• Title/Summary/Keyword: iterative sequence

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DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • Chang, Shih-Sen;Cho, Yeol-Je;Haiyun Zhou
    • Journal of the Korean Mathematical Society
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    • v.38 no.6
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    • pp.1245-1260
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    • 2001
  • A demi-closed theorem and some new weak convergence theorems of iterative sequences for asymptotically nonexpansive and nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corresponding results of [1],[8]-[10],[12],[13],[15],[16], and [18].

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CONVERGENCE THEOREMS OF MODIFIED ISHIKAWA ITERATIVE SEQUENCES WITH MIXED ERRORS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Park, Kwang-Pak;Kim, Ki-Hong;Kim, Kyung-Soo
    • East Asian mathematical journal
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    • v.19 no.1
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    • pp.103-111
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    • 2003
  • In this paper, we will discuss some sufficient and necessary conditions for modified Ishikawa iterative sequence with mixed errors to converge to fixed points for asymptotically quasi-nonexpansive mappings in Banach spaces. The results presented in this paper extend, generalize and improve the corresponding results in Liu [4,5] and Ghosh-Debnath [2].

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Noor Iterations with Error for Non-Lipschitzian Mappings in Banach Spaces

  • Plubtieng, Somyot;Wangkeeree, Rabian
    • Kyungpook Mathematical Journal
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    • v.46 no.2
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    • pp.201-209
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    • 2006
  • Suppose C is a nonempty closed convex subset of a real uniformly convex Banach space X. Let T : $C{\rightarrow}C$ be an asymptotically nonexpansive in the intermediate sense mapping. In this paper we introduced the three-step iterative sequence for such map with error members. Moreover, we prove that, if T is completely continuous then the our iterative sequence converges strongly to a fixed point of T.

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MODIFIED ISHIKAWA ITERATIVE SEQUENCES WITH ERRORS FOR ASYMPTOTICALLY SET-VALUED PSEUCOCONTRACTIVE MAPPINGS IN BANACH SPACES

  • Kim, Jong-Kyu;Nam, Young-Man
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.847-860
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    • 2006
  • In this paper, some new convergence theorems of the modified Ishikawa and Mann iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in uniformly smooth Banach spaces are given.

Stability of Iterative Sequences Approximating Common Fixed Point for a System of Asymptotically Quasi-nonexpansive Type Mappings

  • Li, Jun;Huang, Nan-Jing;Cho, Yeol Je
    • Kyungpook Mathematical Journal
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    • v.47 no.1
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    • pp.81-89
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    • 2007
  • In this paper, we introduce the concept of a system of asymptotically quasinonexpansive type mappings. Furthermore, we define a $k$-step iterative sequence approximating common fixed point for a system of asymptotically quasi-nonexpansive type mappings and study its stability in real Banach spaces.

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ON FIXED POINT OF UNIFORMLY PSEUDO-CONTRACTIVE OPERATOR AND SOLUTION OF EQUATION WITH UNIFORMLY ACCRETIVE OPERATOR

  • Xu, Yuguang;Liu, Zeqing;Kang, Shin-Min
    • East Asian mathematical journal
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    • v.24 no.3
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    • pp.305-315
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    • 2008
  • The purpose of this paper is to study the existence and uniqueness of the fixed point of uniformly pseudo-contractive operator and the solution of equation with uniformly accretive operator, and to approximate the fixed point and the solution by the Mann iterative sequence in an arbitrary Banach space or an uniformly smooth Banach space respectively. The results presented in this paper show that if X is a real Banach space and A : X $\rightarrow$ X is an uniformly accretive operator and (I-A)X is bounded then A is a mapping onto X when A is continuous or $X^*$ is uniformly convex and A is demicontinuous. Consequently, the corresponding results which depend on the assumptions that the fixed point of operator and solution of the equation are in existence are improved.

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CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS

  • Kang, Jung-Im;Cho, Yeol-Je;Zhou, Hai-Yun
    • Communications of the Korean Mathematical Society
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    • v.19 no.2
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    • pp.321-328
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    • 2004
  • In this paper, we will prove the following: Let D be a nonempty of a normed linear space X and T : D -> X be a nonexpansive mapping. Let ${x_n}$ be a sequence in D and ${t_n}$, ${s_n}$ be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;<\;1\;and\;{\sum_{n=1}}^{\infty}\;t_n\;=\;{\infty},\;(ii)\;(a)\;0\;{\leq}\;s_n\;{\leq}\;1,\;s_n\;->\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;<\;{\infty}\;or\;(b)\;s_n\;=\;s\;for\;all\;n\;{\geq}\;1\;and\;s\;{\in}\;[0,1),\;(iii)\;x_{n+1}\;=\;(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n)\;for\;all\;n\;{\geq}\;1.$ Then, if the sequence {x_n} is bounded, then $lim_{n->\infty}\;$\mid$$\mid$x_n-Tx_n$\mid$$\mid$\;=\;0$. This result improves and complements a result of Deng [2]. Furthermore, we will show that certain conditions on D, X and T guarantee the weak and strong convergence of the Ishikawa iterative sequence to a fixed point of T.

AN ITERATIVE METHOD FOR EQUILIBRIUM PROBLEMS, VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS

  • Shang, Meijuan;Su, Yongfu
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.161-173
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    • 2009
  • In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extend and improve the corresponding results announced by many others.

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