• Journal of the Chungcheong Mathematical Society
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• v.20 no.1
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• pp.81-95
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• 2007

We are concerned with the multiplicity of solutions of the nonlinear biharmonic equation with Dirichlet boundary condition, ${\Delta}^2u+c{\Delta}u=g(u)$, in ${\Omega}$, where $c{\in}R$ and ${\Delta}^2$ denotes the biharmonic operator. We show that there exists at least three solutions of the above problem under the suitable condition of g(u).

• Korean Journal of Mathematics
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• v.18 no.4
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• pp.473-487
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• 2010

We investigate the existence of multiple nontrivial solutions (${\xi}$, ${\eta}$) for perturbations $g_1$, $g_2$ of the harmonic system with Dirichlet boundary condition $${\Delta}^2{\xi}+c{\Delta}{\xi}=g_1(2{\xi}+3{\eta})\;in\;{\Omega}\\{\Delta}^2{\eta}+c{\Delta}{\eta}=g_2(2{\xi}+3{\eta})\;in\;{\Omega}$$ where we assume that ${\lambda}_1$ < $c$ < ${\lambda}_2$, $g^{\prime}_1({\infty})$, $g^{\prime}_2({\infty})$ are finite. We prove that the system has at least three solutions under some condition on $g$.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.419-428
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• 2009

We show the existence of at least one nontrivial solution of the homogeneous mixed type nonlinear elliptic problem. Here mixed type nonlinearity means that the nonlinear part contain the jumping nonlinearity and the critical growth nonlinearity. We first investigate the sub-level sets of the corresponding functional in the Soboles space and the linking inequalities of the functional on the sub-level sets. We next investigate that the functional I satisfies the mountain pass geometry in the critical point theory. We obtain the result by the mountain pass method, the critical point theory and variational method.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.311-319
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• 2009

Let H be a Hilbert space which is the direct sum of five closed subspaces $X_0,\;X_1,\;X_2,\;X_3$ and $X_4$ with $X_1,\;X_2,\;X_3$ of finite dimension. Let J be a $C^{1,1}$ functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies $(P.S.)^*_c$ condition and $f|X_0{\otimes}X_4$ has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.693-701
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• 2006

Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems $$(P)\{{\Delta}^2x(k-1)+g(k,y(k))=0,\;k{\in}[1,\;T],\;{\Delta}^2y(k-1)+f(k,\;x(k)=0,\;k{\in}[1,\;T],\;x(0)=y(0)=0,\;x(T+1)=y(T+1)=0$$ where T is a positive integer, [1, T] is the discrete interval {1, 2,..., T}, ${\Delat}x(k)=x(k+1)-x(k)$ is the forward difference operator and ${\Delta}^2x(k)={\Delta}({\Delta}x(k))$.

• ETRI Journal
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• v.9 no.1
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• pp.84-96
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• 1987

The equality relation is very important in mechanical theorem proving procedures. A proposed inference rule called RHU-resolution is intended to extend the hyperparamodulation[23, 9] by introducing a bidirectional proof search that simultaneously employs a forward reasoning and a backward reasoning, and generalize it by incorporating beneflts of extended hyper steps with a preprocessing process, that includes a subsumption check in an equality graph and a high level planning. The forward reasoning in RHU-resolution may replace the role of the function substitution link.[9] That is, RHU-deduction without the function substitution link gets a proof. In order to control explosive generation of positive equalities by the forward reasoning, we haue put some restrictions on input clauses and k-pd links, and also have included a control strategy for a positive-positive linkage, like the set-of-support concept, A linking path between two end terms can be found by simple checking of linked unifiability using the concept of a linked unification. We tried to prevent redundant resolvents from generating by preprocessing using a subsumption check in the subsumption based eauality graph(SPD-Graph)so that the search space for possible RHU-resolution may be reduced.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.19 no.4 s.74
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• pp.357-367
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• 2006

Drift design methods using resizing algorithms have been presented as a practical drift design method since the resizing algorithms proposed easily find drift contribution of each member, called member displacement participation factor, to lateral drift to be designed without calculation of sensitivity coefficient or re-analysis. Weight of material to be redistributed for minimization of the lateral drift is determined according to the member displacement participation factors. However, resizing algorithms based on energy theorem must consider loading conditions because they have different displacement contribution according to different loading conditions. Furthermore, to improve practicality of resizing algorithms, structural member grouping is required in application of resizing algorithms to drift control of high-rise buildings. In this study, three resizing algorithms on considering load condition and structural member grouping are developed and applied to drift design of a 20-story steel-frame shear-wall structure and a 50-story frame shear-wall system with outriggers.