Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups (Q1851318)

Let \(X\) be a real uniformly convex Banach space whose dual space \(X^*\) has the Kadec-Klee property (i.e., for every sequence \(\{x_n\}\subset X^*\), the following implication is true: \((x_n\to x\wedge\| x_n\|\to\| x\|)\Rightarrow (\| x_n- x\|\to 0))\). Let further \(G\) be a commutative semigroup with identity, and \(\{T(t): t\in G\}\) be an asymptotically nonexpansive semigroup acting on a nonempty closed convex subset \(C\subset X\). The main result of the present paper gives three equivalent conditions for the weak convergence of almost orbits of \(\{T(t)\}\) .