• 대한수학회보
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• 제51권4호
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• pp.1041-1054
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• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.

• 대한수학회지
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• 제33권2호
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• pp.291-307
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• 1996

In this paper, we shall show the existence of solutions of the following nonlinear partial differential equation $$ {^{divA(-\Delta u) = f(x, u, \Delta u) in \Omega}^{u = 0 on \partial\Omega} $$ where $f(x, u, \Delta u) = -u$\mid$\Delta u$\mid$^{p-2} + h, p \geq 2, h \in L^\infty$.

• 호남수학학술지
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• 제40권4호
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• pp.771-783
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• 2018

This paper, we investigate a strongly damped nonlinear Klein-Gordon equation with nonlinearities of variable exponent type $$u_{tt}-{\Delta}u-{\Delta}u_t+m^2u+{\mid}u_t{\mid}^{p(x)-2}u_t={\mid}u{\mid}^{q(x)-2}u$$ associated with initial and Dirichlet boundary conditions in a bounded domain. We obtain a nonexistence of solutions if variable exponents p (.), q (.) and initial data satisfy some conditions.

• 대한수학회보
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• 제53권6호
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

• 자원리싸이클링
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• 제22권3호
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• pp.57-64
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• 2013

희토류산화물인 $Eu_2O_3$, 우라늄산화물인 $UO_2$ 및 $U_3O_8$, $Eu_2O_3$와 우라늄산화물의 혼합물에 대한 선택적 황화반응을 조사한 후에, $(U,Eu)O_2$ 및 $(U,Eu)_3O_8$와 같은 Eu 고용 우라늄산화물, Eu 고용 우라늄산화물의 고온 산화열처리 상분리 생성물인 Eu 농도가 높은 $(U,Eu)_4O_9$와 $U_3O_8$의 혼합상에 대한 황화반응 특성을 $400-800^{\circ}C$에서 조사하였다. $Eu_2O_3$ 및 우라늄산화물의 혼합물의 경우에는 $450^{\circ}C$에서 Eu와 우라늄 산화물간의 반응이 없이 $Eu_2O_3$만 $Eu_3S_4$로 전환되었다. $(U,Eu)_3O_8$ 및 $(U,Eu)O_2$에서는 반응온도 $600^{\circ}C$까지는 우라늄산화물과 동일한 황화반응 거동을 보였으며, $800^{\circ}C$에서는 Eu 농도가 높은 $(U,Eu)S_x$과 ${\alpha}-US_2$ 상이 생성되었다. 고온 산화열처리 상분리 생성물은 $600^{\circ}C$에서 $(U,Eu)S_x$과 UOS 상이 생성되었다. 상분리 생성물을 환원하여 얻은 Eu 농도가 높은 $(U,Eu)O_2$와 $UO_2$의 혼합상은 $450^{\circ}C$에서 $(U,Eu)O_2$은 산황화물인 (U,Eu)OS로 전환되고 $UO_2$는 반응하지 않았다.

• 대한수학회논문집
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• 제16권2호
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• pp.287-290
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• 2001

Let X$_1$, X$_2$, … be a sequence of independent and identically distributed random variables with continuous cumulative distribution function F(x). X(sub)j is an upper record value of this sequence if X(sub)j > max {X$_1$, X$_2$, …, X(sub)j-1}. We define u(n) = min {j│j > u(n-1), X(sub)j > X(sub)u(n-1), n $\geq$ 2} with u(1) = 1. Then F(x) = 1 - e(sup)-x/c, x > 0 if and only if E[X(sub)n(n+1) - X(sub)u(n)│X(sub)u(m) = y] = c or E[X(sub)u(n+2) - X(sub)u(n)│X(sub)u(m) = y] = 2c, n $\geq$ m+1.

• 대한수학회지
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• 제57권6호
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• pp.1551-1571
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• 2020

This paper is concerned with the following Kirchhoff-Schrödinger-Poisson system $$\begin{cases} -(a+b{\displaystyle\smashmargin{2}\int\nolimits_{\mathbb{R}^3}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+{\mu}{\phi}u={\lambda}f(x){\mid}u{\mid}^{p-2}u+g(x){\mid}u{\mid}^{p-2}u,&{\text{ in }}{\mathbb{R}}^3,\\-{\Delta}{\phi}={\mu}{\mid}u{\mid}^2,&{\text{ in }}{\mathbb{R}}^3, \end{cases}$$ where a > 0, b, µ ≥ 0, p ∈ (1, 2), q ∈ [4, 6) and λ > 0 is a parameter. Under some suitable assumptions on V (x), f(x) and g(x), we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.

• 대한수학회지
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• 제51권1호
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• pp.67-86
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• 2014

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on ${\Omega}{\subset}R^d$, $1{\leq}d{\leq}3$. We construct the semidiscrete approximations of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u in $L^2$ normed space.

• Korean Journal of Mathematics
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• 제17권2호
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• pp.197-210
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• 2009

We prove the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the elliptic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}A{\xi}+g_1({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\\A{\xi}+g_2({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\end{array}$$ where $lim_{u{\rightarrow}{\infty}}\frac{gj(u)}{u}={\beta}_j$, $lim_{u{\rightarrow}-{\infty}}\frac{gj(u)}{u}={\alpha}_j$ are finite and the nonlinearity $g_1+2g_2$ crosses eigenvalues of A.

• Korean Journal of Mathematics
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• 제21권2호
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• pp.203-212
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• 2013

We investigate the existence of solutions of the one-dimensional wave system $$u_{tt}-u_{xx}+{\mu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\v_{tt}-v_{xx}+{\nu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ with Dirichlet boundary condition. We find them by applying linking inequlaities.