• Korean Journal of Mathematics
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• v.19 no.1
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• pp.101-109
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• 2011

We get a theorem which shows that there exist at least two or three nontrivial weak solutions for the nonlinear parabolic boundary value problem with the variable coefficient jumping nonlinearity. We prove this theorem by restricting ourselves to the real Hilbert space. We obtain this result by approaching the topological method. We use the Leray-Schauder degree theory on the real Hilbert space.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.813-837
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• 2022

In this paper we consider inverse problem for a general class of nonlinear stochastic differential equations on Hilbert spaces whose generating operators (drift, diffusion and jump kernels) are unknown. We introduce a class of function spaces and put a suitable topology on such spaces and prove existence of optimal generating operators from these spaces. We present also necessary conditions of optimality including an algorithm and its convergence whereby one can construct the optimal generators (drift, diffusion and jump kernel).

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.145-152
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• 2006

In this paper, we study a class of strongly nonlinear implicit complementarity problems in the setting of Hilbert spaces H (not necessarily Hilbert lattices). By using the property of the projection and a suitable change of variables, we establish the equivalence between the strongly nonlinear implicit complementarity problem and the fixed point problem in H. Moreover, we use this equivalence and the fixed point theorem of Boyd and Wong to prove the existence and uniqueness of solutions for the strongly nonlinear implicit complementarity problem in H.

• Kyungpook Mathematical Journal
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• v.62 no.4
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• pp.673-687
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• 2022

In this study, we solve the Cayley inclusion problem and the fixed point problem in real Hilbert space using Tseng's technique with inertial extrapolation in order to obtain more efficient results. We provide a strong convergence theorem to approximate a common solution to the Cayley inclusion problem and the fixed point problem under some appropriate assumptions. Finally, we present a numerical example that satisfies the problem and shows the computational performance of our suggested technique.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.533-555
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• 2022

In this paper, we introduce a new algorithm for finding a solution of an equilibrium problem in a real Hilbert space. Our paper extends the single projection method to pseudomonotone variational inequalities, from a 2018 paper of Shehu et. al., to pseudomonotone equilibrium problems in a real Hilbert space. On the basis of the given algorithm for the equilibrium problem, we develop a new algorithm for finding a common solution of a equilibrium problem and fixed point problem. The strong convergence of the algorithm is established under mild assumptions. Several of fundamental experiments in finite (infinite) spaces are provided to illustrate the numerical behavior of the algorithm for the equilibrium problem and to compare it with other algorithms.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.227-233
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• 2002

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation AX$\sub$i/=Y$\sub$i/, for i=1,2, ‥‥, R. In this article, we investigate Hilbert-Schmidt interpolation for operators in tridiagonal algebras.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1347-1359
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• 2017

The purpose of this paper is to give some new iterative methods for finding a common zero of a finite family of monotone operators in Hilbert spaces. We also give the applications of the obtained result for the convex feasibility problem and constrained convex optimization problem in Hilbert spaces.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.36 no.6
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• pp.665-672
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• 2012

Interfacial cracks bonded with dissimilar transversely isotropic piezoelectric media that are subjected to combined anti-plane mechanical and in-plane electrical loading are analyzed. The problem is formulated using complex function theory, from which the Hilbert problem is derived. By solving the Hilbert problem, the general form solution is obtained. Using this solution, closed-form solutions for one or two finite cracks as well as a semi-infinite crack are obtained, for the problem in which one concentrated mechanical and electrical load is imposed on the crack surface. This solution could be used as a Green's function to generate solutions to other problems with the same geometry but different loading conditions.

• Proceedings of the Korea Water Resources Association Conference
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• 2022.05a
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• pp.205-205
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• 2022

본 연구에서는 심해 선형파 조건에서 수중 투과성 방파제 주변의 유속장에 대해 nonhomogeneous Riemann-Hilbert problem을 이용한 해석해 및 수치해를 도출하고, 이를 반사계수와 투과계수를 산정하는 데에 활용한다. 여러 개의 얇은 투과성 판이 일렬로 배열되어 수중에 고정되어있고 규칙파가 작용하는 경우, Riemann-Hilbert problem을 정의할 수 있다. 본 연구에서는 얇은 판으로 이루어진 수중 방파제에 대한 homogeneous Riemann-Hilbert problem을 푸는 것을 넘어, 투과성 판으로 이루어진 수중 방파제에 대해 nonhomogeneous Riemann-Hilbert problem을 정의하고, 이에 대해 무한경계조건과 판 근처에서의 유속장 경계조건을 이용해 해석해를 유도하였다. 투과성 방파제의 경우 permeable boundary를 가지므로 제시한 상황은 기하학적 비선형성을 지닌다. 이에 대해 투수성을 기초로 미소 매개변수를 정의하고, 섭동법(perturbation method)을 이용해 유속장에 대한 leading order solution과 first order solution을 도출하였다. Leading order solution은 Evans (1970) 등의 선행연구에서 제시한 해와의 비교를 통해 그 타당성을 검증하였고, First order solution을 이용해 반사계수와 투과계수를 산정하여 방파제의 투수성이 유속장에 미치는 영향을 고려하였다. 아울러 수치해를 도출하여 해석해의 결과와 비교 및 분석하였다. 본 연구에서 제시한 해석해는 방파제에 가해지는 힘을 산정하는 등 다양한 방향으로 활용 가능하며, 향후 수치해나 실험값을 비교, 검증하기 위한 기초 자료로써 활용될 수 있다.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.251-259
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• 2009

We investigate the multiple solutions for the nonlinear parabolic boundary value problem with jumping nonlinearity crossing two eigenvalues. We show the existence of at least four nontrivial periodic solutions for the parabolic boundary value problem. We restrict ourselves to the real Hilbert space and obtain this result by the geometry of the mapping.