A remark on fixed point theorems for Lipschitzian mappings (Q1329282)

Let \(C\) be a bounded closed convex subset of a \(p\)-uniformly convex Banach space \((p>1)\) and \((a_{n,k})\) be a strongly ergodic matrix. The author proves that every map \(T: C\to C\) satisfying \[ \liminf_{n\to\infty} \inf_ m \sum^ \infty_{k=1} a_{n,k}\| T^{k+m}\|^ p< 1+c \] \((c>0)\) has a fixed point in \(C\).