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

Let Ω be a nonempty closed convex subset of a real Hilbert space H. Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences {ψn}_n=1^∞ and {ϕn}_n=1^∞ as follows: (Formula presented.) for n∈N, where 0≤π_n≤1, and π_n→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 (η,β)-enriched strictly pseudononspreading ((η,β)-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.