@article{Chuiko_Nesmelova_2023, title={Periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative}, volume={75}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7362}, DOI={10.3842/umzh.v75i10.7362}, abstractNote={<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>UDC 517.9</p> </div> </div> </div> <p>We establish constructive necessary and sufficient conditions for the solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative.<span class="Apple-converted-space"> </span>The urgency of studying nonautonomous boundary-value problems, unsolved with respect to the derivative is explained by the fact that the investigation of traditional problems resolved with respect to the derivative is sometimes complicated, e.g., in the case of nonlinearities that are not integrable in elementary functions.<span class="Apple-converted-space"> </span>We consider the critical case in which the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation does not turn into the identity.<span class="Apple-converted-space"> </span>The least-squares method is used to find constructive conditions for the solvability and obtain convergent iterative schemes for constructing approximate solutions to a nonautonomous nonlinear boundary-value problem unsolved with respect to the derivative. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems unsolved with respect to the derivative for the case of periodic problem for the equation used to describer the motion of a satellite on the elliptic orbit.<span class="Apple-converted-space"> </span>We obtain an estimate for the range of values of a small parameter for which the iterative procedure of construction of the solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent.<span class="Apple-converted-space"> </span>To check the accuracy of the presented approximations, we evaluate the discrepancies in the equation used to model the motion of satellites along the elliptic orbit.</p>}, number={10}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Chuiko , S. and Nesmelova, O.}, year={2023}, month={Oct.}, pages={1429 - 1440} }