Abstract

This chapter presents some major developments in the solvability theory of semilinear and fully nonlinear operator equations based on the A-proper mapping approach. A part of the theory is based solely on the Brouwer degree theory, where applicable. The chapter shows how the finite dimensional Morse theory can be used in conjunction with the A-proper mapping approach to study the existence of nontrivial solutions. It introduces a more general class of A-proper maps relative to two spaces and develops a method of studying semilinear and nonlinear equations involving strong nonlinearities. The chapter presents a fixed point theory for maps T with A-proper based only on the notion of approximation-essential maps.

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