• 제목/요약/키워드: uniqueness of positive solution

검색결과 14건 처리시간 0.021초

UNIQUENESS OF POSITIVE STEADY STATES FOR WEAK COMPETITION MODELS WITH SELF-CROSS DIFFUSIONS

  • Ko, Won-Lyul;Ahn, In-Kyung
    • 대한수학회보
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    • 제41권2호
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    • pp.371-385
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    • 2004
  • In this paper, we investigate the uniqueness of positive solutions to weak competition models with self-cross diffusion rates under homogeneous Dirichlet boundary conditions. The methods employed are upper-lower solution technique and the variational characterization of eigenvalues.

BIFURCATION PROBLEM FOR A CLASS OF QUASILINEAR FRACTIONAL SCHRÖDINGER EQUATIONS

  • Abid, Imed
    • 대한수학회지
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    • 제57권6호
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    • pp.1347-1372
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    • 2020
  • We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝn, (-∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.

CONTRACTION MAPPING PRINCIPLE AND ITS APPLICATION TO UNIQUENESS RESULTS FOR THE SYSTEM OF THE WAVE EQUATIONS

  • Jung, Tack-Sun;Choi, Q-Heung
    • 호남수학학술지
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    • 제30권1호
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    • pp.197-203
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    • 2008
  • We show the existence of the unique solution of the following system of the nonlinear wave equations with Dirichlet boundary conditions and periodic conditions under some conditions $U_{tt}-U_{xx}+av^+=s{\phi}_{00}+f$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, ${\upsilon}_{tt}-{\upsilon}_{xx}+bu^+=t{\phi}_{00}+g$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, where $u^+$ = max{u, 0}, s, t ${\in}$ R, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ of the wave operator. We first show that the system has a positive solution or a negative solution depending on the sand t, and then prove the uniqueness theorem by the contraction mapping principle on the Banach space.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY VALUE PROBLEMS

  • Miao, Chunmei;Ge, Weigao
    • Journal of applied mathematics & informatics
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    • 제27권3_4호
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    • pp.895-902
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    • 2009
  • In this paper, the singular three-point boundary value problem $$\{{{u"(t)\;+\;f(t,\;u)\;=\;0,\;t\;{\in}\;(0,\;1),}\atop{u(0)\;=\;0,\;u(1)\;=\;{\alpha}u(\eta),}}\$$ is studied, where 0 < $\eta$ < 1, $\alpha$ > 0, f(t,u) may be singular at u = 0. By mixed monotone method, the existence and uniqueness are established for the above singular three-point boundary value problems. The theorems obtained are very general and complement previous know results.

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UNIQUE POSITIVE SOLUTION FOR A CLASS OF THE SYSTEM OF THE NONLINEAR SUSPENSION BRIDGE EQUATIONS

  • Jung, Tacksun;Choi, Q-Heung
    • 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

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STABILITY OF POSITIVE STEADY-STATE SOLUTIONS IN A DELAYED LOTKA-VOLTERRA DIFFUSION SYSTEM

  • Yan, Xiang-Ping;Zhang, Cun-Hua
    • 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.

THE EXISTENCE, NONEXISTENCE AND UNIQUENESS OF GLOBAL POSITIVE COEXISTENCE OF A NONLINEAR ELLIPTIC BIOLOGICAL INTERACTING MODEL

  • Kang, Joon Hyuk;Lee, Jungho;Oh, Yun Myung
    • Korean Journal of Mathematics
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    • 제12권1호
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    • pp.77-90
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    • 2004
  • The purpose of this paper is to give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system of the Dirichlet problem on the bounded domain ${\Omega}$ in $R^n$. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. This result yields an algebraically computable criterion for the positive coexistence of competing species of animals in many biological models.

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