Existence of Periodic Solution for a Class of Linear Third Order ODE

In this paper, we will consider third order linear differential equation y 000 + αy00 + βy0 + γy + f(t, y) = e(t), where α, β, γ are constant coefficients, f(t, y) is continuous, e(t) is discontinuous, and f and e are periodic functions with respect to t of period w. We will introduce sufficient conditions under which the above equation have at least one non-trivial periodic solution of period w. We will see that under the so called conditions, all the solutions of the equation will be bounded. It must be mentioned that e in this equation is called “controller” in the engineering problems and it was always considered to be continuous to ensure us that periodic solution exists. In this paper, we will show the existence of periodic solution without supposing that e to be continuous