Manfred Krüppel
Ein aymptotischer Fixpunktsatz für Lipschitz-stetige Operatoren in uniform konvexen Banach-Räumen
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 11-24 (1995)
47H10 Fixed-point theorems, See also {54H25, 55M20, 58C30}
Abstract: For every uniformly conex Banach space X and for every p>1 there exists
a constant $\gamma_p>1$ such that holds: If C\subset X is nonempty,
bounded closed and convex and T:C\to C is a Lipschitzian mapping such
that the Lipschitzian norms ||T^n|| of the iterates T^n fullfilles the
\[ \liminf_{n\to\infty}(||T||^p+||T^2||^p+\cdots+||T^n||^p)/n < \gamma_p \]
then T has a fixed point in C. In Hilbert space \gamma_2 and in L^p-space
with p\ge 2 the constant is \gamma_p\ge 1+1/(2^{p-1}-1).