Periodic Vibrations of the Nonhomogeneous Strings With Linear and Nonlinear Couplings

This paper is concerned with the existence of periodic vibrations of the nonhomogeneous strings with linear and nonlinear couplings under some Sturm–Liouville boundary conditions. Such a model is governed by the variable coefficients wave equations and can also be used to describe the simultaneous propagation of seismic waves in nonisotropic media. In the case of linear couplings, the nonlinear self‐interactions exhibit sublinear growth with different powers. In the case of nonlinear coupling, the nonlinear self‐interactions are characterized by superlinear growth with the same powers as the coupling terms. We prove the existence of infinitely many periodic solutions for these two classes of problems by variational methods and Galerkin approximations under the same working space. Our results are applicable to the uncoupled problems of either homogeneous or nonhomogeneous strings and the coupled problems of the homogeneous strings.