• Title, Summary, Keyword: variational

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GENERALIZED VECTOR MINTY'S LEMMA

  • Lee, Byung-Soo
    • The Pure and Applied Mathematics
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    • v.19 no.3
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    • pp.281-288
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    • 2012
  • In this paper, the author defines a new generalized ${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

Approximation Solvability for a System of Nonlinear Variational Type Inclusions in Banach Spaces

  • Salahuddin, Salahuddin
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.101-123
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    • 2019
  • In this paper, we consider a system of nonlinear variational type inclusions involving ($H,{\varphi},{\eta}$)-monotone operators in real Banach spaces. Further, we define a proximal operator associated with an ($H,{\varphi},{\eta}$)-monotone operator and show that it is single valued and Lipschitz continuous. Using proximal point operator techniques, we prove the existence and uniqueness of a solution and suggest an iterative algorithm for the system of nonlinear variational type inclusions. Furthermore, we discuss the convergence of the iterative sequences generated by the algorithms.

Design Sensitivity Analysis of Gate Valve Using the Variational Technology (변동 기법을 이용한 게이트 밸브의 설계민감도해석)

  • Kim, Se-Hun;Kim, Seung-Gyu;Jo, Young-Jik;Kang, Jung-Ho;Park, Young-Chul
    • Journal of the Korean Society of Manufacturing Process Engineers
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    • v.7 no.1
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    • pp.38-46
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    • 2008
  • Design technology and speciality production technology to manufacture high quality valve are insufficient in Korea. In order to design the experiments using Taguchi method and Variational Technology Also, from verification of the response model with optimized results was confirmed that usefulness and reliance of application Taguchi method and Variational Technology to structural's optimum design using Taguchi method and Variational Technology.

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A SYSTEM OF VARIATIONAL INCLUSIONS IN BANACH SPACES

  • Liu, Zeqing;Zhao, Liangshi;Hwang, Hong-Taek;Kang, Shin-Min
    • East Asian mathematical journal
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    • v.26 no.5
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    • pp.681-691
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    • 2010
  • A system of variational inclusions with (A, ${\eta}$, m)-accretive operators in real q-uniformly smooth Banach spaces is introduced. Using the resolvent operator technique associated with (A, ${\eta}$, m)-accretive operators, we prove the existence and uniqueness of solutions for this system of variational inclusions and propose a Mann type iterative algorithm for approximating the unique solution for the system of variational inclusions.

A HYBRID PROXIMAL POINT ALGORITHM AND STABILITY FOR SET-VALUED MIXED VARIATIONAL INCLUSIONS INVOLVING (A, ${\eta}$)-ACCRETIVE MAPPINGS

  • Kim, Jong-Kyu;Li, Hong Gang
    • East Asian mathematical journal
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    • v.26 no.5
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    • pp.703-714
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    • 2010
  • A new class of nonlinear set-valued mixed variational inclusions involving (A, ${\eta}$)-accretive mappings in Banach spaces is introduced and studied, which includes many kind of variational inclusion (inequality) and complementarity problems as special cases. By using the resolvent operator associated with (A, ${\eta}$)-accretive operator due to Lan-Cho-Verma, the existence of solution for this kind of variational inclusion is proved, and a new hybrid proximal point algorithm is established and suggested, the convergence and stability theorems of iterative sequences generated by new iterative algorithms are also given in q-uniformly smooth Banach spaces.

GENERAL NONLINEAR VARIATIONAL INCLUSIONS WITH H-MONOTONE OPERATOR IN HILBERT SPACES

  • Liu, Zeqing;Zheng, Pingping;Cai, Tao;Kang, Shin-Min
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.263-274
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    • 2010
  • In this paper, a new class of general nonlinear variational inclusions involving H-monotone is introduced and studied in Hilbert spaces. By applying the resolvent operator associated with H-monotone, we prove the existence and uniqueness theorems of solution for the general nonlinear variational inclusion, construct an iterative algorithm for computing approximation solution of the general nonlinear variational inclusion and discuss the convergence of the iterative sequence generated by the algorithm. The results presented in this paper improve and extend many known results in recent literatures.

TWO STEP ALGORITHM FOR SOLVING REGULARIZED GENERALIZED MIXED VARIATIONAL INEQUALITY PROBLEM

  • Kazmi, Kaleem Raza;Khan, Faizan Ahmad;Shahza, Mohammad
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.675-685
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    • 2010
  • In this paper, we consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly $\theta$-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al. [9]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problem. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly $\theta$-pseudomonotonicity. The theorems presented in this paper represent improvement and generalization of the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [13], Pang et al. [14], and Xia and Ding [19].

Variational Data Assimilation for Optimal Initial Conditions in Air Quality Modeling

  • Park, Seon-Ki
    • Journal of Korean Society for Atmospheric Environment
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    • v.19 no.E2
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    • pp.75-81
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    • 2003
  • Variational data assimilation, which is recently introduced to the air quality modeling, is a promising tool for obtaining optimal estimates of initial conditions and other important parameters such as emission and deposition rates. In this paper. two advanced techniques for variational data assimilation, based on the adjoint and quasi-inverse methods, are tested for a simple air quality problem. The four-dimensional variational assimilation (4D-Var) requires to run an adjoint model to provide the gradient information in an iterative minimization process, whereas the inverse 3D-Var (I3D-Var) seeks for optimal initial conditions directly by running a quasi -inverse model. For a process with small dissipation, I3D-Vu outperforms 4D-Var in both computing time and accuracy. Hybrid application which combines I3D-Var and standard 4D-Var is also suggested for efficient data assimilation in air quality problems.

FUZZY GENERAL NONLINEAR ORDERED RANDOM VARIATIONAL INEQUALITIES IN ORDERED BANACH SPACES

  • Salahuddin, Salahuddin;Lee, Byung-Soo
    • East Asian mathematical journal
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    • v.32 no.5
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    • pp.685-700
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    • 2016
  • The main object of this work to introduced and studied a new class of fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces. By using the random B-restricted accretive mapping with measurable mappings ${\alpha},{\alpha}^{\prime}:{\Omega}{\rightarrow}(0,1)$, an existence of random solutions for this class of fuzzy general nonlinear ordered random variational inequality (equation) with fuzzy mappings is established, a random approximation algorithm is suggested for fuzzy mappings, and the relation between the first value $x_0(t)$ and the random solutions of fuzzy general nonlinear ordered random variational inequality is discussed.