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VERMA, RAM U. 한국산업정보응용수학회 2003 한국산업정보응용수학회 Vol.7 No.2
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*∈H such that g(x*), g(y*)∈K and < pT(y*) + g(x*) - g(y*), g(x) - g(x*) ≥ 0 ∀g(x)∈K and for p > 0 < ηT(x*) + g(y*) - g(x*), g(x) - g(y*) ≥ 0 ∀g(x)∈K and for η > 0, where T: H→H is a relaxed g-γ-r-cocoercive and g-μ-Lipschitz continuous nonlinear mapping on H and g: H→H is any mapping on H. In recent years general variational inequalities and their algorithmic applications 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.
Verma, Ram U. Korean Mathematical Society 2011 대한수학회논문집 Vol.26 No.4
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.
Verma, Ram U. Korean Mathematical Society 2010 대한수학회논문집 Vol.25 No.2
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.
APPROXIMATION-SOLVABILITY OF A CLASS OF A-MONOTONE VARIATIONAL INCLUSION PROBLEMS
RAM U. VERMA 한국산업응용수학회 2004 Journal of the Korean Society for Industrial and A Vol.8 No.1
First the notion of the A-monotonicity is applied to the approximation-solvability of a class of nonlinear variational inclusion problems, and then the convergence analysis is given based on a projection-like method. Results generalize nonlinear variational inclusions involving H-monotone mappings in the Hilbert space setting.
Ravi P. Agarwal,Ram U. Verma 영남수학회 2011 East Asian mathematical journal Vol.27 No.5
Abstract. Based on the A-maximal (m)-relaxed monotonicity frame-works, the approximation solvability of a general class of variational in-clusion problems using the relaxed proximal point algorithm is explored,while generalizing most of the investigations, especially of Xu (2002) on strong convergence of modied version of the relaxed proximal point al-gorithm, 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 con-vergence results on proximal point algorithms in real Hilbert space set-tings. Furthermore, the main result has been applied to the context of the H-maximal monotonicity frameworks for solving a general class of vari-ational inclusion problems. It seems the obtained results can be used to generalize the Yosida approximation that, in turn, can be applied to rst-order evolution inclusions, and can also be applied to Douglas-Rachford splitting methods for nding the zero of the sum of two A-maximal (m)-relaxed monotone mappings.
Agarwal, Ravi P.,Verma, Ram U. The Youngnam Mathematical Society Korea 2011 East Asian mathematical journal Vol.27 No.5
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.