• Title, Summary, Keyword: variational methods

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GENERALIZED SYSTEMS OF RELAXED $g-{\gamma}-r-COCOERCIVE$ NONLINEAR VARIATIONAL INEQUALITIES AND PROJECTION METHODS

  • Verma, Ram U.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.7 no.2
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    • pp.83-94
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    • 2003
  • Let K be a nonempty closed convex subset of a real Hilbert space H. Approximation solvability of a system of nonlinear variational inequality (SNVI) problems, based on the convergence of projection methods, is given as follows: find elements $x^*,\;y^*{\in}H$ such that $g(x^*),\;g(y^*){\in}K$ and $$<\;{\rho}T(y^*)+g(x^*)-g(y^*),\;g(x)-g(x^*)\;{\geq}\;0\;{\forall}\;g(x){\in}K\;and\;for\;{\rho}>0$$ $$<\;{\eta}T(x^*)+g(y^*)-g(x^*),\;g(x)-g(y^*)\;{\geq}\;0\;{\forall}g(x){\in}K\;and\;for\;{\eta}>0,$$ where T: $H\;{\rightarrow}\;H$ is a relaxed $g-{\gamma}-r-cocoercive$ and $g-{\mu}-Lipschitz$ continuous nonlinear mapping on H and g: $H{\rightarrow}\;H$ is any mapping on H. In recent years general variational inequalities and their algorithmic have assumed a central role in the theory of variational methods. This two-step system for nonlinear variational inequalities offers a great promise and more new challenges to the existing theory of general variational inequalities in terms of applications to problems arising from other closely related fields, such as complementarity problems, control and optimizations, and mathematical programming.

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MIXED QUASI VARIATIONAL INEQUALITIES INVOLVING FOUR NONLINEAR OPERATORS

  • Pervez, Amjad;Khan, Awais Gul;Noor, Muhammad Aslam;Noor, Khalida Inayat
    • Honam Mathematical Journal
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    • v.42 no.1
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    • pp.17-35
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    • 2020
  • In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the fixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

  • Lee, Byung-Soo
    • East Asian mathematical journal
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    • v.26 no.3
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    • pp.415-421
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    • 2010
  • Many kinds of Minty's lemmas show that Minty-type variational inequality problems are very closely related to Stampacchia-type variational inequality problems. Particularly, Minty-type vector variational inequality problems are deeply connected with vector optimization problems. Liu et al. [10] considered vector variational inequalities for setvalued mappings by using scalarization approaches considered by Konnov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vector variational inequalities by using four kinds of Stampacchia-type scalar variational inequalities and obtain the relations of the solution sets between the six variational inequalities, which are more generalized results than those considered in [10]. In this paper, the author considers the Minty-type case corresponding to the Stampacchia-type case considered in [9].

APPROXIMATION OF SOLUTIONS OF A GENERALIZED VARIATIONAL INEQUALITY PROBLEM BASED ON ITERATIVE METHODS

  • Cho, Sun-Young
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.207-214
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    • 2010
  • In this paper, a generalized variational inequality problem is considered. An iterative method is studied for approximating a solution of the generalized variational inequality problem. Strong convergence theorem are established in a real Hilbert space.

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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    • v.24 no.1
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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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CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF GENERALIZED VARIATIONAL INEQUALITIES

  • Jeong, Jae Ug
    • Korean Journal of Mathematics
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    • v.21 no.3
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    • pp.213-222
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    • 2013
  • In this paper, we introduce and consider a new system of generalized variational inequalities involving five different operators. Using the sunny nonexpansive retraction technique we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

PROXIMAL POINTS METHODS FOR GENERALIZED IMPLICIT VARIATIONAL-LIKE INCLUSIONS IN BANACH SPACES

  • He, Xin-Feng;Lou, Jian;He, Zhen
    • East Asian mathematical journal
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    • v.28 no.1
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    • pp.37-47
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    • 2012
  • In this paper, we study generalized implicit variational-like inclusions and $J^{\eta}$-proximal operator equations in Banach spaces. It is established that generalized implicit variational-like inclusions in real Banach spaces are equivalent to fixed point problems. We also establish relationship between generalized implicit variational-like inclusions and $J^{\eta}$-proximal operator equations. This equivalence is used to suggest a iterative algorithm for solving $J^{\eta}$-proximal operator equations.

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.

THE USE OF ITERATIVE METHODS FOR SOLVING NAVEIR-STOKES EQUATION

  • Behzadi, Shadan Sadigh;Fariborzi Araghi, Mohammad Ali
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.381-394
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    • 2011
  • In this paper, a Naveir-Stokes equation is solved by using the Adomian's decomposition method (ADM), modified Adomian's decomposition method (MADM), variational iteration method (VIM), modified variational iteration method (MVIM), modified homotopy perturbation method (MHPM) and homotopy analysis method (HAM). The approximate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods.