Strong Convergence of a Unified General Iteration fork-Strictly Pseudononspreading Mapping in Hilbert Spaces

Dao-Jun Wen,Yi-An Chen,Yan Tang

doi:10.1155/2014/219695

Dao-Jun Wen, Yi-An Chen + Show 1 more

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https://doi.org/10.1155/2014/219695

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Abstract

We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively
Where T is a self-nonexpansive mapping on H, f is a contraction of H into itself, {αn} ⊆ (0, 1) satisfies certain conditions, and B is a strongly positive bounded linear operator on H, converges strongly to x∗ ∈ F(T), which is the unique solution of the following variational inequality:
Inspired and motivated by research going on in this area, we introduce a modified general iterative method for kstrictly pseudononspreading mapping, which is defined in the following way: xn+1 = αnγf + βnxn + [(1 − βn) I − αnB] Tλn xn, (11) n ≥ 1, where Tλn sequences

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Where T is a self-nonexpansive mapping on H, f is a contraction of H into itself, {αn} ⊆ (0, 1) satisfies certain conditions, and B is a strongly positive bounded linear operator on H, converges strongly to x∗ ∈ F(T), which is the unique solution of the following variational inequality:

Results

Conclusion