A geometrical index for the group Sn and its applications in periodic solutions of differential and differential-difference equations

Abstract

In this paper, we construct a new index theory, Sn index theory, which is an improvement of the S1 index theory given by Benci in [1]. Also S1 index theory is a powerful tool in the study of the multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional Φ is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.