Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative

Abstract

We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleightype equation unsolved with respect to the derivative. The urgency of investigation of nonautonomous boundary-value problems unsolved with respect to the derivative is explained by the fact that the analysis of traditional problems solved with respect to the derivative is sometimes significantly complicated, e.g., in the presence of nonlinearities that are not integrable in elementary functions. 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. The least-squares method is used to establish constructive conditions for the solvability and propose convergent iterative schemes for the construction of 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 in the case of periodic problem for the equation that describes the motion of a satellite on the elliptic orbit. We obtain an estimate for the range of values of a small parameter in which the iterative procedure used for the construction of solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent. To check the accuracy of the proposed approximations, we estimate the discrepancies appearing in the equation used to simulate the motion of satellites along the elliptic orbits.