Bifurcation of periodics and subharmonics in abstract nonlinear undamped wave equations (Q1284426)

The author studies the existence of periodic and subharmonic solutions to a boundary value problem for the beam equation. In his model equation there is no damping term in contrast to several papers about this topic. The periodic force term with a small parameter \(\varepsilon\) and parameters in the beam equation have an influence on possible results: (1) there exist arbitrary many, but a finite number of subharmonic solutions, (2) there exists a small T-periodic solution. The author obtains these results from an abstract problem of the form \[ u''+ Au+ F(u)Bu= \varepsilon h(t) \] in a Hilbert space framework. The existence of T-periodic weak solutions are studied. Using special orthogonal projections, the author decomposes the above problem and reduces it to a finite-dimensional one.