On Periodic Solutions of a Beam Vibration Equation

Periodic Solutions to the Vibration Equation of a Beam with a Rigidly Clamped Endpoint and an Elastically Fixed Endpoint

“…This paper continues the study of [6]- [8]. The main results are asymptotic formulas for eigenvalues of the Sturm-Liouville problem and a theorem on the existence of countably many solutions to the problem (1.1)-(1.3) under the assumption that the nonlinear term has power growth and is not necessarily homogeneous.…”

“…In the case a > 0, the problem (1.1),(1.2) with different boundary conditions and nonlinear terms of different kinds was studied in [5]- [8]. The case of hinged endpoints of the beam u(0, t) = u xx (0, t) = u(π, t) = u xx (π, t) = 0 (1.6) was studied in [5,6]. Equation (1.1) with the conditions u(0, t) = u xx (0, t) = u(π, t) = u x (π, t) = 0, (1.7) under the assumption that the nonlinear term has at most linear growth in u, was considered in [7].…”

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

“…It is well known that the variational method pioneered by Rabinowitz [31,32] is a powerful tool for dealing with nonlinear problems, and it is closely related to compactness. Recently, by variational method, Rudakov [28] proved that, under the boundary conditions (1.5), there is a sequence of periodic solutions to the nonlinear beam equation…”

“…where c satisfies the condition (A) in [28] and the nonlinear term g has a polynomial growth in u. The condition (A) can make sure that the inverse of the linearized operator of this problem is compact on its range.…”