| Abstrakt: |
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 { ψ n } n = 1 ∞ and { ϕ n } n = 1 ∞ as follows: { ψ n + 1 = π n ψ n + (1 − π n) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , 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. [ABSTRACT FROM AUTHOR] |
