On p-parameter bifurcation of an n-dimensional function-germ.
Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...
The author examined non-zero -periodic (in time) solutions for a semilinear beam equation under the condition that the period is an irrational multiple of the length. It is shown that for a.e. (in the sense of the Lebesgue measure on ) the solutions do exist provided the right-hand side of the equation is sublinear.