PROC
Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian
John R. Graef Lingju Kong
We consider the boundary value problem with nonhomogeneous three-point boundary condition Sufficient conditions are obtained for the existence and uniqueness of a positive solution. The dependence of the solution on the parameter $\lambda$ is also studied. This work extends and improves some recent results in the literature on the above problem, especially those in the paper [L. Kong, D. Piao, and L. Wang, Positive solutions for third order boundary value problems with $p$-Laplacian, Result. Math. 55 (2009) 111{128].
keywords: boundary value problems Positive solutions uniqueness dependence on a parameter
PROC
Positive solutions of nonlocal fractional boundary value problems
John R. Graef Lingju Kong Qingkai Kong Min Wang
The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
keywords: fractional calculus. positive solution Green's function
DCDS-S
Positive solutions of a third order nonlocal boundary value problem
John R. Graef Bo Yang
We consider a nonlocal boundary value problem for a third order differential equation. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. The results are illustrated with some examples.
keywords: existence of positive solutions nonexistence of positive solutions nonlinear equations Boundary value problems nonlocal problems.
PROC
Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle
Yu Tian John R. Graef Lingju Kong Min Wang
In this paper, by using the dual least action principle, the authors investigate the existence of solutions to a multi-point boundary value problem for a second-order differential system with $p$-Laplacian.
keywords: dual least action principle. Multi-point boundary value problem critical point theory
PROC
Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy
John R. Graef János Karsai
Please refer to Full Text.
keywords: impulses. Oscillation second order nonlinear systems
PROC
Attractivity properties of oscillator equations with superlinear damping
János Karsai John R. Graef
The authors investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation $$x''+ a(t)|x'|^\alpha \sgn(x') + f(x)=0, $$ where $uf(u) > 0$ for $u \neq 0$, $a(t)\geq 0$, and $\alpha\geq 1$. The case $\alpha=1$ has been investigated by a number of other authors. There are also some results for the case $\alpha>1$, but they are not really based on the power $\alpha$, although it plays an essential role in the behavior of the solutions. In this paper, we give new attractivity results for the large damping case, $a(t) \geq a_0 > 0$, that improve previously known results. Our conditions involve the power $\alpha$ in such a way that our results reduce directly to known conditions in the case $\alpha=1$. Some open problems for future research are also indicated.
keywords: attractivity nonlinear damping monotonicity. Second order equations
PROC
Multiple positive solutions to a three point third order boundary value problem
John R. Graef Bo Yang
The authors consider the boundary value problem $$ \begin{cases} u'''(t) = q(t)f(u), \quad 0 < t < 1, \\ u(0) = u'(p) = u''(1) = 0, \end{cases} $$ where $p \in(\frac{1}{2},1)$ is a constant. They give sufficient conditions for the existence of multiple positive solutions to this problem. In so doing, they are able to improve some recent results on this problem. Examples are included to illustrate the results.
keywords: third order equations. three point problems nonlinear equations existence of positive solutions Boundary value problems nonexistence of positive solutions
PROC
Some limit-point/limit-circle results for third order differential equations
M . Bartušek John R. Graef
Please refer to Full Text.
keywords: nonlinear limit-point square integrable solutions. nonoscillatory solutions nonlinear limit-circle oscillatory solutions
PROC
A study on the effects of disease caused death in a simple epidemic model
John R. Graef Michael Y. Li Liancheng Wang
Please refer to Full Text.
keywords:
PROC
Existence of multiple solutions to a discrete fourth order periodic boundary value problem
John R. Graef Lingju Kong Min Wang
Sufficient conditions are obtained for the existence of multiple solutions to the discrete fourth order periodic boundary value problem \begin{equation*} \begin{array}{l} \Delta^4 u(t-2)-\Delta(p(t-1)\Delta u(t-1))+q(t) u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb{Z}},\\ \Delta^iu(-1)=\Delta^iu(N-1),\quad i=0, 1,2, 3. \end{array} \end{equation*} Our analysis is mainly based on the variational method and critical point theory. One example is included to illustrate the result.
keywords: fourth order variational methods. critical points Discrete boundary value problem multiple solutions

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