On eventually and asymptotically Lipschitzian mappings. (Q1862689)

This article deals with mappings \(T:M\to M\) in a metric space \((M, d)\) satisfying the condition \[ D(T^nx,T^ny)\leq kd(x,y)\;\bigl(x,y\in M,n \geq n_0(T)\bigr); \] such mappings are called eventually \(k\)-Lipschitzian. Asymptotically \(k\)-Lipschitzian and uniformly \(k\)-Lipschitzian mappings are partial classes of the eventually \(k\)-Lipschitzian ones. The authors prove the following results: (I) for a bounded closed convex set \(M\) in a uniformly convex Banach space \(X\), each eventually \(k\)-Lipschitzian operator with \(k<\kappa(M)\) has a fixed point in \(M\); here \(\kappa(M)\) is the Lifschitz characteristics of \(M\); (II) each uniformly \(k\)-Lipschitzian operator with \(k<\gamma\) \((\gamma\) is the solution to the equation \(\gamma(1-\delta_X(1/ \gamma))=1\), \(\delta_X(\cdot)\) the modulus of convexity of \(X)\) has a fixed point in \(M\). In the case of compact metric spaces, some modifications of the Freudenthal and Hurewicz theorem for asymptotically \(k\)-Lipschitzian mappings are also given.