scholarly journals Convergence of iterative algorithms for continuous pseudocontractive mappings

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1767-1777 ◽  
Author(s):  
Jong Jung
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Convex Combination ◽  
Iterative Algorithms ◽  
Duality Mapping ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Implicit Iterative Algorithm

In this paper, we prove strong convergence of a path for a convex combination of a pseudocontractive type of operators in a real reflexive Banach space having a weakly continuous duality mapping J? with gauge function ?. Using path convergency, we establish strong convergence of an implicit iterative algorithm for a pseudocontractive mapping combined with a strongly pseudocontractive mapping in the same Banach space.

scholarly journals Convergence Theorems for Modified Implicit Iterative Methods with Perturbation for Pseudocontractive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 72
Author(s):  
Jong Soo Jung
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Reflexive Banach Space ◽  
Convex Combination ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Pseudocontractive Mappings ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping

In this paper, first, we introduce a path for a convex combination of a pseudocontractive type of mappings with a perturbed mapping and prove strong convergence of the proposed path in a real reflexive Banach space having a weakly continuous duality mapping. Second, we propose two modified implicit iterative methods with a perturbed mapping for a continuous pseudocontractive mapping in the same Banach space. Strong convergence theorems for the proposed iterative methods are established. The results in this paper substantially develop and complement the previous well-known results in this area.

scholarly journals The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Methods ◽  
Reflexive Banach Space ◽  
Iterative Scheme ◽  
Duality Mapping ◽  
Nonexpansive Semigroups ◽  
Weakly Continuous ◽  
Asymptotically Nonexpansive ◽  
Weakly Continuous Duality Mapping

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

scholarly journals The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Rabian Wangkeeree
Keyword(s):  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Linear Operators ◽  
Duality Mapping ◽  
Approximation Methods ◽  
Iterative Approximation ◽  
Bounded Linear ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Hybrid Approximation

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

scholarly journals A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Duality Mapping ◽  
Strong Convergence Theorem ◽  
Iterative Approximation ◽  
Nonexpansive Semigroups ◽  
Weakly Continuous ◽  
Weakly Continuous Duality Mapping ◽  
Iterative Approximation Method

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

scholarly journals A strong convergence theorem for contraction semigroups in Banach spaces

2005 ◽  
Vol 72 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Contraction Semigroup ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Weakly Continuous ◽  
Duality Map ◽  
Image Position

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined byfor all n ≥ 1 converges strongly to a member of Fix(F).

scholarly journals Strong Convergence for Hybrid ImplicitS-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shin Min Kang ◽  
Arif Rafiq ◽  
Faisal Ali ◽  
Young Chel Kwun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Pseudocontractive Mappings

LetKbe a nonempty closed convex subset of a real Banach spaceE, letS:K→Kbe nonexpansive, and let  T:K→Kbe Lipschitz strongly pseudocontractive mappings such thatp∈FS∩FT=x∈K:Sx=Tx=xandx-Sy≤Sx-Sy and x-Ty≤Tx-Tyfor allx, y∈K. Letβnbe a sequence in0, 1satisfying (i)∑n=1∞βn=∞; (ii)limn→∞⁡βn=0.For arbitraryx0∈K, letxnbe a sequence iteratively defined byxn=Syn, yn=1-βnxn-1+βnTxn, n≥1.Then the sequencexnconverges strongly to a common fixed pointpofSandT.

scholarly journals A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Kamonrat Nammanee ◽  
Suthep Suantai ◽  
Prasit Cholamjiak
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Reflexive Banach Space ◽  
Duality Mapping ◽  
Nonexpansive Semigroup ◽  
Gauge Functions ◽  
Implicit And Explicit ◽  
General Iterative Method

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mappingJφ, whereφis a gauge function on[0,∞). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

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