<p>Constructive solvability conditions and a scheme for constructing solutions of a nonlinear periodic problem for a Liènard-type equation with switchings at non-fixed points in time have been obtained. Using the Adomian decomposition method, solvability conditions were obtained and a new iterative technique was constructed for finding solutions of a weakly non-linear periodic problem for Liènard-type equations with switchings at non-fixed points in time. Constructive conditions for the convergence of the iterative scheme to the solution of a weakly non-linear boundary value problem, as well as switching points, were constructed. The obtained iterative scheme was applied to find approximations to the periodic solution of the equation that models a chain non-isothermal chemical reaction.</p>