Periodic oscillations of an unhomogeneous string with fixed ends

The paper considers the problem of time-periodic solutions to the quasi-linear wave equation with x-dependent coefficients of a general form. The author proves the existence of a denumerable number of periodic solutions, if there are homogeneous Dirichlet boundary conditions on the segment when the nonlinear term features a power-law growth. The proof is based on a variational method. Periodic solutions are energy functional critical points, the existence of which is proved with the help of the Feireisle method. The author formulates a theorem about the existence and the regularization of at least one periodic solution in the case when the nonlinear term satisfies a non-resonance condition at infinity. The author also describes the conditions under which the periodic solution is unique. The proof of the theorem is obtained using the Lera-Schauder principle of a fixed point and it is based on the author’s previous research. Keywords: wave equation, periodic solution, Sturm -Liouville problem, functional critical point.

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