• 제목/요약/키워드: strong and weak convergence theorems

검색결과 38건 처리시간 0.027초

WEAK AND STRONG CONVERGENCE THEOREMS FOR THE MODIFIED ISHIKAWA ITERATION FOR TWO HYBRID MULTIVALUED MAPPINGS IN HILBERT SPACES

  • Cholamjiak, Watcharaporn;Chutibutr, Natchaphan;Weerakham, Siwanat
    • 대한수학회논문집
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    • 제33권3호
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    • pp.767-786
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    • 2018
  • In this paper, we introduce new iterative schemes by using the modified Ishikawa iteration for two hybrid multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. We use CQ and shrinking projection methods with Ishikawa iteration for obtaining strong convergence theorems. Furthermore, we give examples and numerical results for supporting our main results.

WEAK AND STRONG CONVERGENCE THEOREMS FOR A SYSTEM OF MIXED EQUILIBRIUM PROBLEMS AND A NONEXPANSIVE MAPPING IN HILBERT SPACES

  • Plubtieng, Somyot;Sombut, Kamonrat
    • 대한수학회보
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    • 제50권2호
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    • pp.375-388
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    • 2013
  • In this paper, we introduce an iterative sequence for finding solution of a system of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Then, the weak and strong convergence theorems are proved under some parameters controlling conditions. Moreover, we apply our result to fixed point problems, system of equilibrium problems, general system of variational inequalities, mixed equilibrium problem, equilibrium problem and variational inequality.

MONOTONE CQ ALGORITHM FOR WEAK RELATIVELY NONEXPANSIVE MAPPINGS AND MAXIMAL MONOTONE OPERATORS IN BANACH SPACES

  • Kang, Jinlong;Su, Yongfu;Zhang, Xin
    • Journal of applied mathematics & informatics
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    • 제29권1_2호
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    • pp.293-309
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    • 2011
  • The purpose of this article is to prove strong convergence theorems for weak relatively nonexpansive mapping which is firstly presented in this article. In order to get the strong convergence theorems for weak relatively nonexpansive mapping, the monotone CQ iteration method is presented and is used to approximate the fixed point of weak relatively nonexpansive mapping, therefore this article apply above results to prove the strong convergence theorems of zero point for maximal monotone operators in Banach spaces. Noting that, the CQ iteration method can be used for relatively nonexpansive mapping but it can not be used for weak relatively nonexpansive mapping. However, the monotone CQ method can be used for weak relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W.Takahashi, and some others.

CONVERGENCE THEOREMS FOR TWO FAMILIES OF WEAK RELATIVELY NONEXPANSIVE MAPPINGS AND A FAMILY OF EQUILIBRIUM PROBLEMS

  • Zhang, Xin;Su, Yongfu
    • 대한수학회논문집
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    • 제25권4호
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    • pp.583-607
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    • 2010
  • The purpose of this paper is to prove strong convergence theorems for common fixed points of two families of weak relatively nonexpansive mappings and a family of equilibrium problems by a new monotone hybrid method in Banach spaces. Because the hybrid method presented in this paper is monotone, so that the method of the proof is different from the original one. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^2$. Our results improve and extend the corresponding results announced in [W. Takahashi and K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008), Article ID 528476, 11 pages; doi:10.1155/2008/528476] and [Y. Su, Z. Wang, and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009), no. 11, 5616?5628] and some other papers.

WEAK AND STRONG CONVERGENCE CRITERIA OF MODIFIED NOOR ITERATIONS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE

  • Banerjee, Shrabani;Choudhury, Binayak Samadder
    • 대한수학회보
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    • 제44권3호
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    • pp.493-506
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    • 2007
  • In this paper weak and strong convergence theorems of modified Noor iterations to fixed points for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces are established. In one theorem where we establish strong convergence we assume an additional property of the operator whereas in another theorem where we establish weak convergence assume an additional property of the space.

WEAK AND STRONG CONVERGENCE OF THREE-STEP ITERATIONS WITH ERRORS FOR TWO ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • Jeong, Jae-Ug
    • Journal of applied mathematics & informatics
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    • 제26권1_2호
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    • pp.325-336
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    • 2008
  • In this paper, we prove the weak and strong convergence of the three-step iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize some previous results.

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CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Wu, Changqun;Xu, Sumei
    • Journal of applied mathematics & informatics
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    • 제27권3_4호
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    • pp.661-672
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    • 2009
  • In this paper, we introduce a modified three-step iteration scheme with errors for asymptotically nonexpansive mappings in the framework of uniformly convex Banach spaces. Weak and strong convergence theorems are established.

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STRONG CONVERGENCE THEOREMS FOR GENERALIZED VARIATIONAL INEQUALITIES AND RELATIVELY WEAK NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Liu, Ying
    • East Asian mathematical journal
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    • 제28권3호
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    • pp.265-280
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    • 2012
  • In this paper, we introduce an iterative sequence by using a hybrid generalized $f$-projection algorithm for finding a common element of the set of fixed points of a relatively weak nonexpansive mapping an the set of solutions of a generalized variational inequality in a Banach space. Our results extend and improve the recent ones announced by Y. Liu [Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings, J. Glob. Optim. 46 (2010), 319-329], J. Fan, X. Liu and J. Li [Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Analysis 70 (2009), 3997-4007], and many others.

CONVERGENCE TO COMMON FIXED POINTS FOR A FINITE FAMILY OF GENERALIZED ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Saluja, G.S.
    • East Asian mathematical journal
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    • 제29권1호
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    • pp.23-37
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    • 2013
  • The purpose of this paper is to study an implicit iteration process with errors and establish weak and strong convergence theorems to converge to common fixed points for a finite family of generalized asymptotically quasi-nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve and generalize some known results from the existing literature.