On existence theorems for nonlinear equations involving noncompact mappings.

In this note we present two new theorems concerning the solvability of nonlinear equations in normed linear spaces involving either K-coercive pseudo-A-proper mappings (Theorem 1) or their uniform limits (Theorem 2). These theorems are then used to obtain certain known as well as new existence results for various special classes of equations involving, for example, A-proper and P-compact operators; K-monotone, quasi-K-monotone and pseudo-K-monotone operators; monotone and others.