• Title/Summary/Keyword: Gateaux differentiable

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CONVERGENCE THEOREMS ON VISCOSITY APPROXIMATION METHODS FOR FINITE NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Jung, Jong-Soo
    • The Pure and Applied Mathematics
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    • v.16 no.1
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    • pp.85-98
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    • 2009
  • Strong convergence theorems on viscosity approximation methods for finite nonexpansive mappings are established in Banach spaces. The main theorem generalize the corresponding result of Kim and Xu [10] to the viscosity approximation method for finite nonexpansive mappings in a reflexive Banach space having a uniformly Gateaux differentiable norm. Our results also improve the corresponding results of [7, 8, 19, 20].

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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 ̄.

Range of Operators and an Application to Existence of a Periodic Solution

  • Bae, Jong Sook;Sung, Nak So
    • Journal of the Chungcheong Mathematical Society
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    • v.1 no.1
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    • pp.19-26
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    • 1988
  • In this paper, we calculate the precise estimation of range of a Gateaux differentiable operator, and apply to the existence of a periodic solution of the second order nonlinear differential equation $$z^{{\prime}{\prime}}+Az^{\prime}+G(z)=e(t)=e(t+2{\pi})$$.

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ITERATIVE ALGORITHMS WITH ERRORS FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Jung, Jong-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.771-790
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    • 2006
  • The iterative algorithms with errors for nonexpansive mappings are investigated in Banach spaces. Strong convergence theorems for these algorithms are obtained. Our results improve the corresponding results in [5, 13-15, 23, 27-29, 32] as well as those in [1, 16, 19, 26] in framework of a Hilbert space.

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE

  • Kum, Sang-Ho;Lee, Gue-Myung
    • Journal of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.683-695
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    • 2001
  • In this paper we are concerned with theoretical properties of gap functions for the extended variational inequality problem (EVI) in a general Banach space. We will present a correction of a recent result of Chen et. al. without assuming the convexity of the given function Ω. Using this correction, we will discuss the continuity and the differentiability of a gap function, and compute its gradient formula under tow particular situations in a general Banach space. Our results can be regarded as infinite dimensional generalizations of the well-known results of Fukushima, and Zhu and Marcotte with soem modifications.

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STRONG CONVERGENCE OF PATHS FOR NONEXPANSIVE SEMIGROUPS IN BANACH SPACES

  • Kang, Shin Min;Cho, Sun Young;Kwun, Young Chel
    • Korean Journal of Mathematics
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    • v.19 no.3
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    • pp.279-289
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    • 2011
  • Let E be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, C be a nonempty closed convex subset of E and f : $C{\rightarrow}C$ be a fixed bounded continuous strong pseudocontraction with the coefficient ${\alpha}{\in}(0,1)$. Let $\{{\lambda}_t\}_{0<t<1}$ be a net of positive real numbers such that ${\lim}_{t{\rightarrow}0}{\lambda}_t={\infty}$ and S = {$T(s)$ : $0{\leq}s$ < ${\infty}$} be a nonexpansive semigroup on C such that $F(S){\neq}{\emptyset}$, where F(S) denotes the set of fixed points of the semigroup. Then sequence {$x_t$} defined by $x_t=tf(x_t)+(1-t)\frac{1}{{\lambda}_t}{\int_{0}}^{{\lambda}_t}T(s)x{_t}ds$ converges strongly as $t{\rightarrow}0$ to $\bar{x}{\in}F(S)$, which solves the following variational inequality ${\langle}(f-I)\bar{x},\;p-\bar{x}{\rangle}{\leq}0$ for all $p{\in}F(S)$.