• 대한수학회보
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• 제55권2호
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• pp.591-598
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• 2018

This paper is concerned with global existence of weak solutions to the relativistic Vlasov-Klein-Gordon system. The energy of this system is conserved, but the interaction term ${\int}_{{\mathbb{R}}^n}\;{\rho}{\varphi}dx$ in it need not be positive. So far existence of global weak solutions has been established only for small initial data [9, 14]. In two dimensions, this paper shows that the interaction term can be estimated by the kinetic energy to the power of ${\frac{4q-4}{3q-2}}$ for 1 < q < 2. As a consequence, global existence of weak solutions for general initial data is obtained.

• 대한수학회보
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• 제57권2호
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• pp.331-343
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• 2020

This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems -Δv = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ R^N : r₁ < |x| < r₂} with 0 < r₁ < r₂, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s₁(x)) = h(x, s₂(x)) = 0 for suitable positive, concave function s₁ and negative, convex function s₂, as well as sh(x, s) > 0 for s ∈ ℝ \ {0, s₁(x), s₂(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.

• 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 $x_{-(k+2)}$,$x_{-(k+1)}$, ${\dots}$, $x_{-1}$, $x_0{\in}\mathb{R}^+$.

• 대한수학회논문집
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• 제37권1호
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• pp.137-161
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• 2022

In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation (1) u⁽⁴⁾(x) + ωu''(x) + a(x)u(x) = f(x, u(x)), ∀x ∈ ℝ where a(x) is not required to be either positive or coercive, and F(x, u) = ∫^u₀ f(x, v)dv is of subquadratic or superquadratic growth as |u| → ∞, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as |u| → ∞). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.

• 대한수학회논문집
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• 제37권3호
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• pp.693-703
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• 2022

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$(1){\hspace{32}}{\ddot{u}}(t)+q(t){\dot{u}}(t)-L(t)u(t)+{\nabla}W(t,u(t))=0,\;{\forall}t{\in}{\mathbb{R}},$$ where q ∈ C(ℝ, ℝ), L ∈ C(ℝ, ${\mathbb{R}}^{N^2}$) is a symmetric and positive definite matix-valued function and W ∈ C¹(ℝ×ℝ^N, ℝ). The novelty of this paper is that, assuming that L is bounded from below unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

• 대한수학회보
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• 제51권1호
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• pp.55-66
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• 2014

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation $u_t=div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+a{\int}_{\Omega}u^q(y,t)dy$, 1 < p < 2, in a bounded domain ${\Omega}{\subset}R^N$ with $N{\geq}1$. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in ${\Omega}$ for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of $a{\mu}$, where ${\mu}{\int}_{\Omega}{\phi}^{p-1}(x)dx$ and ${\phi}$ is the unique positive solution of the elliptic problem -div(${\mid}{\nabla}{\phi}{\mid}^{p-2}{\nabla}{\phi}$) = 1, $x{\in}{\Omega}$; ${\phi}(x)$=0, $x{\in}{\partial}{\Omega}$. This is a main difference between equations with local and nonlocal sources.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1994년도 Proceedings of the Korea Automatic Control Conference, 9th (KACC) ; Taejeon, Korea; 17-20 Oct. 1994
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• pp.364-368
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• 1994

