Time periodic solutions of one-dimensional forced Kirchhoff equations with x -dependent coefficients

Abstract

In this paper, we consider the one-dimensional Kirchhoff equation with -dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2/ and amplitude . By using the Nash-Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2/ and order . Such results hold for parameters (,) in a positive measure Cantor set that has asymptotically full measure as the amplitude goes to zero.

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

“…Subsequently, the research on periodic solutions of variable coefficient wave equations has gained more attention (see e.g. [10,16,19,21,24,28,29]). It is worth mentioning that, in [20], Ji and Li obtained the existence of periodic solution of the nonlinear wave equation with sublinear nonlinearity and constant or variable coefficients which may not satisfy the assumption η u (x) > 0.…”

Periodic solutions of a semilinear Euler-Bernoulli beam equation with variable coefficients

“…Here, ρ(x) and η(x) respectively denote the rock density and the elasticity coefficient. Under some homogeneous boundary conditions, the existence of the periodic solutions to the equation (1.7) equipped with various nonlinearity has been extensively studied in recent decades (see [4,7,9,13,18,19,20,23,24,27,37,38]).…”