• Title, Summary, Keyword: nonexpansive mapping

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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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    • v.29 no.1_2
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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.

Fixed Point Theorems in Product Spaces

  • Bae, Jong Sook;Park, Myoung Sook
    • Journal of the Chungcheong Mathematical Society
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    • v.6 no.1
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    • pp.53-57
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    • 1993
  • Let E and F be Banach spaces with $X{\subset}E$ and $Y{\subset}F$. Suppose that X is weakly compact, convex and has the fixed point property for a nonexpansive mapping, and Y has the fixed point property for a multivalued nonexpansive mapping. Then $(X{\oplus}Y)_p$, $1{\leq}$ P < ${\infty}$ has the fixed point property for a multi valued nonexpansive mapping. Furthermore, if X has the generic fixed point property for a nonexpansive mapping, then $(X{\oplus}Y)_{\infty}$ has the fixed point property for a multi valued nonexpansive mapping.

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NEW HYBRID ALGORITHM FOR WEAK RELATIVELY NONEXPANSIVE MAPPING AND INVERSE-STRONGLY MONOTONE MAPPING IN BANACH SPACE

  • Zhang, Xin;Su, Yongfu;Kang, Jinlong
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.87-102
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    • 2011
  • The purpose of this paper is to prove strong convergence theorems for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping by a new hybrid method in a Banach space. 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 by Ying Liu[Ying Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. -Engl. Ed. 30(7)(2009), 925-932] and some others.

STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE MAPPINGS

  • He, Xin-Feng;Xu, Yong-Chun;He, Zhen
    • East Asian mathematical journal
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    • v.27 no.1
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    • pp.1-9
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    • 2011
  • In this paper, we consider an iterative scheme for finding a common element of the set of fixed points of a asymptotically quasi nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a asymptotically quasi-nonexpansive mapping and strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a asymptotically quasi-nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.

AN ITERATION SCHEMES FOR NONEXPANSIVE MAPPINGS AND VARIATIONAL INEQUALITIES

  • Wang, Hong-Jun;Song, Yi-Sheng
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.991-1002
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    • 2011
  • An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.

APPROXIMATING FIXED POINTS OF NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES WITHOUT UNIFORM CONVEXITY

  • Sahu, Daya Ram;Khan, Abdul Rahim;Kang, Shin Min
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.3
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    • pp.1007-1020
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    • 2013
  • Approximate fixed point property problem for Mann iteration sequence of a nonexpansive mapping has been resolved on a Banach space independent of uniform (strict) convexity by Ishikawa [Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71]. In this paper, we solve this problem for a class of mappings wider than the class of asymptotically nonexpansive mappings on an arbitrary normed space. Our results generalize and extend several known results.

STRONG CONVERGENCE THEOREMS FOR FIXED POINT PROBLEMS OF ASYMPTOTICALLY QUASI-𝜙-NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE

  • Jeong, Jae Ug
    • Journal of applied mathematics & informatics
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    • v.32 no.5_6
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    • pp.621-633
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    • 2014
  • In this paper, we introduce a general iterative algorithm for asymptotically quasi-${\phi}$-nonexpansive mappings in the intermediate sense to have the strong convergence in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors.

COMMON FIXED POINTS FOR SINGLE-VALUED AND MULTI-VALUED MAPPINGS IN COMPLETE ℝ-TREES

  • Phuengrattana, Withun;Sopha, Sirichai
    • Communications of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.507-518
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    • 2016
  • The aim of this paper is to prove some strong convergence theorems for the modified Ishikawa iteration process involving a pair of a generalized asymptotically nonexpansive single-valued mapping and a quasi-nonexpansive multi-valued mapping in the framework of $\mathbb{R}$-trees under the gate condition.

A NEW GENERALIZED RESOLVENT AND APPLICATION IN BANACH MAPPINGS

  • Wang, Xian;Chen, Jun-Min;He, Zhen
    • East Asian mathematical journal
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    • v.30 no.1
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    • pp.69-77
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    • 2014
  • In this paper, we introduce a new generalized resolvent in a Banach space and discuss its some properties. Using these properties, we obtain an iterative scheme for finding a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping. Furthermore, strong convergence of the scheme to a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping is proved.