1990/06/01 by Robert Sine · 2 citations
Computer Science · Mathematics · #Optimization and Variational Analysis #Advanced Optimization Algorithms Research #Matrix Theory and Algorithms
paper · doi:10.1090/s0002-9939-1990-0948156-x
If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a nonexpansive map on a domain in a finite-dimensional sup norm space then there is a universal bound on the periods of periodic points. This yields the same result for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nonexpansive on a domain in a finite-dimensional Banach space which has a polyhedral unit ball. Similar results are obtained for certain nonexpansive maps defined on all of an infinite-dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Lp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p > ∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .