On the isolation of the roots of a non-linear operator

Abstract

Certain problems in optimal control may be formulated by the theory of Lagrange multipliers. Stationary controls are thus roots of an operator equation. See, e.g., Ref. [1, 2]. We give below a measure of the isolation of these roots. We will follow the notational convention tha t f ' (x ) represents the Gateaux derivative o f f evaluated at x, and that f ' (x , h) is the differential o f f at x in the direction h.