• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.139-146
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• 2008

We investigate the uniqueness and multiplicity of solutions for the nonlinear elliptic system with Dirichlet boundary condition $$\{-{\Delta}u+g_1(u,v)=f_1(x){\text{ in }}{\Omega},\\-{\Delta}v+g_2(u,v)=f_2(x){\text{ in }}{\Omega},$$ where ${\Omega}$ is a bounded set in $R^n$ with smooth boundary ${\partial}{\Omega}$. Here $g_1$, $g_2$ are nonlinear functions of u, v and $f_1$, $f_2$ are source terms.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.261-267
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• 2007

We show the existence of the positive solution for the system of the following nonlinear wave equations with Dirichlet boundary conditions $$u_{tt}-u_{xx}+av^+=s{\phi}_{00}+f$$, $$v_{tt}-v_{xx}+bu^+=t{\phi}_{00}+g$$, $$u({\pm}\frac{\pi}{2},t)=v({\pm}\frac{\pi}{2},t)=0$$, where $u_+=max\{u,0\}$, s, $t{\in}R$, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}=1$ of the eigenvalue problem $u_{tt}-u_{xx}={\lambda}_{mn}u$ with $u({\pm}\frac{\pi}{2},t)=0$, $u(x,t+{\pi})=u(x,t)=u(-x,t)=u(x,-t)$ and f, g are ${\pi}$-periodic, even in x and t and bounded functions in $[-\frac{\pi}{2},\frac{\pi}{2}]{\times}[-\frac{\pi}{2},\frac{\pi}{2}]$ with $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}f{\phi}_{00}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}g{\phi}_{00}=0$.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.695-706
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• 1996

In the control system $ \dot{x} = f(t,x(t)) + g(t,x(t))\dot{u}, x(0) = \bar{x}, t \in [0,T], $ this paper shows that the map from u with $L^1(m)$-topology to $x_u$ with $L^1(\mu)$-topology is Lipschitz continuous where f is $C^1$, $\mu$ is the Stieltjes measure derived from the function g which is not smooth in the variable t and $x_u$ is the solution of the above system corresponding to u under the assumption that $\dot{u}$ is bounded.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.153-161
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• 2015

In this work, we consider an elliptic system $$\left{\array {-{\Delta}u=au+bv+{\delta}_1u+-{\delta}_2u^-+f_1(x,u,v) && in\;{\Omega},\\-{\Delta}v=bu+cv+{\eta}_1v^+-{\eta}_2v^-+f_2(x,u,v) && in\;{\Omega},\\{\hfill{70}}u=v=0{\hfill{90}}on\;{\partial}{\Omega},}$$, where ${\Omega}{\subset}R^N$ be a bounded domain with smooth boundary. We prove that the system has at least two nontrivial solutions by applying linking theorem.

• Korean Journal of Mathematics
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• v.21 no.2
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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.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.1-37
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• 2004

We consider the following system of discrete equations $u_i(\kappa)\;=\;{\Sigma{N}{\ell=0}}g_i({\kappa},\;{\ell})f_i(\ell,\;u_1(\ell),\;u_2(\ell),\;{\cdots}\;,\;u_n(\ell)),\;{\kappa}\;{\in}\;\{0,\;1,\;{\cdots}\;,\;T\},\;1\;{\leq}\;i\;{\leq}\;n\;where\;T\;{\geq}\;N\;>\;0,\;1\;{\leq}i\;{\leq}\;n$. Existence criteria for single, double and multiple constant-sign solutions of the system are established. To illustrate the generality of the results obtained, we include applications to several well known boundary value problems. The above system is also extended to that on $\{0,\;1,\;{\cdots}\;\}\;u_i(\kappa)\;=\;{\Sigma{\infty}{\ell=0}}g_i({\kappa},\;{\ell})f_i(\ell,\;u_1(\ell),\;u_2(\ell),\;\cdots\;,\;u_n(\ell)),\;{\kappa}\;{\in}\;\{0,\;1,\;{\cdots}\;\},\;1\;{\leq}\;i\;{\leq}\;n$ for which the existence of constant-sign solutions is investigated.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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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.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.693-707
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• 1996

Consider a specific class of scalar-valued reaction diffusion equations of the form $$ (1.1) u_t = \nu\Delta u + f(u), u \in R $$ where $\nu$ < 0 is viscosity parameter and $f : R \to R$ is sufficiently smooth.

• Clean Technology
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• v.9 no.3
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• pp.107-113
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• 2003

Printing process generates a vast amount of toxical waste and wastewater by the development of printing and publishing industry. The regulations for various environmental pollution material, which were indispensably used in printing industries, were getting stronger. The printing industries should develop the cleaner technologies in order to avoid the regulations. In this paper, the separation characteristics of microfiltration, ultrafiltration, reverse osmosis were surveyed to make basic data for the optimization of process as cleaner technologies for printing industries. The $2kg/cm^2$ of operation pressure were suitable to the U/F System. Because of the permeate of U/F was below 3 NTU as turbidity, which was probed to be possible using the rinsing water in printing process. U/F System,

• Proceedings of the KIEE Conference
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• 1987.07a
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• pp.125-128
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• 1987

We investigate the feedback linearizability of nonlinear discrete-time system s of a specific form, $x_k=G_{u_k}oF(x_k)$ where F is a diffeomorphism and [$G_{u_k}$] forms an one parameter group of diffeomorphisms. This structure represents a class of systems which are state equivalent to linear ones and approximates the sampled data model of a continuous-time system. It is also considered a relationship between linearizability and discretization.