• East Asian mathematical journal
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• 제27권5호
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• pp.545-555
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• 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.

• 대한수학회보
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• 제40권3호
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• pp.385-397
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• 2003

In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{*},\;y^{*},\;z^{*}\;\in\;E$ such that ${\theta}\;{\in}\;{\alpha}T(y^{*})\;+\;g(x^{*})\;-\;g(y^{*})\;+\;A(g(x^{*}))\;\;\;for\;{\alpha}\;>\;0,\;{\theta}\;{\in}\;{\beta}T(z^{*})\;+\;g(y^{*})\;-\;g(z^{*})\;+\;A(g(y^{*}))\;\;\;for\;{\beta}\;>\;0,\;{\theta}\;{\in}\;{\gamma}T(x^{*})\;+\;g(z^{*})\;-\;g(x^{*})\;+\;A(g(z^{*}))\;\;\;for\;{\gamma}\;>\;0,$ where T, g : $E\;{\rightarrow}\;E,\;{\theta}$ is zero element in Banach space E, and A : $E\;{\rightarrow}\;{2^E}$ be m-accretive mapping. By using resolvent operator technique for n-secretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in q-uniformly smooth Banach spaces and in real Banach spaces, respectively.

• 전자공학회논문지S
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• 제35S권10호
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• pp.85-93
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• 1998

선형 시스템의 출력 조절 문제에 있어서, 출력 조절가능성은 일반적으로 보존되지 않는다. 따라서, 연속시간 선형시스템에 대해 설계된 아날로그 조절기를 샘플링하여 구현한 디지털 출력조절기는, 샘플링 시간에 대해 1차 근사적인 근사 샘플치 조절기에 불과하다. 그런데, 주어진 시스템의 내재적 구조를 잘 이용하면, 샘플링 시간에 대해 개선된 근사 샘플치 출력 조절기 설계가 가능하다. 본 논문은 샘플링 시간에 대해 더 개선된 근사 샘플치 출력 조절기를 설계하기 위한 시스템의 구조를 연구하였다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제7권2호
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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.