Application of the product net technique and Kadec–Klee property to study nonlinear ergodic theorems and weak convergence theorems in uniformly convex multi-Banach spaces

Abstract

Let Y be a uniformly convex multi-Banach space which has not a Frechet differentiable norm. We use the technique of product net to obtain the nonlinear ergodic theorems in Y. Finally, let the dual of uniformly convex multi-Banach space have the Kadec–Klee property, we instate the weak convergence theorem in the case of reversible semi-group.