On uniformly Lipschitzian multivalued mappings in Banach and metric spaces

Let (X,d)(X,d) be a metric space. A mapping T:X→XT:X→X is said to be uniformly Lipschitzian if there exists a constant kk such that d(Tn(x),Tn(y))≤kd(x,y)d(Tn(x),Tn(y))≤kd(x,y) for all x,y∈Xx,y∈X and n≥1n≥1. It is known that such mappings always have fixed points in certain metric spaces for k>1k>1, provided kk is sufficiently near 11. These spaces include uniformly convex metric and Banach spaces, as well as metric spaces having ‘Lifšic characteristic’ greater than 11. A uniformly Lipschitzian concept for multivalued mappings is introduced in this paper, and multivalued analogues of these results are obtained.