Regularized Semigroups, Iterated Cauchy Problems and Equipartition of Energy.
The present note discusses an interesting positive linear operator which was recently introduced by J.P. King. New estimates in terms of the first and second modulus of continuity are given, and iterates of the operators are considered as well. For general King operators the second moments are minimized.
Let be a nonempty closed convex subset of a Banach space and a -Lipschitzian rotative mapping, i.eṡuch that and for some real , and an integer . The paper concerns the existence of a fixed point of in -uniformly convex Banach spaces, depending on , and .
We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on -almost everywhere Fréchet differentiability of Lipschitz functions on (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at -almost every at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are -almost everywhere Fréchet differentiable....