Properties of Bounded Solutions of Linear and Nonlinear Evolution Equations: Homoclinics of a Beam Equation*

The objective of this paper is to discuss the existence, bifurcation, and regularity, with respect to time and parameters, of bounded solutions of infinite dimensional equations. As an application of our results, we study homoclinic solutions of a nonlinear equation. Chow et al. [2, 31, using the Liapunov-Schmidt method, studied periodic and homoclinic solutions of j; + g(x) = -1f + pf(t), where f is periodic, and ;1 and ,U are small parameters, with suitable conditions on g. Holmes and Marsden [9] extended results of Melnikov [l l] and discussed the solutions described above for infinite dimensional equations. Their main application involved a forced beam equation. Concerning the regularity of solutions, some technical difficulties arise