Israel Journal of Mathematics

, Volume 17, Issue 4, pp 339–346 | Cite as

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

  • W. A. Kirk


LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup i→∞{sup y∈K t ix−Tiy∼−‖x−y‖}≦0. IfT N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k ix−y∼,x,y∈K, wherek i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ0 (X), be less than one.


Banach Space Convex Subset Fixed Point Theorem Nonexpansive Mapping Lipschitz Constant 
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Copyright information

© Hebrew University 1974

Authors and Affiliations

  • W. A. Kirk
    • 1
  1. 1.Department of MathematicsThe University of IowaIowa CityU.S.A.

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