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

• Zaifu Yang
Part of the Theory and Decision Library book series (TDLC, volume 21)

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 ℝn 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 xø(x,t) for every (x,t) ∈ C, CX × {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 iI 2 and D = {x ∈ ℝ n | u lx ≪ υ2} where u 1 ≪ υl, u 2υ 2, u lu 2, υ 1υ 2, and cl(D 1) ∩ cl(D 2) = Ø. Let f : cl(D) ↦ ℝn 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 iI 2. Then there exists at least one fixed point in each set D 1, D2, 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
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Authors and Affiliations

• Zaifu Yang
• 1
1. 1.Yokohama National UniversityJapan