This project seeks to illuminate interconnections between the use of generalized degree theory and minimax methods for nonconvex functionals applied to bifurcation problems in the mathematical theory of nonlinear functional analysis. Plans include the adaptation of arguments concerning A-proper mappings for finding multiple solutions of operator equations to possible use in numerical approximations. The concept of A-proper mapping has been used in proving results of Landesman-Lazer type for semilinear equations whose linear parts have finite-dimensional kernels. Work will be done seeking to obtain extensions to the infinite-dimensional case. Various models in mathematical physics arise in this context. A second thrust of this work involves the application of a recently established, and very precise, form of the Morse deformation lemma. It will be used to obtain information concerning differential equations with discontinuous nonlinearities and to produce indirect existence results for free boundary problems. Using Clarke's concept of subgradient, variational inequalities and nonconvex problems will be analyzed via the mountain pass lemma and more general saddle point techniques to develop new applications to mechanics.
