for a second-order integro-differential equation is reduced to one and one integral equation to identify the Green's solution. The coefficients of the equation are assumed to be generally nonsmooth functions satisfying some general properties such as p-integrability and boundedness. We obtain new adjoint system and Green's functional for second-order linear integro-differential equation with nonlocal conditions. An application illustrate the adjoint system and the Green's functional. Another application shows when the Green's functional does not exist