Approximating common fixed points by the Mann iteration procedure in Banach spaces (Q2708297)

This article deals with the convergence properties for the Mann iterates NEWLINE\[NEWLINEx_{n+1}= \alpha_n x_n+ (1- \alpha_n) T_{\mu_n} x_n\qquad (n= 0,1,\dots).NEWLINE\]NEWLINE Here is \(S= \{T(t): t\in S\}\) a semigroup of nonexpansive mappings leaving invariant a nonempty closed convex subset in a Banach space \(E\), the values \(T_{\mu_n}x_n\) are defined from the equations \(\langle T_{\mu_n} x_n, x^*\rangle= (\mu_n)_t\langle T(t) x_n, x^*\rangle\) \((x^*\in E^*)\), \(\mu_n\) is a sequence of means, \(\{\alpha_n\}\) is a sequence in \([0,1]\). The authors prove the weak convergence of these iterations when \(E\) is uniformly convex and the norm in \(E\) is Fréchet differentiable or satisfies the Opial condition, the strong convergence if \(S\) contains compact operators and \(E\) is strictly convex.