• 제목/요약/키워드: critical point theorem

검색결과 55건 처리시간 0.02초

THE EXISTENCE OF THE SOLUTION OF ELLIPTIC SYSTEM APPLYING TWO CRITICAL POINT THEOREM

  • Nam, Hyewon
    • 충청수학회지
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    • 제31권1호
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    • pp.53-64
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    • 2018
  • This paper deals with the study of solutions for the elliptic system with jumping nonlineartity and growth nonlinearity and Dirichlet boundary condition. We apply the two critical point theorem when proving the existence of nontrivial solutions for the elliptic system. We define the energy functional associated to the elliptic system and prove that the functional has two critical values.

NONTRIVIAL PERIODIC SOLUTION FOR THE SUPERQUADRATIC PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • 제17권1호
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    • pp.53-66
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    • 2009
  • We show the existence of a nontrivial periodic solution for the superquadratic parabolic equation with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear term at infinity which have continuous derivatives. We use the critical point theory on the real Hilbert space $L_2({\Omega}{\times}(0 2{\pi}))$. We also use the variational linking theorem which is a generalization of the mountain pass theorem.

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EXISTENCE OF SIX SOLUTIONS OF THE NONLINEAR HAMILTONIAN SYSTEM

  • Jung, Tack-Sun;Choi, Q-Heung
    • 호남수학학술지
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    • 제30권3호
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    • pp.443-468
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    • 2008
  • We give a theorem of existence of six nontrivial solutions of the nonlinear Hamiltonian system $\.{z}$ = $J(H_z(t,z))$. For the proof of the theorem we use the critical point theory induced from the limit relative category of the torus with three holes and the finite dimensional reduction method.

'피타고라스 정리'의 명칭과 활용에 대한 비판적 고찰 (Rethinking the Name and Use of Pythagorean Theorem from the Perspectives of History of Mathematics and Mathematics Education)

  • 장혜원
    • 한국수학사학회지
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    • 제34권6호
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    • pp.205-223
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    • 2021
  • It has been argued that as for the origin of the Pythagorean theorem, the theorem had already been discovered and proved before Pythagoras, and the historical records of ancient mathematics have confirmed various uses of this theorem. The purpose of this study is to examine the relevance of its name caused by Eurocentrism and the weakness of its use in Korean school mathematics and to seek improvements from a critical point of view. To this end, the Pythagorean theorem was reviewed from the perspectives of the history of mathematics and mathematics education. In addition, its name in relation to objective mathematical contents regardless of any specific civilization and its use as a starting point for teaching the theorem in school mathematics were suggested.

MULTIPLICITY RESULTS AND THE M-PAIRS OF TORUS-SPHERE VARIATIONAL LINKS OF THE STRONGLY INDEFINITE FUNCTIONAL

  • Jung, Tack-Sun;Choi, Q-Heung
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제12권4호
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    • pp.239-247
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    • 2008
  • Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies two pairs of Torus-Sphere variational linking inequalities and when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities. We show that I has at least four critical points when I satisfies two pairs of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. Moreover we show that I has at least 2m critical points when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities with $(P.S.)^*_c$ condition. We prove these results by Theorem 2.2 (Theorem 1.1 in [1]) and the critical point theory on the manifold with boundary.

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APPLICATIONS OF CRITICAL POINT THEOREMS TO NONLINEAR BEAM PROBLEMS

  • Choi, Q-Heung;Jin, Ying-Hua;Choi, Kyung-Pyo
    • 호남수학학술지
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    • 제29권1호
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    • pp.19-40
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    • 2007
  • Let L be the differential operator, Lu = $u_{tt}+u_{xxxx}$. We consider nonlinear beam equations, Lu + $bu^+$ = j, in H, where H is the Hilbert space spanned by eigenfunctions of L. We reveal the existence of multiple solutions of the nonlinear beam problems by critical point theorems.

ASYMPTOTICALLY LINEAR BEAM EQUATION AND REDUCTION METHOD

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • 제19권4호
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    • pp.481-493
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    • 2011
  • We prove a theorem which shows the existence of at least three ${\pi}$-periodic solutions of the wave equation with asymptotical linearity. We obtain this result by the finite dimensional reduction method which reduces the critical point results of the infinite dimensional space to those of the finite dimensional subspace. We also use the critical point theory and the variational method.