• Title, Summary, Keyword: contractive mapping

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STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS

  • Jung, Jong-Soo
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1151-1164
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    • 2009
  • Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\rightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $\rightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${\neq}{\emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, ($n{\geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\lambda_n$} $\subset$ (0, 1) satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n$ = 0 and $\sum_{n=0}^{\infty}{\lambda}_n={\infty}$, {$\beta_n$} $\subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.

CONVERGENCE THEOREMS OF THREE-STEP ITERATION METHODS FOR QUASI-CONTRACTIVE MAPPINGS

  • Hao, Jinbiao;Wang, Li;Kang, Shin-Min;Shim, Soo-Hak
    • East Asian mathematical journal
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    • v.18 no.2
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    • pp.261-270
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    • 2002
  • We obtain the convergence of three-step iteration methods and generalized three-step iteration methods for quasi-contractive and generalized quasi-contractive mappings, respectively, in Banach spaces. Our results extend the corresponding results in [1], [4]-[6].

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ITERATION PROCESSES OF ASYMPTOTICALLY PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

  • Park, Jong-Yeoul;Jeong, Jae-Ug
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.611-622
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    • 2001
  • Some convergence theorems of modified Ishikawa and Mann iteration processes with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corresponding results in Liu [7] and Schu [10].

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Convergence of an Iterative Algorithm for Systems of Variational Inequalities and Nonlinear Mappings in Banach Spaces

  • JEONG, JAE UG
    • Kyungpook Mathematical Journal
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    • v.55 no.4
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    • pp.933-951
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    • 2015
  • In this paper, we consider the problem of convergence of an iterative algorithm for a general system of variational inequalities, a nonexpansive mapping and an ${\eta}$-strictly pseudo-contractive mapping. Strong convergence theorems are established in the framework of real Banach spaces.

ITERATIVE APPROXIMATION OF FIXED POINTS FOR STRONGLY PSEUDO-CONTRACTIVE MAPPINGS

  • Sharma, Sushil;Deshpande, Bhavana
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.43-51
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    • 2002
  • The aim of this paper is to prove a convergence theorem of a generalized Ishikawa iteration sequence for two multi-valued strongly pseudo-contractive mappings by using an approximation method in real uniformly smooth Banach spaces. We generalize and extend the results of Chang and Chang, Cho, Lee, Jung, and Kang.

FIXED POINT AND PERIODIC POINT THEOREMS ON METRIC SPACES

  • Cho, Seong-Hoon;Park, Dong-Gon
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.1
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    • pp.1-16
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    • 2013
  • The aim of this paper is to establish a new fixed point theorem for a set-valued mapping defined on a metric space satisfying a weak contractive type condition and to establish a new common fixed point theorem for a pair of set-valued mappings defined on a metric space satisfying a weak contractive type inequality. And we give periodic point theorems for single-valued mappings defined on a metric space satisfying weak contractive type conditions.

STRONG CONVERGENCE THEOREMS FOR LOCALLY PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

  • Jung, Jong-Soo
    • Communications of the Korean Mathematical Society
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    • v.17 no.1
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    • pp.37-51
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    • 2002
  • Let X be a reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty bounded open subset of X, and T a continuous mapping from the closure of C into X which is locally pseudo-contractive mapping on C. We show that if the closed unit ball of X has the fixed point property for nonexpansive self-mappings and T satisfies the following condition: there exists z $\in$ C such that ∥z-T(z)∥<∥x-T(x)∥ for all x on the boundary of C, then the trajectory tlongrightarrowz$_{t}$$\in$C, t$\in$[0, 1) defined by the equation z$_{t}$ = tT(z$_{t}$)+(1-t)z is continuous and strongly converges to a fixed point of T as t longrightarrow 1 ̄.ow 1 ̄.

ON THE CONVERGENCE OF HYBRID PROJECTION METHODS FOR ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN THE INTERMEDIATE SENSE

  • Cho, Sun-Young;Kang, Shin-Min;Qin, Xiaolong
    • Communications of the Korean Mathematical Society
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    • v.26 no.3
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    • pp.473-482
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    • 2011
  • In this paper, mappings which are asymptotically pseudo-contractive in the intermediate sense are considered based on a hybrid projection method. Strong convergence theorems of fixed points are established in the framework of Hilbert spaces.

A FIXED POINT THEOREM FOR ASYMPTOTICALLY C-CONTRACTIVE MAPPINGS

  • Park, Jong An
    • Korean Journal of Mathematics
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    • v.11 no.2
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    • pp.111-115
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    • 2003
  • Under some asymptotical conditions we obtain a fixed point theorem for a mapping(not necessarily nonexpansive) from an unbounded closed convex subset of a reflexive Banach space into itself. Also we give an example for applying it.

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