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

Abstract

Abstract Let Ω be a nonempty closed convex subset of a real Hilbert space H $\mathfrak{H}$ . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ $\{\psi _{{n}}\}_{n=1}^{\infty}$ and { ϕ n } n = 1 ∞ $\{\phi _{{n}}\}_{n=1}^{\infty}$ as follows: { ψ n + 1 = π n ψ n + ( 1 − π n ) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , $$\begin{aligned} \textstyle\begin{cases} \psi _{n+1}=\pi _{n}\psi _{{n}}+(1-\pi _{n})\Im \psi _{{n}}, \\ \phi _{{n}}=\dfrac{1}{n}\underset{t=1}{\overset{n}{\sum}}\psi _{t}, \end{cases}\displaystyle \end{aligned}$$ for n ∈ N $n\in \mathit{N}$ , where 0 ≤ π n ≤ 1 $0\leq \pi _{n}\leq 1$ , and π n → 0 $\pi _{n} \rightarrow 0$ . 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 ( η , β ) $(\eta ,\beta )$ -enriched strictly pseudononspreading ( ( η , β ) $(\eta ,\beta )$ -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.

Keywords

• Enriched nonlinear map
• Pseudocontractive map
• Banach space
• Lipschitizian
• Quasi-nonexpansive map
• Hilbert space