A strong convergence theorem for relatively nonexpansive mappings in a Banach space

By S. Matsushita, Wataru Takahashi. Journal of Approximation Theory, 2005

441 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3ef5a18a350a9498c422097b9f6c5b30fa326293

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Citations in the corpus

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 ).”
Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces (2008, background): “After that, we prove that every nonspreading mapping in a Banach space with a fixed point is relatively nonexpansive in the sense of Matsushita and Takahashi [21, 22 ] (Theorem 3.3).”
Minimum-norm solution of variational inequality and fixed point problem in banach spaces (2015, background): “3 [18] Let E be a strictly convex and smooth Banach space, let C be a closed convex subset of E, and let T be a relatively non-expansive mapping from C into itself.”
Strong convergence theorem for a common point of solution of variational inequality and fixed point problem (2012, positive): “In spaces more general than Hilbert spaces, Matsushita and Takahashi [11] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping T in a Banach space E : They proved that if the sequence { α n } is bounded above from one then the sequence { x n…”
Parallel and sequential hybrid methods for a finite family of asymptotically quasi $$\phi $$ϕ-nonexpansive mappings (2015, background): “For the case N = 1, Theorem 1 gives the following monotone hybrid method, which modifies the corresponding algorithms in Kim and Xu [17], as well as Kim and Takahashi (Theorems 3.1, 3.7, 4.1 [16]).”
Convergence theorems of a modified hybrid algorithm for a family of quasi-φ-asymptotically nonexpansive mappings (2010, background): “Let us consider the functional defined as [8, 9 ]b y”
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 Projection Algorithms for Nonlinear Operators (2007, background): “@Let E be a strictly convex and smooth Banach space, let C be a closed convex subset of E , and let T be a relatively nonexpansive mapping from C into itself.”
Fixed Point Theorems and Ergodic Theorems for Nonlinear Mappings in Hilbert Spaces (2011, positive): “78 P. Kocourek et al. Lemma 4.2 (Matsushita and Takahashi [ 28 ]).”