Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces

By Dan Butnariu, Simeon Reich, Alexander J. Zaslavski. 2001

267 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e558b95717345a4b0b2c0635b8e381a878b93e85

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Citations in the corpus

Fast and Simple Bregman Projection Methods for Solving Variational Inequalities and Related Problems in Banach Spaces (2020, unknown): “In this connection, please see also [5].”
A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces (2010, background): “The class of quasi-φ-nonexpansivemappings ismore general than the class of relatively nonexpansivemappings [36–38]which requires the strong restriction: F(T ) = F̂(T ).”
Generalized Projection Algorithms for Nonlinear Operators (2007, unknown): “‖Tx − Ty‖ ≤ ‖x − y‖, ∀x , y ∈ C , and relatively nonexpansive [4, 5, 7] if F̂ (T ) = F (T ) and (p,Tx) ≤ (p, x), ∀x ∈ C , p ∈ F (T )”
Bregman Monotone Optimization Algorithms (2003, methodology): “3(i) below, there is some overlap between the “paracontractions” introduced in [31, 75] (see also [24, 26]) and operators in B.”
Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces (2004, background): “In [16], Rockafellar proved that if liminfn→∞ rn > 0 and A−10 = ∅, then the sequence {xn} defined by (1.2) converges weakly to an element of solutions of (1.1).”
Generalized mixed equilibrium and fixed point problems in a Banach space (2016, unknown): “Recall that T is said to be relatively nonexpansive [5] if Fix(T ) = F̃ ix(T ) 6= ∅ and φ(p, Tx) ≤ φ(p, x), ∀x ∈ C,∀p ∈ Fix(T ).”
Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings (2011, background): “The asymptotic behavior of a relatively nonexpansive mapping was studied in [21-23].”
Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces (2008, background): “... fixed points of T by F (T ), that is, F (T )= {z ∈ C : Tz = z}. A point u ∈ C is said to be an asymptotic fixed point (Reich [23]) of T if there exists a sequence {xn} of C such that xn �u and xn − Tx n → 0. We denote the set of asymptotic fixed points of T by F (T ). Following Matsushita and Ta”
Weak Convergence of Orbits of Nonlinear Operators in Reflexive Banach Spaces (2003, background): “The results presented in this article also relate to previous results we have presented in Butnariu et al. (2001a).”