A new index theory for GL+(2)-paths with applications to asymptotically linear systems

In this article, we establish a new index theory defined for the general non-degenerate matrix paths in GL+(2). This is done by the complete homotopy classification for such paths. The parity relation theorem is established for relating this index to the Morse index of the corresponding differential operator. As an application, we establish the existence result of a non-trivial periodic solution of the asymptotically linear differential equation systems under an index twist condition.