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1 results

  • Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem

  • Helga Schirmer
  • Journal:
    Canadian Mathematical Bulletin / Volume 24 / Issue 3 / 01 September 1981
    Published online by Cambridge University Press:
    20 November 2018, pp. 351-357
    Print publication:
    01 September 1981
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  • Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.