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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

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

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