Abstract
Let H be a real Hilbert space and f(x,λ) be a C2 operator mapping a small neighborhood U of (x0,λ0) ε (H × R1) into itself. We investigate the solutions of the equation f(x,λ) = 0 near a solution (x0,λ0), assuming that f(x,λ) is a gradient mapping and 0 < dim Ker fx(x0,λ0) < ∞. In particular, we show that the type numbers of Marston Morse for an isolated critical point can be used to prove the existence of a point of bifurcation at (x0,λ0). An application of this result is given to the discovery of periodic motions near a stationary point for a large class of nonlinear Hamiltonian systems in “resonant” cases.
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