• Korean Journal of Mathematics
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• 제18권3호
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• pp.299-309
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• 2010

In this paper, we study the first order nonlinear neutral difference equation: $${\Delta}(x(n)+px(n-{\tau}))+f(n,x(n-c),x(n-d))=r(n),\;n{\geq}n_0$$. Using the Banach fixed point theorem, we prove the existence of bounded positive solutions of the equation, suggest Mann iterative schemes of bounded positive solutions, and discuss the error estimates between bounded positive solutions and sequences generated by Mann iterative schemes.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.245-253
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• 2007

In this paper, the existence of bounded positive solution of the second-order nonlinear neutral differential equations are studied. Some new sufficient conditions are given. Furthermore, examples given show that the results here are almost sharp.

• Journal of applied mathematics & informatics
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• 제29권3_4호
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• pp.1011-1016
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• 2011

In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equation on $R^N$. we obtained the results under different suitable conditions on the locally H$\"{o}$lder continuous nonlinearity f(x, u), we needn't any mono-tonicity condition about the nonlinearity.

• Korean Journal of Mathematics
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• 제5권1호
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• pp.29-33
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• 1997

In this paper we investigate the existence of positive solutions of a nonlinear biharmonic equation under Dirichlet boundary condition in a bounded open set ${\Omega}$ in $\mathbf{R}^n$, i.e., $${\Delta}^2u+c{\Delta}u=bu^{+}+s\;in\;{\Omega},\\u=0,\;{\Delta}u=0\;on\;{\partial}{\Omega}$$.

• East Asian mathematical journal
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• 제39권3호
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• pp.355-367
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• 2023

Extending [14], we establish the existence of multiple positive solutions for a Schrödinger-type singular elliptic equation: $$\{{-{\Delta}u+V(x)u={\lambda}{\frac{f(u)}{u^{\beta}}},\;x{\in}{\Omega}, \atop u=0,\;x{\in}{\partial}{\Omega},$$ where 0 ∈ Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂Ω, β ∈ [0, 1), f ∈ C[0, ∞), V : Ω → ℝ is a bounded function and λ is a positive parameter. In particular, when f(s) > 0 on [0, σ) and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• Korean Journal of Mathematics
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• 제16권3호
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• pp.355-362
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• 2008

We prove the existence of a unique positive solution for a class of systems of the following nonlinear suspension bridge equation with Dirichlet boundary conditions and periodic conditions $$\{{u_{tt}+u_{xxxx}+\frac{1}{4}u_{ttxx}+av^+={\phi}_{00}+{\epsilon}_1h_1(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\{v_{tt}+v_{xxxx}+\frac{1}{4}u_{ttxx}+bu^+={\phi}_{00}+{\epsilon}_2h_2(x,t)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ where $u^+={\max}\{u,0\},\;{\epsilon}_1,\;{\epsilon}_2$ are small number and $h_1(x,t)$, $h_2(x,t)$ are bounded, ${\pi}$-periodic in t and even in x and t and ${\parallel} h_1{\parallel}={\parallel} h_2{\parallel}=1$. We first show that the system has a positive solution, and then prove the uniqueness by the contraction mapping principle on a Banach space

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1986년도 한국자동제어학술회의논문집; 한국과학기술대학, 충남; 17-18 Oct. 1986
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• pp.68-72
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• 1986

Upper bounds of the perturbed Co-semigroups of the infinite dimensional systems are investigated by using the algebraic Riccati equation(ARE). In the case that the solution P of the ARE is strictly positive, the perturbed semigroups are uniformly bounded. A sufficient condition for the solution P to be strictly positive is provided. The uniform boundedness plays an important role in extending approximately weak stability to weak stability on th whole space. Exponential Stability of the perturbed semigroups is studied by using the Young's inequlity. Some further discussions on the uniform boundedness of the perturbed semigroups are given.

• 대한수학회지
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• 제49권4호
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• pp.715-731
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• 2012

This paper considers the stability of positive steady-state solutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and subject to the homogeneous Dirichlet boundary conditions on a general bounded open spatial domain with smooth boundary. The existence, uniqueness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit function theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.

• East Asian mathematical journal
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• 제40권3호
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• pp.319-328
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• 2024

We establish an existence result for a positive solution to the Schrödinger-type singular semipositone problem: $-{\Delta}u\,=\,V(x)u\,=\,{\lambda}{\frac{f(u)}{u^{\alpha}}}$ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ^N , N > 2, λ ∈ ℝ is a positive parameter, V ∈ L^∞(Ω), 0 < α < 1, f ∈ C([0, ∞), ℝ) with f(0) < 0. In particular, when ${\frac{f(s)}{s^{\alpha}}}$ is sublinear at infinity, we establish the existence of a positive solutions for λ ≫ 1. The proofs are mainly based on the sub and supersolution method. Further, we extend our existence result to infinite semipositone problems with mixed boundary conditions.

• East Asian mathematical journal
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• 제38권1호
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• pp.13-20
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• 2022

We establish multiplicity results for positive solutions to the Schrödinger-type singular positone problem: -∆u + V (x)u = λf(u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ^N, N > 2, λ is a positive parameter, V ∈ L^∞(Ω) and f : [0, ∞) → (0, ∞) is a continuous function. In particular, when f is sublinear at infinity we discuss the existence of at least three positive solutions for a certain range of λ. The proofs are mainly based on the sub- and supersolution method.