Algebraic matrix Riccati equations of the form, FP+PF$^{T}$ -PRP+Q=0. are analyzed with reference to the stability of closed-loop system F-PR. Here F, R and Q are n * n real matrices with R=R$^{T}$ and Q=Q$^{T}$ .geq.0 (nonnegative-definite). Such equations have been playing key roles in optimal control and filtering problems with R .geq. 0. and also in the solutions of in H$_{\infty}$ control problems with R taking the form R=H$_{1}$$^{T}$ H$_{1}$-H$_{2}$$^{T}$ H$_{2}$. In both cases an existence of stabilizing solution, i.e. the solution yielding asymptotically stable closed-loop system, is an important problem. First, we briefly review the typical results when R is of definite form, namely either R .geq. 0 as in LQG problems or R .leq. 0. They constitute two extrence cases of Riccati to the cases H$_{2}$=0 and H$_{1}$=0. Necessary and sufficient conditions are shown for the existence of nonnegative-definite or positive-definite stabilizing solution. Secondly, we focus our attention on more general case where R is only assumed to be symmetric, which obviously includes the case for H$_{\infty}$ control problems. Here, necessary conditions are established for the existence of nonnegative-definite or positive-definite stabilizing solutions. The results are established by employing consistently the so-called algebraic method based on an eigenvalue problem of a Hamiltonian matrix.x.ix.x.

• Steel and Composite Structures
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• 제13권1호
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• pp.71-89
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• 2012

The paper investigates beam lateral buckling stability according to linear and non-linear models. Closed form solutions for single-symmetric cross sections are first derived according to a non-linear model considering flexural-torsional coupling and pre-buckling deformation effects. The closed form solutions are compared to a beam finite element developed in large torsion. Effects of pre-buckling deflection and gradient moment on beam stability are not well known in the literature. The strength of singly symmetric I-beams under gradient moments is particularly investigated. Beams with T and I cross-sections are considered in the study. It is concluded that pre-buckling deflections effects are important for I-section with large flanges and analytical solutions are possible. For beams with T-sections, lateral buckling resistance depends not only on pre-buckling deflection but also on cross section shape, load distribution and buckling modes. Effects of pre-buckling deflections are important only when the largest flange is under compressive stresses and positive gradient moments. For negative gradient moments, all available solutions fail and overestimate the beam strength. Numerical solutions are more powerful. Other load cases are investigated as the stability of continuous beams. Under arbitrary loads, all available solutions fail, and recourse to finite element simulation is more efficient.

• Parasites, Hosts and Diseases
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• 제54권6호
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• pp.697-702
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• 2016

Acanthamoeba keratitis has been increasing in recent years. Main risk factors are contact lens wear and their cleaning solutions. Most contact lens wearers use multipurpose disinfecting solutions (MPDS) for cleansing and disinfecting microorganisms because of its convenience. We determined amoebicidal effects of MPDS made in Korea and their cytotoxicity on human corneal epithelium cells. Fifteen commercial MPDS (A to O) were tested for their amoebicidal effects on Acanthamoeba castellanii trophozoites and cysts by using a most probable number (MPN) technique. Among them, 7 kinds of MPDS showed little or no amoebicidal effects for 24 hr exposure. Solutions A, B, G, H, L, and O showed positive amoebicidal effects, and solutions M and N killed almost all trophozoites and cysts after 24 hr exposure. However, 50%-N solution showed 56% cytotoxicity on human corneal epithelial cells within 4 hr exposure, and 50%-O solution also showed 62% cytotoxicity on human cells within 4 hr exposure. Solution A did not show any cytotoxicity on human cells. These results revealed that most MPDS made in Korea were ineffective to kill Acanthamoeba. The solutions having amoebicidal activity also showed high levels of cytotoxicity on human corneal epithelial cells. New formulations for improved MPDS that are amoebicidal but safe for host cells are needed to prevent Acanthamoeba keratitis.

• 대한수학회논문집
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• 제12권2호
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• pp.221-236
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• 1997

In this paper we investigate the multiplicity of positive solutions to a quasilinear Neumann problem; $$ {\varepsilon^m div($\mid$\bigtriangledown_u$\mid$^{m-2}\bigtriangledown_u) - u$\mid$u$\mid$^{m-2} + u$\mid$u$\mid$^{m-2} + u$\mid$u$\mid$^{p-2} = 0 in \omega $$ $$ \frac{\partial u}{\partial \nu} = 0 on \partial \omega, $$ making use of Ljusternik Schnirelmann category theory.