The aim of the work is to find critical points of a continuously differentiable function on a Banach manifold X, which ist invariant under the operation of a symmetry (compact Lie) group G. We suppose, there is a compact G-manifold M and a G-equivariant surjection from X onto M with a G-equivariant section, such that all fixed points of X are contained in the image of the section. Such situations occur in symmetric Lagrangian and Hamiltonian systems.The presence of fixed points renders the usual cohomological index theories (Benci, Fadell, Rabinowitz) useless. If the function is twice continuously differentiable, and if we suppose that the fixed point set consists of non-degenerate critical submanifolds, then the existence of one fixed point p with high Morse index implies the existence of further critical points outside the fixed point set and below the level f(p). The results remain valid for continuously differentiable functionals with appropriate modifications. The mulitplicity results are proved by means of products in Borel cohomology. In order to characterise critical points outside the fixed point set by Borel cohomology, we need certain "separating" cohomology classes that restrict to a non-zero class on p and to zero on the fixed point set below the level f(p). The smaller the degree of these classes the more critical points we find. Our work extends a method developed by T. Bartsch and Z. Q. Wang for even Hamiltonian systems on a torus or the cotangent space of a torus, and in many cases improves their multiplicity results. We apply our abstract theorems to T-periodic Lagrangian systems on compact Riemannian manifolds with a fibrewise convex Lagrangian, in order to prove the existence of T-periodic solutions. Under these conditions, the Lagrange functional in general is not twice continuously differentiable, yet a method by Abbondandolo and Schwarz still allows to characterise non-degenerate critical points by the Hessian.
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