| Summary: | Abstract We investigate the existence of solutions of weakly nonlinear periodic boundary value problems for systems of ordinary differential equations with switchings and the construction of these solutions. We consider the critical case where the equation for the generating constants of a weakly nonlinear periodic boundary-value problem with switchings does not turn into an identity. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of weakly nonlinear periodic boundary value problems with switchings in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary value problem for the mathematical model of nonisothermal chemical reactions. To check the accuracy of the proposed approximations, we evaluate discrepancies in the original equation.
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