Abstract
We investigated two new modified inertial Mann Halpern and inertial Mann viscosity algorithms for solving fixed point problems. Strong convergence theorems under some fewer restricted conditions are established in the framework of infinite dimensional Hilbert spaces. Finally, some numerical examples are provided to support our main results. The algorithms and results presented in this paper can generalize and extend corresponding results previously known in the literature.
Highlights
PreliminariesLet C be a nonempty closed convex subset of a real Hilbert space H
T : C → C is said to be nonexpansive if k Tx − Tyk ≤ k x − yk for all x, y ∈ C
The main purpose of this paper is to consider the following fixed point problem: find x ∗ ∈ C, such that Tx ∗ = x ∗, where T : C → C is nonexpansive with Fix( T ) 6= ∅
Summary
Let C be a nonempty closed convex subset of a real Hilbert space H. The Mann iteration algorithm is extremely useful for finding the fixed point problem of nonexpansive mappings, and provides a unified framework for different algorithms. It should be pointed out that even in a Hilbert space, the iterative sequence { xn } defined by (1) has only weak convergence under certain conditions. Nakajo and Takahashi [6] established strong convergence of the Mann iteration by means of projection methods, and proposed the following algorithm in a Hilbert space H: yn = ψn xn + (1 − ψn ) Txn ,. The inertial type technique, which was first proposed by Polyak [18], have attracted considerable attention in the research of fast convergence of algorithms It is a heavy-ball method based on a second-order time dynamic system.
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