Solvability of some pendulum-type equations with linear damping and homogeneous Dirichlet conditions (Q1612551)

Using a Lyapunov-Schmidt reduction, the authors study the solvability of the nonlinear boundary value problem of pendulum-type \[ -u'' - \alpha u' - \lambda_1(\alpha)y + g(u) = h(x), \quad u(0) = u(\pi) = 0, \] where \(\lambda_1(\alpha) = 1 + \alpha^2/4\) and \(g\) is continuous, \(T\)-periodic and has mean value zero. The results are based upon a delicate asymptotic analysis of the bifurcation equations.