Computing Equilibria and Fixed Points pp 171-194 | Cite as

# The Computation of a Continuum of Zero Points of a Point-to-Set Mapping

## Abstract

So far in the literature most numerical methods are designed to search for a single fixed point or zero point of the underlying function. As a matter of fact, most of the existing fixed point theorems only guarantee the existence of a single fixed point. As far as we are aware of, Browder theorem is the only result on the existence of a connected set of fixed points up to now. Recall in Chapter 1 this theorem is stated as follows. Let *X* be a nonempty, compact and convex subset of ℝ^{n} and let *ø* : *X* × [0,1] ↦ *X* be a nonempty-valued, convex-valued and compact-valued u.s.c. point-to-set mapping. Then there exists a connected subset *C* of *X* × [0,1] such that *x* ∈ *ø*(*x*,*t*) for every (*x*,*t*) ∈ *C*, *C* ∩ *X* × {0} ≠ Ø and *C * ∩ *X* × {1} ≠ Ø. Note that for each fixed *t* ∈ [0, 1], there exists a fixed point of *ø* (·, *t*) by Kakutani theorem. Although our intuition tells us there must exist a connected set of fixed points from level zero to level one since *t* is a free variable, it is by no means easy to prove this. Amann [1972] presented a theorem for the existence of three fixed points. This theorem says: Let *D* _{ i } = {*x* ∈ ℝ^{ n } | *u* ^{ i } ≪ *x* ≪ υ ^{ i }} for *i* ∈ *I* _{2} and *D* = {*x* ∈ ℝ^{ n } | *u* ^{l} ≪ *x* ≪ υ^{2}} where *u* ^{1} ≪ υ^{l}, *u* ^{2} ≪ *υ* ^{2}, *u* ^{l} ≤ *u* ^{2}, *υ* ^{1} ≤ *υ* ^{2}, and *cl*(*D* _{1}) ∩ *cl*(*D* _{2}) = Ø. Let *f* : *cl*(*D*) ↦ ℝ^{n} be a function satisfying that (a) *u* ≤ υ implies *f*(*u*) ≤ *f* (*υ*); and (b) *u* ^{ i } ≪ *f*(*u* ^{ i }), *f*(*u* ^{ i }) ≪ *υ* ^{ i } for all *i* ∈ *I* _{2}. Then there exists at least one fixed point in each set *D* _{1}, *D*2, and *D*\(*cl*(*D* _{1})∪*cl*(*D* _{2})). However, neither Browder nor Amann gives constructive proofs for their results.

## Keywords

Zero Point Sign Vector Admissible Solution Constructive Proof Piecewise Linear Approximation## Preview

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