The index of a critical point for densely defined operators of type $(S_+)_L$ in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type $(S_{+})_{L}$ acting from a real, reflexive and separable Banach space $X$ into $X^{*}.$ This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in $l^{p},~p>2,$ illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type $(S_{+})).$ Applications to nonlinear Dirichlet problems have appeared elsewhere.