The closure of the numerical range contains the spectrum

The purpose of this research is to prove that the well-known theorem in the theory of linear operators in Hilbert space [l, p. 147] indicated in the title holds for nonlinear operators and to a certain extent for noncontinuous ones, and to provide a constructive method for solving the equations involved. DEFINITION 1. The numerical range of a mapping T: 3C-->3C of a complex Hilbert space into itself with domain 3D(T) is the set of complex numbers