# Constructive proofs of theorems relating to:*F(x) = y*, with applications

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## Abstract

Some theorems are given which relate to approximating and establishing the existence of solutions to systems*F(x) = y* of*n* equations in*n* unknowns, for various*y*, in a region of euclidean*n*-space E^{ n }. They generalize known theorems.

Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained.

A familiar use is made of homotopies H: E^{ n } × [0, 1]→E^{ n } of the form*H(x, t)* = (1 −*t*)*F*^{0}*(x) + t[F(x) − y]* where the*F*^{0} in this paper is taken to be linear. Simplicial subdivisions*T*^{ k } of E^{ n } × [0, 1] furnish piecewise linear approximates*G*^{ k } to*H.* The basic computation is via the generation of piecewise linear curves*P*^{ k } which satisfy*G*^{ k }*(x, t)* = 0. Visualizing a sequence {*T*^{ k }} of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curves*P* on which*H(x, t)* = 0.

This paper builds upon and continues recent work of C.B. Garcia.

## Keywords

Recent Work Mesh Size Mathematical Method Complementarity Problem Basic Computation## Preview

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