<p>We analyze the relationship between boundary-value problems with impulsive action at fixed times and boundary-value problems with switchings at fixed and nonfixed times. We find the constructive conditions of solvability and a scheme for construction of solutions of a nonlinear periodic boundary-value problem with switchings at nonfixed times. By using the Adomian decomposition method, we establish the solvability conditions and construct a new iterative technique for finding solutions of a weakly nonlinear periodic boundary-value problem with switchings at nonfixed times. In addition, we obtain constructive conditions for convergence of the iterative scheme to the solution of the weakly nonlinear boundary-value problem, as well as the switching points. The obtained iterative scheme is used to find approximations to the periodic solution of the equation with switchings at nonfixed times, which models a nonisothermal chemical reaction.</p>