• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.119-127
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• 2003

The purpose of this paper is to study the necessary and sufficient conditions for the sequences of Ishikawa iterative sequences with mixed errors of asymptotically quasi-nonexpansive type mappings in Banach spaces to converge to a fixed point in Banach spaces. The results presented in this paper extend and improve the corresponding results of［l-4, 7-9].

• 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.

• 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.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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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.