Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space

Quan, Jing; Chang, Shih-sen

doi:10.1186/1687-1812-2012-16

Cited by 5 publications

(4 citation statements)

References 12 publications

(25 reference statements)

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“…(a) Our results extend many other results that have been proved for this important class of general pseudocontractive mappings. For example, we extend the total asymptotically strict pseudocontractive mapping in [1] and Lipschitzian pseudo-contraction semigroup in [8] to the total asymptotically strict pseudocontractive semigroup. (b) In addition, we study the weak convergence of the total asymptotically strict pseudocontractive semigroup by using the demiclosedness of − ( ) which, in some way, extends the result in [11] in Banach spaces.…”

Section: Resultsmentioning

confidence: 99%

“…Miao et al [3] introduced an implicit iteration process for a finite family of total asymptotically pseudocontractive maps. And in recent years, many researchers focused on the convergence of pseudocontractive and asymptotically strict pseudocontractive semigroups; see [4][5][6][7][8] and their references. In [9,10] especially, the authors gave the modified Mann type iteration algorithm and studied its convergence.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Wang

2013

Abstract and Applied Analysis

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The purpose of this paper is to give a new modified Ishikawa type iteration algorithm for common fixed points of total asymptotically strict pseudocontractive semigroups. Under the reduction of some conditions, both strong convergence and weak convergence of the iteration algorithm are proved in Banach spaces with new methods of proofs, respectively. The main results presented in this paper extend and improve the corresponding recent results of many others.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Wang

2013

Abstract and Applied Analysis

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“…In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [1][2][3][4][5][6][7][8][9]. The first interesting algorithm for finding the fixed points of the Lipschitz pseudocontractive mappings in Hilbert spaces was presented by Ishikawa [4] in 1974.…”

Section: Introductionmentioning

confidence: 99%

“…Zhou proved that the sequence { } generated by (5) converges strongly to proj Fix(T) ( 0 ) without the compactness assumption.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Iterations for the Fixed Point Problem and Variational Inequalities

Zhu

Kang

Yao

et al. 2014

Journal of Applied Mathematics

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Some Recent Trends in Variational Inequalities and Optimization Problems with Applications

Bnouhachem

Hamdi

2014

Abstract and Applied Analysis

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Variational inequalities theory, which was introduced in the sixties, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, finance, economics, social, and pure and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. The ideas and techniques of variational inequalities are being applied in a variety of diverse areas of sciences and prove to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems. Variational inequalities have been extended and generalized in several directions using novel and new techniques. In parallel, optimization methods based on proximal point and proximal-like type methods have attracted a large number of researchers in the last three decades. In the same spirit, we can cite, for instance, the alternating direction multipliers method, which is based on the augmented lagrangian algorithm, which itself can be seen as a direct application of the proximal point algorithm to the dual problem of a constrained optimization problem.The aim of this special issue is to present new approaches and theories for variational inequalities arising in mathematics and applied sciences. This special issue includes 14 highquality peer-reviewed papers that deal with different aspects of variational inequalities. These papers contain some new, novel, and innovative techniques and ideas. We hope that all the papers published in this special issue can motivate and foster further scientific works and development of the research in the area of theory, algorithms, and applications of variational inequalities.The summaries of the 14 papers in this issue are listed as follows.The paper of C. Chen et al. considers a class of linearly constrained separable convex programming problems without coupled variables. They weaken some conditions to obtain convergence of the alternating direction method of multipliers and they propose also a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions. The paper of H. Sun and Y. Wang revisits the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP) and sharpens the global error bound for the GNCP under weaker conditions, which improves the existing error bound estimation for the problem.The paper of M. Ma concerns the design and the convergence analysis of algorithms to split variational inequality and equilibrium problems.The paper of Y. Wang and C. Wang gives a new modified Ishikawa type iteration algorithm for common fixed points of total asymptotically strict pseudocontractive semigroups. Abstract and Applied AnalysisStrong and weak convergence are proved under mild conditions. Further...

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Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space

Abstract: The purpose of this article is to study the strong and weak convergence of implicit iterative sequence to a common fixed point for pseudocontractive semigroups in Banach spaces. The results presented in this article extend and improve the corresponding results of many authors.

Cited by 5 publications

References 12 publications

Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Hybrid Iterations for the Fixed Point Problem and Variational Inequalities

Some Recent Trends in Variational Inequalities and Optimization Problems with Applications

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