• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.739-756
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• 2005

The paper presents new collectively fixed point theorems, minimax and quasi-variational inequalities for maps in the S-KKM class.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.491-502
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• 1998

We obtain new fixed point theorems on maps defined on "locally G-convex" subsets of a generalized convex spaces. Our first theorem is a Schauder-Tychonoff type generalization of the Brouwer fixed point theorem for a G-convex space, and the second main result is a fixed point theorem for the Kakutani maps. Our results extend many known generalizations of the Brouwer theorem, and are based on the Knaster-Kuratowski-Mazurkiewicz theorem. From these results, we deduce new results on collectively fixed points, intersection theorems for sets with convex sections and quasi-equilibrium theorems.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.305-320
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• 2013

Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..

• Journal of the Computational Structural Engineering Institute of Korea
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• v.28 no.3
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• pp.301-307
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• 2015

A bond-based peridynamic model has been shown to be capable of analyzing many of dynamic brittle fracture phenomena. However, there have been issued limitations on handling constitutive models of various materials. Especially, it assumes bonds act independently of each other, so that Poisson's ratio for 3D model is fixed as 1/4 as well as taking only account the bond stretching results in a volume change not a shear change. In this paper a state-based peridynamic model of dynamic brittle fracture is presented. The state-based peridynamic model is a generalized peridynamic model that is able to directly use a constitutive model from the standard theory. It permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. Thus, the volume and shear changes of the material can be reproduced by the state-based peridynamic theory. For a linearly elastic solid, a plane stress model is introduced and the damage model suitable for the state-based peridynamic model is discussed. Through a convergence study under decreasing the peridynamic nonlocal region($\delta$-convergence), the dynamic fracture model is verified. It is also shown that the state-based peridynamic model is reliable for modeling dynamic crack propagatoin.