Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

“…Rudakov [25] considered the existence of periodic solutions for such wave equation with power-law growth nonlinearity. Ji and Li [20] obtained an existence result of periodic solution for η ρ (x) = 0, which actually solves an open problem posted in [5]. Chen [7] applied a global inverse function theorem (see [21]) to prove the existence and uniqueness of periodic solutions for a system of nonlinear wave equation with x-dependent coefficients.…”

“…where a, b are relatively prime positive integers. As stated in [4,5,6,15,16,17,18,19,20,25,26], equation (1.1) is a mathematical model to account for the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. More precisely, the vertical displacement u(t, z) at time t and depth z of a plane seismic wave is described by the equation where ρ = (ων) 1/2 denotes the acoustic impedance function.…”

Existence of multiple periodic solutions to asymptotically linear wave equations in a ball

“…have developed topological tools, index theory, and variational methods to get some existence results for the one-dimensional problem with various nonlinearities. The related results are [1,8,9,11,12,15,[17][18][19]24,25] and the references therein.…”

The existence of periodic solutions for nonlinear beam equations on by a para‐differential method

“…Subsequently, many related results have been obtained by Bahri, Brézis, Corn, Nirenberg, etc, see previous works, [4][5][6][7] while some recent papers can be found in other studies. [8][9][10] Among most of these results, period T was required to be a rational multiple of (length of spatial interval). Otherwise, it results in the "small divisor" problem.…”