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Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the typeThe first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

Abstract. We prove the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We provide some analytic and numerical evidence for this periodic orbit to be linearly stable. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. We numerically investigate the linear stability of these periodic orbits through long-term integration of the regularized equations, showing that linear stability occurs when 0.538 ≤ m ≤ 1, and instability occurs when 0 < m ≤ 0.537 with spectral stability for m ≈ 0.537.