It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and homotopy invariance
We give a continuation principle for forced oscillations of second order differential equations on not necessarily compact diίferentiable manifolds. A topological sufficient condition for an equilibrium point to be a bifurcation point for periodic orbits is a straightforward consequence of such a continuation principle. Known results on open sets of euclidean spaces as well as a recent continuation principle for forced oscillations on compact manifolds with nonzero Euler-Poincare characteristic are also included as particular cases.
We prove a global bifurcation result for T -periodic solutions of the T -periodic delay differential equation x (t) = λf (t, x(t), x(t − 1)) depending on a real parameter λ 0. The approach is based on the fixed point index theory for maps on ANRs.
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