Interval Test and Existence Theorem

Interval methods have been introduced for computationally verifiable sufficient conditions for existence, uniqueness and convergence, for solving finite dimensional nonlinear systems [2], [7], [8], [9], [13] and for nonlinear operator equations in infinite dimensional spaces [11]. The methods can also be used to discuss some classical existence theorems [20], [21]. The conditions, like bounded inverse of the de­ rivative, norm coercivity, or uniform monotonicity guarantee the homeomorphism of a differentiable function, and also, the nonsingu­ larity assumption of the partial derivative guarantee the existence of a implicit function [14], [19]. In this paper, using interval methods, precisely, the centred form of the Newton-transform of f [3] and the Moore-like test for the Krawczyk operator [2], [8], we will derive some computationally verifiable sufficient conditions for a function f to be a homeomorphism and the global implicit function theorem. The cases of the function f(x) or f(x,y) to be local Lipschitz continuous and continuously differentiable are of special interest. Some further results for these classes of functions and a sufficient condition for the feasibility of the numerical continuation method are given.