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1. Introduction. In  Kaniel proved a fixed point theorem for a nonlinear quasi-compact operator in a Banach space. The purpose of this paper is to generalize and simplify Kaniel's main result and its proof to a more general class of nonlinear operators which we call projectionally-compact (P-compact) and which, among others, contains completely continuous, quasi-compact, and monotone operators. From our general fixed point theorem for P-compact operators we then deduce in a simple way thedoi:10.1090/s0002-9904-1966-11519-7 fatcat:xqrjquycgfe2vhlapc7coioxlq