Let $K$ be a nonempty closed convex subset of a uniformly convex Banach space E, let $\{T(t) : t \geq 0\}$ be a strongly continuous semigroup of nonexpansive mappings from $K$ into itself such that $F := \cap_{T\geq 0} F(T (t)) \not=\emptyset$. Assuming that $\{\alpha_n\}$ and $\{t_n\}$ are sequences of real numbers satisfying appropriate conditions, we show that the sequence $x_n$ defined by $$x_n=\alpha_nx_{n-1}+(1-\alpha_n)T(t_n)x_n$$ converges weakly to an element of $F$. This extends Thong’s result (Thong, Nonlinear Anal. 74, 6116–6120, 2011) from a Hilbert space setting to a Banach space setting. Next, theorems of weak convergence of an implicit iterative algorithm with errors for treating a strongly continuous semigroup of Lipschitz pseudocontractions are established in the framework of a real Banach space.