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On Fixed Points of Non‐Expansive Piecewise Isometric Mappings
Author(s) -
Lawrence Jim,
Spingarn Jonathan E.
Publication year - 1987
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-55.3.605
Subject(s) - mathematics , isometric exercise , expansive , piecewise , fixed point , pure mathematics , mathematical analysis , physical therapy , physics , medicine , compressive strength , thermodynamics
For a non‐expansive piecewise isometric mapping q on R d having a fixed point, we study the iterates of ½( I + q ). Such iterates are shown to behave, eventually, as if q were actually an isometry. Under certain circumstances, a finite number of iterations are guaranteed to produce a fixed point. Examples dealing with systems of linear inequalities and with network flows are examined. A new constructive proof is presented that for any real antisymmetric matrix A , there exists x ⩾0 such that Ax ⩾0 and x + Ax ⩾0.
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