Let \(C\) be a nonempty closed convex subset of a real Hilbert space \(H\) and let \(S\) be a commutative semigroup. Let \({\mathcal S}=\{T(t):t\in S\}\) be a nonexpansive semigroup on \(C\) such that \(\text{Fix}({\mathcal S})\neq\emptyset\). Let \(X\) be a subspace of \(B(S)\) such that \(1\in X\) and for each \(x\in C\), \(y\in H\), the function \(t\mapsto(T(t)x,y)\) is an element of \(X\).
In the present paper, the authors investigate the following iteration procedure: \[ \begin{cases} x_0=x\in C,\\ y_n=\alpha_nx_n+(1-\alpha_n)T_{\mu_n}x_n,\;\alpha_n\in[0,1],\\ C_n=\bigl\{z\in C:\| y_n-z\|\leq\| x_n-z\|\bigr\},\\ Q_n=\bigl\{z\in C:(x_n-z,x_0-c_n)\geq 0\bigr\},\\ x_{n+1}=P_{C_n\cap Q_n}(x_0),\end{cases} \] where \(P\) is the well-known metric projection and \(\{\mu_n\}\) is a sequence of means on \(X\). They prove the strong convergence of the sequence \(\{x_n\}\) to \(P_{\text{Fix}(S)}(x_0)\) by the hybrid method which is used in mathematical programming. As a consequence, the authors gives various strong convergence theorems for nonexpansive mappings and one-parameter nonexpansive semigroups in a real Hilbert space.