Existence and continuation of periodic solutions of autonomous Newtonian systems

In this article, we study the existence and the continuation of periodic solutions of autonomous Newtonian systems. To prove the results we apply the infinite-dimensional version of the degree for SO(2)-equivariant gradient operators defined by the third author in Nonlinear Anal. Theory Methods Appl. 23(1) (1994) 83-102 and developed in Topol. Meth. Nonlinear Anal. 9(2) (1997) 383-417. Using the results due to Rabier [Symmetries, Topological degree and a Theorem of Z.Q. Wang, J. Math. 24(3) (1994) 1087-1115] and Wang [Symmetries and calculation of the degree, Chinese Ann. Math. 10 (1989) 520-536] we show that the Leray-Schauder degree is not applicable in the proofs of our theorems, because it vanishes.