• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.463-472
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• 2011

In this paper we study the solvability of parabolic equations governed by the difference of time dependent subdifferential and time independent subdifferential in reflexive Banach spaces.

• East Asian mathematical journal
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• v.14 no.2
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• pp.299-309
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• 1998

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.335-341
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• 2009

In this paper, a system of nonsmooth equations reformulated from a nonlinear complementarity problem with nonsmooth data is studied. The formulas of some subdifferentials for related functions in this system of nonsmooth equations are developed. The present work can be applied to Newton methods for solving this kind of nonlinear complementarity problem.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.371-381
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• 1997

Our basic concern in this paper is to investigate some geometric properties of the graph of a maximal monotone operator in the one dimensional case. Using a well-known theorem of Minty, we answer S. Simon's questions affirmatively in the one dimensional case. Further developments of these results are also treated. In addition, we provide a new proof of Rockafellar's characterization of maximal monotone operators on R: every maximal monotne operator from R to $2^R$ is the subdifferential of a proper convex lower semicontinuous function.

• East Asian mathematical journal
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• v.24 no.2
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• pp.177-186
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• 2008

In this paper, we introduce and study a new class of system of parametric general quasivariational-like inequalities. Using $\eta$-subdifferential and $\eta$-proximal mappings of proper functionals in Hilbert spaces, we prove the equivalence between the system of parametric general quasivariational-like inequalities and a xed point problem and construct two iterative algorithms. A few existence and uniqueness results as well as the sensitivity analysis of solutions are also established for the system of parametric general quasivariational-like inequalities, and some convergence results of iterative sequence generated by the algorithms are proved. Our results extend a few known results in the literature.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.271-276
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• 2014

Recently, Taji [7] and Harms et al. [4] studied the regularized gap function of QVI analogous to that of VI by Fukushima [2]. Discussions are made in a finite dimensional Euclidean space. In this note, an infinite dimensional generalization is considered in the framework of a reflexive Banach space. To do so, we introduce an extended quasi-variational inequality problem (in short, EQVI) and a generalized regularized gap function of EQVI. Then we investigate some basic properties of it. Our results may be regarded as an infinite dimensional extension of corresponding results due to Taji [7].

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.145-150
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• 2015

Aussel et al. [1] introduced the notion of inverse quasi-variational inequalities (IQVI) by combining quasi-variational inequalities and inverse variational inequalities. Discussions are made in a finite dimensional Euclidean space. In this note, we develop an infinite dimensional version of IQVI by investigating some basic properties of the regularized gap function of IQVI in a Banach space.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.647-665
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• 2004

The existence of solutions for the nonlinear functional differential equation with nonlocal conditions governed by the variational inequality is studied. The regularity and a variation of solutions of the equation are also given.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.881-891
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• 2012

In this paper we investigate the approximate controllability for the following nonlinear functional differential control problem: $$x^{\prime}(t)+Ax(t)+{\partial}{\phi}(x(t)){\ni}f(t,x(t))+h(t)$$ which is governed by the variational inequality problem with nonlinear terms.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.173-181
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• 2012

In this paper, we study the control problems governed by the semilinear parabolic type equation in Hilbert spaces. Under the Lipschitz continuity condition of the nonlinear term, we can obtain the sufficient conditions for the approximate controllability of nonlinear functional equations with nonlinear monotone hemicontinuous and coercive operator. The existence, uniqueness and a variation of solutions of the system are also given.