Let Ω be a nonempty closed convex subset of a real Hilbert space $\mathfrak{H}$ . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences $\{\psi _{{n}}\}_{n=1}^{\infty}$ and $\{\phi _{{n}}\}_{n=1}^{\infty}$ as follows: \(\begin{aligned} \textstyle\begin{cases} \psi _{n+1}=\pi _{n}\psi _{{n}}+(1-\pi _{n})\Im \psi _{{n}}, \\ \phi _{{n}}=\dfrac{1}{n}\underset{t=1}{\overset{n}{\sum}}\psi _{t}, \end{cases}\displaystyle \end{aligned}\) for $n\in \mathit{N}$ , where $0\leq \pi _{n}\leq 1$ , and $\pi _{n} \rightarrow 0$ . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of $(\eta ,\beta )$ -enriched strictly pseudononspreading ( $(\eta ,\beta )$ -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.
