• Korean Journal of Mathematics
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• 제16권3호
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• pp.259-269
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• 2008

The non-existence and existence of the positive solution for the generalized cooperation biological model for two species of animals $${\Delta}u+u(a-bu+g(v))=0\;in\;{\Omega}\\{\Delta}v+v(d+h(u)-cv)=0\;in\;{\Omega}\\u=v=0\;on\;{\partial}{\Omega}$$ are investigated. The techniques used in this paper are elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

• 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.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.877-882
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• 2010

In this paper, a third-order three-point boundary value problem on time scales is considered. We establish criteria for the existence of a solution and a positive solution by using the Leray-Schauder fixed point theorem. An example is also given to illustrate our results.

• 대한수학회지
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• 제37권3호
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• pp.339-358
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• 2000

Let be a bonded domain in N with smooth boundary . In this paper, we consider the existence of solutions of the following problem: (1.1)-div{} - + = , , , , , , where q > 1, p$\geq$1, $\delta$>0, , the Laplacian in N and is a positive function like as .

• 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.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.697-710
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• 2010

In this paper, we prove existence of infinitely many positive and concave solutions, by means of a simple approach, to $3^{th}$ order three-point singular boundary value problem {$x^{\prime\prime\prime}(t)=\alpha(t)f(t,x(t))$, 0 < t < 1, $x(0)=x'(\eta)=x^{\prime\prime}(1)=0$, (1/2 < $\eta$ < 1). Moreover with respect to multiplicity of solutions, we don't assume any monotonicity on the nonlinearity. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Knesser's type properties of the solutions funnel and the well-known Krasnosel'ski$\breve{i}$'s fixed point theorem. The later is applied on a new very simple cone K, just on the plane $R^2$. These extensions justify the efficiency of our new approach compared to the commonly used one, where the cone $K\;{\subset}\;C$ ([0, 1], $\mathbb{R}$) and the existence of a positive Green's function is a necessity.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.183-194
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• 2009

In this paper, we considered the following four-point boundary value problem $\{{x"(t)+h(t)f(t,x(t),x'(t))=0,\;0<t<1\atop%20x'(0)=ax(\xi),\;x'(1)=bx(\eta)}\$. where $0\;<\;{\xi}\;<\;{\eta}\;<\;1,\;{\delta}\;=\;ab{\xi}\;-\;ab{\eta}\;+\;a\;-\;b\;<\;0,\;0\;<\;a\;<\;\frac{1}{\xi},\;0\;<\;b\;<\;\frac{1}{\eta}$. After the discussion of the Green function of the corresponding homogeneous system, we establish some criteria for the existence of positive solutions by using the generalized Leggett-William's fixed point theorem. The interesting point is the expression of the Green function, which is a difficulty for multi-point BVP.

• 대한수학회보
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• 제45권1호
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• pp.23-31
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• 2008

In this paper, we investigate nonoscillatory solutions of a class of higher order neutral nonlinear difference equations with positive and negative coefficients $\Delta^m(x(n)+p(n)x(\tau(n)))+f_1(n,x(\sigma_1(n)))-f_2(n,x(\sigma_2(n)))=0,\;n{\geq}n_0$. Some sufficient conditions for the existence of nonoscillatory solutions are obtained.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.79-87
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• 2009

This paper deals with the multipoint boundary value problem for one dimensional p-Laplacian $({\phi}_p(u'))'(t)$ + f(t,u(t)) = 0, $t{\in}$ (0, 1), subject to the boundary value conditions: u'(0) - $\sum\limits^n_{i=1}{\alpha_i}u({\xi}_i)$ = 0, u'(1) + $\sum\limits^n_{i=1}{\alpha_i}u({\eta}_i)$ = 0. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple (at least three) positive solutions to the above boundary value problem.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권3호
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• pp.147-163
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• 2014

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) where the nonlinearity f(t, u, v) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.