• 제목/요약/키워드: smooth Banach space

검색결과 28건 처리시간 0.019초

GENERALIZED PROJECTION AND APPROXIMATION FOR GENERALIZED VARIATIONAL INEQUALITIES SYSTEM IN BANACH SPACES

  • He, Xin-Feng;Xu, Yong-Chun;He, Zhen
    • East Asian mathematical journal
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    • 제24권1호
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    • pp.57-65
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    • 2008
  • The approximate solvability of a generalized system for non-linear variational inequality in Hilbert spaces was studied, based on the convergence of projection methods. But little research was done in Banach space. The primary reason was that projection mapping lacked preferably property in Banach space. In this paper, we introduced the generalized projection methods. By using these methods, the results presented in this paper extended the main results of S. S. Chang [3] from Hilbert spaces to Banach space.

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STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS IN q-UNIFORMLY SMOOTH BANACH SPACES

  • Jeong, Jae Ug
    • Korean Journal of Mathematics
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    • 제20권2호
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    • pp.225-237
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    • 2012
  • In this paper, we introduce a new iterative scheme to investigate the problem of nding a common element of nonexpansive mappings and the set of solutions of generalized variational inequalities for a $k$-strict pseudo-contraction by relaxed extra-gradient methods. Strong convergence theorems are established in $q$-uniformly smooth Banach spaces.

ON THE EXISTENCE OF SOLUTIONS OF EXTENDED GENERALIZED VARIATIONAL INEQUALITIES IN BANACH SPACES

  • He, Xin-Feng;Wang, Xian;He, Zhen
    • East Asian mathematical journal
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    • 제25권4호
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    • pp.527-532
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    • 2009
  • In this paper, we study the following extended generalized variational inequality problem, introduced by Noor (for short, EGVI) : Given a closed convex subset K in q-uniformly smooth Banach space B, three nonlinear mappings T : $K\;{\rightarrow}\;B^*$, g : $K\;{\rightarrow}\;K$, h : $K\;{\rightarrow}\;K$ and a vector ${\xi}\;{\in}\;B^*$, find $x\;{\in}\;K$, $h(x)\;{\in}\;K$ such that $\xi$, g(y)-h(x)> $\geq$ 0, for all $y\;{\in}\;K$, $g(y)\;{\in}\;K$. [see [2]: M. Aslam Noor, Extended general variational inequalities, Appl. Math. Lett. 22 (2009) 182-186.] By using sunny nonexpansive retraction $Q_K$ and the well-known Banach's fixed point principle, we prove existence results for solutions of (EGVI). Our results extend some recent results from the literature.

CONVERGENCE AND ALMOST STABILITY OF ISHIKAWA ITERATION METHOD WITH ERRORS FOR STRICTLY HEMI-CONTRACTIVE OPERATORS IN BANACH SPACES

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제11권4호
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    • pp.293-308
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    • 2004
  • Let K be a nonempty convex subset of an arbitrary Banach space X and $T\;:\;K\;{\rightarrow}\;K$ be a uniformly continuous strictly hemi-contractive operator with bounded range. We prove that certain Ishikawa iteration scheme with errors both converges strongly to a unique fixed point of T and is almost T-stable on K. We also establish similar convergence and almost stability results for strictly hemi-contractive operator $T\;:\;K\;{\rightarrow}\;K$, where K is a nonempty convex subset of arbitrary uniformly smooth Banach space X. The convergence results presented in this paper extend, improve and unify the corresponding results in Chang [1], Chang, Cho, Lee & Kang [2], Chidume [3, 4, 5, 6, 7, 8], Chidume & Osilike [9, 10, 11, 12], Liu [19], Schu [25], Tan & Xu [26], Xu [28], Zhou [29], Zhou & Jia [30] and others.

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Fixed point iterations for quasi-contractive maps in uniformly smooth banach spaces

  • Chidume, C.E.;Osilike, M.O.
    • 대한수학회보
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    • 제30권2호
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    • pp.201-212
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    • 1993
  • It is our purpose in this paper to first establish an inequality in real uniformly smooth Banach spaces with modulus of smoothness of power type q > 1 that generalizes a well known Hilbert space inequality. Using our inequality, we shall then extend the above result of Qihou [15] on the Ishikawa iteration process from Hilbert spaces to these much more general Banach spaces. Furthermore, we shall prove that the Mann iteration process converges strongly to the unique fixed point of a quasi-contractive map in this general setting. No compactness assumption on K is required in our theorems.

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ISHIKAWA AND MANN ITERATION METHODS FOR STRONGLY ACCRETIVE OPERATORS

  • JAE UG JEONG
    • Journal of applied mathematics & informatics
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    • 제4권2호
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    • pp.477-485
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    • 1997
  • Let E be a smooth Banach space. Suppose T:$E \rightarrow E$ is a strongly accretive map. It is proved that each of the two well known fixed point iteration methods (the Mann and ishikawa iteration methods), under suitable conditions converges strongly to a solution of the equation $T_x=f$.

ISHIKAWA AND MANN ITERATION METHODS FOR STRONGLY ACCRETIVE OPERATORS

  • JONG YEOUL PARK;JAE UG JEONG
    • 대한수학회논문집
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    • 제13권4호
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    • pp.765-773
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    • 1998
  • Let E be a smooth Banach space. Suppose T : E longrightarrow E is a strongly accretive map. It is proved that each of the two well known fixed point iteration methods (the Mann and Ishikawa iteration methods), under suitable conditions, converges strongly to a solution of the equation Tx = f.

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FUNCTIONS ATTAINING THE SUPREMUM AND ISOMORPHIC PROPERTIES OF A BANACH SPACE

  • D. Acosta, Maria ;Becerra Guerrero, Julio ;Ruiz Galan, Manuel
    • 대한수학회지
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    • 제41권1호
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    • pp.21-38
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    • 2004
  • We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace Μ containing u, it happens that the subset of norm attaining functionals on Μ is second Baire category in $M^{*}$ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to $\ell$$_1$, where the norm is the restriction of a Luxembourg norm on $L_1$. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.m.