Second-order linear differential equations with piecewise constant arguments subject to nonlocal boundary conditions

We analyze the existence of solution for the class of second-order linear differential equations with functional dependencex″(t)+ax′(t)+bx(t)+cx′([t])+dx([t])=σ(t),t∈J=[0,T],where a,b,c,d∈R,T>0, and σ is a piecewise continuous function, subject to a certain type of nonlocal mixed boundary conditions which include the value of the derivative of x at the initial point x′(0).