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

Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space En. 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: En × [0, 1]→En of the formH(x, t) = (1 −t)F0(x) + t[F(x) − y] where theF0 in this paper is taken to be linear. Simplicial subdivisionsTk of En × [0, 1] furnish piecewise linear approximatesGk toH. The basic computation is via the generation of piecewise linear curvesPk which satisfyGk(x, t) = 0. Visualizing a sequence {Tk} of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0.This paper builds upon and continues recent work of C.B. Garcia.