We develop some relative index theories for abstract operator equations. As applications, we prove a new Galerkin approximation formula and a new saddle point reduction formula for the $P$-index. We apply these new formulas to the minimal periodic problem for $P$-symmetric periodic solutions of nonlinear Hamiltonian systems.
"Relative index theories and applications." Topol. Methods Nonlinear Anal. 49 (2) 587 - 614, 2017. https://doi.org/10.12775/TMNA.2016.093