On uniformly Lipschitzian multivalued mappings in Banach and metric spaces

Abstract Let ( X , d ) be a metric space. A mapping T : X → X is said to be uniformly Lipschitzian if there exists a constant k such that d ( T n ( x ) , T n ( y ) ) ≤ k d ( x , y ) for all x , y ∈ X and n ≥ 1 . It is known that such mappings always have fixed points in certain metric spaces for k > 1 , provided k is sufficiently near 1 . These spaces include uniformly convex metric and Banach spaces, as well as metric spaces having ‘Lifsic characteristic’ greater than 1 . A uniformly Lipschitzian concept for multivalued mappings is introduced in this paper, and multivalued analogues of these results are obtained.