Weak and strong convergence theorems for a new class of enriched strictly pseudononspreading mappings in Hilbert spaces

Abstract

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 ${\{{\psi }_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{{\varphi }_n\}}_{n=1}^{\mathrm{\infty }}$ as follows: $\{\begin{array}{c}{\psi }_{n+1}={\pi }_n{\psi }_n+(1-{\pi }_n)\mathrm{\Im }{\psi }_n,\hfill \\ {\varphi }_n=\frac{1}{n}\underset{t=1}{\sum ^n}{\psi }_t,\hfill \end{array}$ for $n\in \mathit{N}$, where $0\le {\pi }_n\le 1$, and ${\pi }_n\to 0$. 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.