TY - JOUR

T1 - Weak and strong convergence theorems for a new class of enriched strictly pseudononspreading mappings in Hilbert spaces

AU - Agwu, Imo Kalu

AU - Işık, Hüseyin

AU - Igbokwe, Donatus Ikechi

N1 - Publisher Copyright:
© The Author(s) 2024.

PY - 2024/12

Y1 - 2024/12

N2 - 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≤πn≤1, and πn→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.

AB - 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≤πn≤1, and πn→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.

KW - 47H09

KW - 47H10

KW - 47J05

KW - 65J15

KW - Banach space

KW - Enriched nonlinear map

KW - Hilbert space

KW - Lipschitizian

KW - Pseudocontractive map

KW - Quasi-nonexpansive map

UR - http://www.scopus.com/inward/record.url?scp=85203351179&partnerID=8YFLogxK

U2 - 10.1186/s13663-024-00770-5

DO - 10.1186/s13663-024-00770-5

M3 - Article

AN - SCOPUS:85203351179

SN - 1687-1820

VL - 2024

JO - Fixed Point Theory and Algorithms for Sciences and Engineering

JF - Fixed Point Theory and Algorithms for Sciences and Engineering

IS - 1

M1 - 14

ER -