• Title, Summary, Keyword: general variational inclusion

Search Result 13, Processing Time 0.038 seconds

VARIATIONAL-LIKE INCLUSION SYSTEMS VIA GENERAL MONOTONE OPERATORS WITH CONVERGENCE ANALYSIS

  • Dadashi, Vahid;Roohi, Mehdi
    • East Asian mathematical journal
    • /
    • v.26 no.1
    • /
    • pp.95-103
    • /
    • 2010
  • In this paper using Lipschitz continuity of the resolvent operator associated with general H-maximal m-relaxed $\eta$-monotone operators, existence and uniqueness of the solution of a variational inclusion system is proved. Also, an iterative algorithm and its convergence analysis is given.

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
    • /
    • v.47 no.2
    • /
    • pp.263-274
    • /
    • 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.

GENERAL VARIATIONAL INCLUSIONS AND GENERAL RESOLVENT EQUATIONS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • Bulletin of the Korean Mathematical Society
    • /
    • v.41 no.2
    • /
    • pp.241-256
    • /
    • 2004
  • In this paper, we introduce and study a new class of variational inclusions, called the general variational inclusion. We prove the equivalence between the general variational inclusions, the general resolvent equations, and the fixed-point problems, using the resolvent operator technique. This equivalence is used to suggest and analyze a few iterative algorithms for solving the general variational inclusions and the general resolvent equations. Under certain conditions, the convergence analyses are also studied. The results presented in this paper generalize, improve and unify a number of recent results.

A RESOLVENT APPROACH FOR SOLVING A SET-VALUED VARIATIONAL INCLUSION PROBLEM USING WEAK-RRD SET-VALUED MAPPING

  • Ahmad, Iqbal;Ahmad, Rais;Iqbal, Javid
    • Korean Journal of Mathematics
    • /
    • v.24 no.2
    • /
    • pp.199-213
    • /
    • 2016
  • The resolvent operator approach of [2] is applied to solve a set-valued variational inclusion problem in ordered Hilbert spaces. The resolvent operator under consideration is called relaxed resolvent operator and we demonstrate some of its properties. To obtain the solution of a set-valued variational inclusion problem, an iterative algorithm is developed and weak-RRD set-valued mapping is used. The problem as well as main result of this paper are more general than many previous problems and results available in the literature.

STRONG CONVERGENCE OF EXTENDED GENERAL VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS

  • Chen, Jun-Min;Zhang, Li-Juan;He, Zhen
    • East Asian mathematical journal
    • /
    • v.26 no.1
    • /
    • pp.59-67
    • /
    • 2010
  • In this paper, we suggest and analyze some three step iterative scheme for finding the common elements of the set of the solutions of the extended general variational inequalities involving three operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of suggested iterative schemes under some mild conditions. Since the extended general variational inequalities include general variational inequalities and several other classes of variational inequalities as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.

GENERAL FRAMEWORK FOR PROXIMAL POINT ALGORITHMS ON (A, η)-MAXIMAL MONOTONICIT FOR NONLINEAR VARIATIONAL INCLUSIONS

  • Verma, Ram U.
    • Communications of the Korean Mathematical Society
    • /
    • v.26 no.4
    • /
    • pp.685-693
    • /
    • 2011
  • General framework for proximal point algorithms based on the notion of (A, ${\eta}$)-maximal monotonicity (also referred to as (A, ${\eta}$)-monotonicity in literature) is developed. Linear convergence analysis for this class of algorithms to the context of solving a general class of nonlinear variational inclusion problems is successfully achieved along with some results on the generalized resolvent corresponding to (A, ${\eta}$)-monotonicity. The obtained results generalize and unify a wide range of investigations readily available in literature.

RELAXED PROXIMAL POINT ALGORITHMS BASED ON A-AXIMAL RELAXED MONOTONICITY FRAMEWORKS WITH APPLICATIONS

  • Agarwal, Ravi P.;Verma, Ram U.
    • East Asian mathematical journal
    • /
    • v.27 no.5
    • /
    • pp.545-555
    • /
    • 2011
  • Based on the A-maximal(m)-relaxed monotonicity frameworks, the approximation solvability of a general class of variational inclusion problems using the relaxed proximal point algorithm is explored, while generalizing most of the investigations, especially of Xu (2002) on strong convergence of modified version of the relaxed proximal point algorithm, Eckstein and Bertsekas (1992) on weak convergence using the relaxed proximal point algorithm to the context of the Douglas-Rachford splitting method, and Rockafellar (1976) on weak as well as strong convergence results on proximal point algorithms in real Hilbert space settings. Furthermore, the main result has been applied to the context of the H-maximal monotonicity frameworks for solving a general class of variational inclusion problems. It seems the obtained results can be used to generalize the Yosida approximation that, in turn, can be applied to first- order evolution inclusions, and can also be applied to Douglas-Rachford splitting methods for finding the zero of the sum of two A-maximal (m)-relaxed monotone mappings.

A GENERAL ITERATIVE METHOD BASED ON THE HYBRID STEEPEST DESCENT SCHEME FOR VARIATIONAL INCLUSIONS, EQUILIBRIUM PROBLEMS

  • Tian, Ming;Lan, Yun Di
    • Journal of applied mathematics & informatics
    • /
    • v.29 no.3_4
    • /
    • pp.603-619
    • /
    • 2011
  • To the best of our knowledge, it would probably be the first time in the literature that we clarify the relationship between Yamada's method and viscosity iteration correctly. We design iterative methods based on the hybrid steepest descent algorithms for solving variational inclusions, equilibrium problems. Our results unify, extend and improve the corresponding results given by many others.

A PROXIMAL POINT ALGORITHM FOR SOLVING THE GENERAL VARIATIONAL INCLUSIONS WITH M(·, ·)-MONOTONE OPERATORS IN BANACH SPACES

  • Chen, Junmin;Wang, Xian;He, Zhen
    • East Asian mathematical journal
    • /
    • v.29 no.3
    • /
    • pp.315-326
    • /
    • 2013
  • In this paper, a new monotonicity, $M({\cdot},{\cdot})$-monotonicity, is introduced in Banach spaces, and the resolvent operator of an $M({\cdot},{\cdot})$-monotone operator is proved to be single valued and Lipschitz continuous. By using the resolvent operator technique associated with $M({\cdot},{\cdot})$-monotone operators, we construct a proximal point algorithm for solving a class of variational inclusions. And we prove the convergence of the sequences generated by the proximal point algorithms in Banach spaces. The results in this paper extend and improve some known results in the literature.

A Mixed Variational Principle of Fully Anisotropic Linear Elasticity (이방성탄성문제의 혼합형변분원리)

  • 홍순조
    • Computational Structural Engineering
    • /
    • v.4 no.2
    • /
    • pp.87-94
    • /
    • 1991
  • In this paper, a mixed variational principle applicable to the linear elasticity of inhomogeneous anisotropic materials is presented. For derivation of the general variational principle, a systematic procedure for the variational formulation of linear coupled boundary value problems developed by Sandhu et al. is employed. Consistency condition of the field operators with the boundary operators results in explicit inclusion of boundary conditions in the governing functional. Extensions of admissible state function spaces and specialization to a certain relation in the general governing functional lead to the desired mixed variational principle. In the physical sense, the present variational principle is analogous to the Reissner's recent formulation obtained by applying Lagrange multiplier technique followed by partial Legendre transform to the classical minimum potential energy principle. However, the present one is more advantageous for the application to the general anisotropic materials since Reissner's principle contains an implicit function which is not easily converted to an explicit form.

  • PDF