• 대한수학회보
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• 제55권6호
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• pp.1869-1889
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• 2018

We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.

• 대한수학회보
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• 제50권4호
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• pp.1061-1067
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• 2013

In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space $\mathbb{R}^3$, satisfying the condition ${\Delta}^{II}G=f(G+C)$, where ${\Delta}^{II}$ is the Laplace operator with respect to the second fundamental form, $f$ is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in $\mathbb{R}^3$ which satisfy the condition ${\Delta}^{II}G=fG$, coincide with surfaces of revolution with non-zero constant Gaussian curvature.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.909-931
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• 2015

This paper presents some new paranormed sequence spaces $X(r,s,t,p;{\Delta})$ where $X{\in}\{l_{\infty}(p),c(p),c_0(p),l(p)\}$ defined by using generalized means and difference operator. It is shown that these are complete linear metric spaces under suitable paranorms. Furthermore, the ${\alpha}$-, ${\beta}$-, ${\gamma}$-duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r,s,t,p;{\Delta})$ to X. Finally, it is proved that the sequence space $l(r,s,t,p;{\Delta})$ is rotund when $p_n$ > 1 for all n and has the Kadec-Klee property.

• 대한수학회논문집
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• 제36권2호
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• pp.247-256
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• 2021

Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: $$(\mathcal{P}_{\lambda})\;\{\begin{array}{lll}{\Delta}^2u-(a{\int}_{\Omega}{\mid}{\nabla}u{\mid}^2dx+b){\Delta}u+cu=f(x){\mid}u{\mid}^{-{\gamma}}-{\lambda}{\mid}u{\mid}^{p-2}u&&\text{ in }{\Omega},\\{\Delta}u=u=0&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω is a smooth bounded domain of ℝⁿ (n ≥ 5), ∆² is the biharmonic operator, and ∇u denotes the spatial gradient of u and 0 < γ < 1, λ > 0, 0 < p ≤ 2^# and a, b, c are three positive constants with a + b > 0 and f belongs to a given Lebesgue space.

• 충청수학회지
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• 제12권1호
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• pp.125-131
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• 1999

We obtain some sufficient conditions for oscillation of the neutral difference equation with positive and negative coefficients $${\Delta}(x_n-cx_{n-m})+px_{n-k}-qx_{n-l}=0$$, where ${\Delta}$ denotes the forward difference operator, m, k, l, are nonnegative integers, and $c{\in}[0,1),p,q{\in}\mathbb{R}^+$.

• 대한수학회지
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• 제32권2호
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• pp.341-350
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• 1995

Let (M,g) be a compact manifold of dimension n with metric tensor g. Let $\Delta^p = d\delta + \deltad$ be the Laplace-Beltrami operator acting on the space of smooth p-forms.

• 한국지능시스템학회논문지
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• 제22권6호
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• pp.700-705
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• 2012

본 논문은 미리 설계된 타카기-수게노 퍼지 모델 기반 아날로그 제어기를 상태 정합의 의미에서 등가인 샘플치 제어기로 효율적으로 변환하기 위해, 관측기 기반 출력 궤환 퍼지 제어기에 대한 지능형 디지털 재설계 기법을 제안한다. 아날로그 제어 시스템과 샘플치 제어 시스템 사이의 점근적 연관성을 위해 델타 연산자를 사용한다. 지능형 디지털 재설계 문제는 정합될 선형 연산자 간의 놈의 거리를 최소화하는 문제로 생각한다. 제어기 설계 조건은 선형행렬부등식의 형태로 유도되며, 디지털 재설계시 관측기와 제어기에 대한 분리 설계 조건이 만족함을 보인다.

• 대한수학회보
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• 제5권1호
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• pp.67-71
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• 1968

• Korean Journal of Mathematics
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• 제16권2호
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• pp.165-172
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• 2008

We investigate the multiplicity of solutions of the nonlinear biharmonic equation with Dirichlet boundary condition,${\Delta}^2u+c{\Delta}u=bu^{+}+s$, in ­${\Omega}$, where $c{\in}R$ and ${\Delta}^2$ denotes the biharmonic operator. We show by degree theory that there exist at least three solutions of the problem.

• 대한수학회지
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• 제31권3호
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• pp.401-416
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• 1994

We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {\Delta u = f(x,u,\Delta u), in \Omega {Bu(x) = \phi(x), on \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $R^n, n \geq 1, and \partial\Omega \in C^{2,\alpha}, (0 < \alpha < 1), \Delta$ is the Laplacian operator, $\nabla u = (D_1u, D_2u, \cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)\frac{d\nu}{du} (x), $$ where $\frac{d\nu}{du} denotes the outward normal derivative of u on $\partial\Omega$.