• 호남수학학술지
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• 제42권4호
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• pp.667-680
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• 2020

Our paper deals with the following neutral nonlinear Levin-Nohel integro-differential with variable delay $${\frac{d}{dt}x(t)}+{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{t-r(t)}}^t}a(t,s)x(s)ds+{\frac{d}{dt}}g(t,x(t-{\tau}(t)))=0.$$ By using Krasnoselskii's fixed point theorem we obtain the existence of periodic and positive periodic solutions and by contraction mapping principle we obtain the existence of a unique periodic solution. An example is given to illustrate this work.

• 충청수학회지
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• 제20권3호
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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 $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}f{\phi}_{00}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}g{\phi}_{00}=0$.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.507-516
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• 2016

In this paper, we introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation $$x_{n+1}={\frac{ax_{n-3}}{b-cx_{n-1}x_{n-3}}}$$, $n=0,1,{\ldots}$ where a, b, c are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

• Korean Journal of Mathematics
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• 제19권3호
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• pp.331-342
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• 2011

We consider the multiplicity of the solutions for a class of a system of critical growth beam equations with periodic condition on t and Dirichlet boundary condition $$\{u_{tt}+u_{xxxx}=av+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{{\alpha}-1}v_{+}^{\beta}+s{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,\\u_{tt}+v_{xxxx}=bu+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{\alpha}v_{+}^{{\beta}-1}+t{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,$$ where ${\alpha}$, ${\beta}$ > 1 are real constants, $u_+=max\{u,0\}$, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_00=1$ of the eigenvalue problem $u_{tt}+u_{xxxx}={\lambda}_{mn}u$. We show that the system has a positive solution under suitable conditions on the matrix $A=\(\array{0&a\\b&0}\)$, s > 0, t > 0, and next show that the system has another solution for the same conditions on A by the linking arguments.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.193-207
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• 2010

In this paper, a predator-prey model with stage structure and distributed delay is investigated. Mathematical analyses of the model equation with regard to boundedness of solutions, nature of equilibria, permanence, extinction and stability are performed. By the comparison theorem, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Taking the product of the per-capita rate of predation and the rate of conversing prey into predator as the bifurcating parameter, we prove that there exists a threshold value beyond which the positive equilibrium bifurcates towards a periodic solution.

• Journal of applied mathematics & informatics
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• 제35권3_4호
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• pp.267-276
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• 2017

The purpose of this paper is to investigate the local asymptotic stability of equilibria, the periodic nature of solutions, the existence of unbounded solutions and the global behavior of solutions of the fractional difference equation $$x_{n+1}=\frac{^{{\alpha}x}n-1(k+1)}{{\beta}+{\gamma}x^p_{n-k}x^q_{n-(k+2)}}$$, $$n=0,1,{\dots}$$ where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-(k+2)}$,$x_{-(k+1)}$, ${\dots}$, $x_{-1}$, $x_0{\in}\mathb{R}^+$.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.831-844
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• 2016

In this paper, we consider a discrete predator-prey system with Watt-type functional response and impulsive controls. First, we find sufficient conditions for stability of a prey-free positive periodic solution of the system by using the Floquet theory and then prove the boundedness of the system. In addition, a condition for the permanence of the system is also obtained. Finally, we illustrate some numerical examples to substantiate our theoretical results, and display bifurcation diagrams and trajectories of some solutions of the system via numerical simulations, which show that impulsive controls can give rise to various kinds of dynamic behaviors.

• 대한치과교정학회지
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• 제10권1호
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• pp.7-13
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• 1980

To study some informations on the morphogenesis and developmental process of the coronal suture in rats, the author performed daily oral administrations of demethylchlorotetracycline, a kind of tetracycline group, in the amount of 30mg/kg of body weight to the female rats from the 7th day of pregnancy to the time of delivery. Microscopic evaluation was undertaken on the fetal rats in the experimental group. The subject of this experiment were defined to the fetal rats of each group at 1st, 3rd, 7th and 14th day after birth. All these fetal rats were sacrificed and the heads were removed. All the tissue sections were fixed with $10\%$ formalin, Bouin' and Carnoy' solution and then stained by Hematoxylin-Van Gieson stain, or Feulgen and Rossenbeck, Periodic acid Schiff, and prepared for alcian blue reaction. The results were as follows; 1. The directions of osteogenic fibers were arranged irregulary during first 3 days, but after the 7th day they tended to change radial directions like control group. 2. The density of deep stained cells by Feulgen-Rossenbeck reaction were shown leu in the experimental group than that in the control group in first 3 days, but there was shown no significant difference between both groups after the 7th day. 3. PAS reaction in early stage was generally negative in the experimental group unlike as in the control group, but diffuse reaction was observed in the loose middle zone like as in the control group after 14th day. 4. Alcian blue reaction was negative in cambial zone, and slightly positive in uniting zone compared with control group in early stage. After 14th day, however, there was observed a tendency of moderately positive reaction.

• 한국전기전자재료학회:학술대회논문집
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• 한국전기전자재료학회 2008년도 하계학술대회 논문집 Vol.9
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• pp.241-241
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• 2008

Macrofore formation in silicon and other semiconductors using electrochemical etching processes has been, in the last years, a subject of great attention of both theory and practice. Its first reason of concern is new areas of macropore silicone applications arising from microelectromechanical systems processing (MEMS), membrane techniques, solar cells, sensors, photonic crystals, and new technologies like a silicon-on-nothing (SON) technology. Its formation mechanism with a rich variety of controllable microstructures and their many potential applications have been studied extensively recently. Porous silicon is formed by anodic etching of crystalline silicon in hydrofluoric acid. During the etching process holes are required to enable the dissolution of the silicon anode. For p-type silicon, holes are the majority charge carriers, therefore porous silicon can be formed under the action of a positive bias on the silicon anode. For n-type silicon, holes to dissolve silicon is supplied by illuminating n-type silicon with above-band-gap light which allows sufficient generation of holes. To make a desired three-dimensional nano- or micro-structures, pre-structuring the masked surface in KOH solution to form a periodic array of etch pits before electrochemical etching. Due to enhanced electric field, the holes are efficiently collected at the pore tips for etching. The depletion of holes in the space charge region prevents silicon dissolution at the sidewalls, enabling anisotropic etching for the trenches. This is correct theoretical explanation for n-type Si etching. However, there are a few experimental repors in p-type silicon, while a number of theoretical models have been worked out to explain experimental dependence observed. To perform ordered macrofore formaion for p-type silicon, various kinds of mask patterns to make initial KOH etch pits were used. In order to understand the roles played by the kinds of etching solution in the formation of pillar arrays, we have undertaken a systematic study of the solvent effects in mixtures of HF, N-dimethylformamide (DMF), iso-propanol, and mixtures of HF with water on the macrofore structure formation on monocrystalline p-type silicon with a resistivity varying between 10 ~ 0.01 $\Omega$ cm. The etching solution including the iso-propanol produced a best three dimensional pillar structures. The experimental results are discussed on the base of Lehmann's comprehensive model based on SCR width.