A Z. Index Theory

In this paper, we introduce a Z. index theory. For any given positive integer p. we introduce a subset E. of positive integers and define a family of index mappings .. , .n . E.. We prove that this index theory possesses the similar properties as Z. and S' index theories do. In par ticular, by means of a Z. Borsuk-Ulam theorem given in one of our recent papers we prove that under some suitable conditions this theory also possesses dimensional property which is important in applications. As a simple application we study the bifurcation problem of periodic solutions of nonautonomous Hamiltonian