• Title, Summary, Keyword: relaxed Lipschitz mapping

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GENERALIZED RELAXED PROXIMAL POINT ALGORITHMS INVOLVING RELATIVE MAXIMAL ACCRETIVE MODELS WITH APPLICATIONS IN BANACH SPACES

  • Verma, Ram U.
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.313-325
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    • 2010
  • General models for the relaxed proximal point algorithm using the notion of relative maximal accretiveness (RMA) are developed, and then the convergence analysis for these models in the context of solving a general class of nonlinear inclusion problems differs significantly than that of Rockafellar (1976), where the local Lipschitz continuity at zero is adopted instead. Moreover, our approach not only generalizes convergence results to real Banach space settings, but also provides a suitable alternative to other problems arising from other related fields.

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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GENERAL VARIATIONAL INCLUSIONS AND GENERAL RESOLVENT EQUATIONS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.241-256
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    • 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.

ON GENERALIZED NONLINEAR QUASI-VARIATIONAL-LIKE INCLUSIONS DEALING WITH (h,η)-PROXIMAL MAPPING

  • Liu, Zeqing;Chen, Zhengsheng;Shim, Soo-Hak;Kang, Shin-Min
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1323-1339
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    • 2008
  • In this paper, a new class of $(h,{\eta})$-proximal for proper functionals in Hilbert spaces is introduced. The existence and Lip-schitz continuity of the $(h,{\eta})$-proximal mappings for proper functionals are proved. A class of generalized nonlinear quasi-variational-like inclusions in Hilbert spaces is introduced. A perturbed three-step iterative algorithm with errors for the generalized nonlinear quasi-variational-like inclusion is suggested. The existence and uniqueness theorems of solution for the generalized nonlinear quasi-variational-like inclusion are established. The convergence and stability results of iterative sequence generated by the perturbed three-step iterative algorithm with errors are discussed.

GENERALIZED MULTIVALUED QUASIVARIATIONAL INCLUSIONS FOR FUZZY MAPPINGS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • The Pure and Applied Mathematics
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    • v.14 no.1
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    • pp.37-48
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    • 2007
  • In this paper, we introduce and study a class of generalized multivalued quasivariational inclusions for fuzzy mappings, and establish its equivalence with a class of fuzzy fixed-point problems by using the resolvent operator technique. We suggest a new iterative algorithm for the generalized multivalued quasivariational inclusions. Further, we establish a few existence results of solutions for the generalized multivalued quasivariational inclusions involving $F_r$-relaxed Lipschitz and $F_r$-strongly monotone mappings, and discuss the convergence criteria for the algorithm.

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