This chapter discusses the (approximation) solvability of operator equations of the form f ∈ Tx with T A-proper or strongly A-closed. Applications to operator equations, involving monotone like and other classes of mappings and to elliptic BV problems, are also presented in the chapter. The chapter proves several new abstract results on the (approximation) solvability of semilinear equations of the form Ax + Nx = f, where A is a Fredholm mapping of index zero and N is a given quasi-bounded nonlinear mapping such that A + N is A-proper or strongly A-closed. The results are then applied to investigating the semilinear elliptic equations exhibiting (double) resonance. The chapter presents the solvability of certain nonlinear noncoercive operator equations. Theory of A-proper mappings, their uniform limits, and of strongly A-closed (that is, pseudo A-proper) mappings unifies and extends theories of compact and ball-condensing vector fields of monotone and accretive like and other classes of mappings and is, more importantly, useful in investigating various classes of mappings.
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