• Title/Summary/Keyword: smooth Banach space

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A SYSTEM OF NONLINEAR VARIATIONAL INCLUSIONS IN REAL BANACH SPACES

  • Bai, Chuan-Zhi;Fang, Jin-Xuan
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.3
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    • pp.385-397
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    • 2003
  • In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{*},\;y^{*},\;z^{*}\;\in\;E$ such that ${\theta}\;{\in}\;{\alpha}T(y^{*})\;+\;g(x^{*})\;-\;g(y^{*})\;+\;A(g(x^{*}))\;\;\;for\;{\alpha}\;>\;0,\;{\theta}\;{\in}\;{\beta}T(z^{*})\;+\;g(y^{*})\;-\;g(z^{*})\;+\;A(g(y^{*}))\;\;\;for\;{\beta}\;>\;0,\;{\theta}\;{\in}\;{\gamma}T(x^{*})\;+\;g(z^{*})\;-\;g(x^{*})\;+\;A(g(z^{*}))\;\;\;for\;{\gamma}\;>\;0,$ where T, g : $E\;{\rightarrow}\;E,\;{\theta}$ is zero element in Banach space E, and A : $E\;{\rightarrow}\;{2^E}$ be m-accretive mapping. By using resolvent operator technique for n-secretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in q-uniformly smooth Banach spaces and in real Banach spaces, respectively.

A FIXED POINT THEOREM FOR NONEXPANSIVE SEMIGROUPS IN P-UNIFORMLY CONVEX BANACH SPACES

  • Jeong, Jae-Ug
    • Journal of applied mathematics & informatics
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    • v.3 no.1
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    • pp.47-54
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    • 1996
  • We prove that if RUC(S) has a left invariant mean ${\rho}={T_{S} : s \;{\in}\; S}$ is a continuous repesentation of S as nonexpansive map-pings on a closed convex subset C of a p-uniformly convex and p-uniformly smooth Banach space and C contains an element of bounded orbit then C contains a common fixed point for ${\rho}$.

ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES

  • SON, TA CONG;THANG, DANG HUNG
    • Journal of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1023-1036
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    • 2015
  • Let {Xn; $n{\succeq}1$} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides conditions under which the series ${\sum}_{i{\preceq}n}\;Xi$ converges almost surely and the tail series {$Tn={\sum}_{i{\gg}n}\;X_i;n{\succeq}1$} satisfies $sup_{k{\succeq}n}{\parallel}T_k{\parallel}=\mathcal{O}p(b_n)$ and ${\frac{sup_{k{\succeq}n}{\parallel}T_k{\parallel}}{B_n}}{\rightarrow\limits^p}0$ for given fields of positive numbers {bn} and {Bn}. This result generalizes results of A. Rosalsky, J. Rosenblatt [7], [8] and S. H. Sung, A. I. Volodin [11].

A TECHNIQUE WITH DIMINISHING AND NON-SUMMABLE STEP-SIZE FOR MONOTONE INCLUSION PROBLEMS IN BANACH SPACES

  • Abubakar Adamu;Dilber Uzun Ozsahin;Abdulkarim Hassan Ibrahim;Pongsakorn Sunthrayuth
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.4
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    • pp.1051-1067
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    • 2023
  • In this paper, an algorithm for approximating zeros of sum of three monotone operators is introduced and its convergence properties are studied in the setting of 2-uniformly convex and uniformly smooth Banach spaces. Unlike the existing algorithms whose step-sizes usually depend on the knowledge of the operator norm or Lipschitz constant, a nice feature of the proposed algorithm is the fact that it requires only a diminishing and non-summable step-size to obtain strong convergence of the iterates to a solution of the problem. Finally, the proposed algorithm is implemented in the setting of a classical Banach space to support the theory established.

BROYDEN'S METHOD FOR OPERATORS WITH REGULARLY CONTINUOUS DIVIDED DIFFERENCES

  • Galperin, Anatoly M.
    • Journal of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.43-65
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    • 2015
  • We present a new convergence analysis of popular Broyden's method in the Banach/Hilbert space setting which is applicable to non-smooth operators. Moreover, we do not assume a priori solvability of the equation under consideration. Nevertheless, without these simplifying assumptions our convergence theorem implies existence of a solution and superlinear convergence of Broyden's iterations. To demonstrate practical merits of Broyden's method, we use it for numerical solution of three nontrivial infinite-dimensional problems.

Strong Convergence Theorems for Common Points of a Finite Family of Accretive Operators

  • Jeong, Jae Ug;Kim, Soo Hwan
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.445-464
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    • 2019
  • In this paper, we propose a new iterative algorithm generated by a finite family of accretive operators in a q-uniformly smooth Banach space. We prove the strong convergence of the proposed iterative algorithm. The results presented in this paper are interesting extensions and improvements of known results of Qin et al. [Fixed Point Theory Appl. 2014(2014): 166], Kim and Xu [Nonlinear Anal. 61(2005), 51-60] and Benavides et al. [Math. Nachr. 248(2003), 62-71].

STRONG CONVERGENCE OF HYBRID PROJECTION METHODS FOR QUASI-ϕ-NONEXPANSIVE MAPPINGS

  • Kang, Shin Min;Rhee, Jungsoo;Kwun, Young Chel
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.4
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    • pp.801-812
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    • 2010
  • In this paper, we consider the convergence of the shrinking projection method for quasi-$\phi$-nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property.

SENSITIVITY ANALYSIS FOR SYSTEM OF PARAMETRIC GENERALIZED QUASI-VARIATIONAL INCLUSIONS INVOLVING R-ACCRETIVE MAPPINGS

  • Kazmi, Kaleem Raza;Khan, Faizan Ahmad;Ahmad, Naeem
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1319-1338
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    • 2009
  • In this paper, using proximal-point mappings technique of Raccretive mappings and the property of the fixed point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of the system of parametric generalized quasi-variational inclusions involving R-accretive mappings in real uniformly smooth Banach space. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to parameters. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [3, 23, 24, 32, 33, 34].

STRONG AND WEAK CONVERGENCE OF THE ISHIKAWA ITERATION METHOD FOR A CLASS OF NONLINEAR EQUATIONS

  • Osilike, M.O.
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.153-169
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    • 2000
  • Let E be a real q-uniformly smooth Banach space which admits a weakly sequentially continuous duality map, and K a nonempty closed convex subset of E. Let T : K -> K be a mapping such that $F(T)\;=\;{x\;{\in}\;K\;:\;Tx\;=\;x}\;{\neq}\;0$ and (I - T) satisfies the accretive-type condition: $\;{\geq}\;{\lambda}$\mid$$\mid$x-Tx$\mid$$\mid$^2$, for all $x\;{\in}\;K,\;x^*\;{\in}\;F(T)$ and for some ${\lambda}\;>\;0$. The weak and strong convergence of the Ishikawa iteration method to a fixed point of T are investigated. An application of our results to the approximation of a solution of a certain linear operator equation is also given. Our results extend several important known results from the Mann iteration method to the Ishikawa iteration method. In particular, our results resolve in the affirmative an open problem posed by Naimpally and Singh (J. Math. Anal. Appl. 96 (1983), 437-446).

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