Autonomous periodic boundary-value problems with switchings at nonfixed points of time

We analyze the relationship between boundary-value problems with impulsive action at fixed points of time and boundary-value problems with switchings at fixed and nonfixed points of time. 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 points of time. By using the Adomian decomposition method, we obtain the solvability conditions and construct a new iterative technique for finding solutions of a weakly nonlinear periodic boundary-value problem with switchings at nonfixed points of time. 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 switchings points. The obtained iterative scheme is applied to find approximations to the periodic solution of the equation with switchings at nonfixed points of time, which models a nonisothermal chemical reaction.