PROC
Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian
John R. Graef Lingju Kong
Conference Publications 2011, 2011(Special): 515-522 doi: 10.3934/proc.2011.2011.515
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
Conference Publications 2013, 2013(special): 283-290 doi: 10.3934/proc.2013.2013.283
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
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
Conference Publications 2013, 2013(special): 759-769 doi: 10.3934/proc.2013.2013.759
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
Existence of nodal solutions of multi-point boundary value problems
Lingju Kong Qingkai Kong
Conference Publications 2009, 2009(Special): 457-465 doi: 10.3934/proc.2009.2009.457
We study the nonlinear boundary value problem consisting of the equation $y^{''}+ w(t)f(y)=0$ on $[a,b]$ and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes.
keywords: multi-point boundary value problems Nodal solutions eigenvalues Sturm-Liouville problems
PROC
Positive solutions of a nonlinear higher order boundary-value problem
John R. Graef Lingju Kong Bo Yang
Conference Publications 2009, 2009(Special): 276-285 doi: 10.3934/proc.2009.2009.276
The authors study a higher order three point boundary value problem. Estimates for positive solutions are given; these estimates improve some recent results in the literature. Using these estimates, new sufficient conditions for the existence and nonexistence of positive solutions of the problem are obtained. An example illustrating the results is included.
keywords: Existence and nonexistence of positive solutions; Guo-Krasnosel'skii fixed point theorem; higher order boundary value problem
PROC
Existence of multiple solutions to a discrete fourth order periodic boundary value problem
John R. Graef Lingju Kong Min Wang
Conference Publications 2013, 2013(special): 291-299 doi: 10.3934/proc.2013.2013.291
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
PROC
Existence of homoclinic solutions for second order difference equations with $p$-laplacian
John R. Graef Lingju Kong Min Wang
Conference Publications 2015, 2015(special): 533-539 doi: 10.3934/proc.2015.0533
Using the variational method and critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation \begin{equation*} -\Delta \big(a(k)\phi_p(\Delta u(k-1))\big)+b(k)\phi_p(u(k))=\lambda f(k,u(k))),\quad k\in\mathbb{Z}, \end{equation*} where $p>1$ is a real number, $\phi_p(t)=|t|^{p-2}t$ for $t\in\mathbb{R}$, $\lambda>0$ is a parameter, $a, b:\mathbb{Z}\to (0,\infty)$, and $f: \mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is continuous in the second variable. Related results in the literature are extended.
keywords: homoclinic solutions Diff erence equations infi nitely many homoclinic solutions variational method.
PROC
Existence of nontrivial solutions to systems of multi-point boundary value problems
John R. Graef Shapour Heidarkhani Lingju Kong
Conference Publications 2013, 2013(special): 273-281 doi: 10.3934/proc.2013.2013.273
In this paper, sufficient conditions are established for the existence of at least one nontrivial solution of the multi-point boundary value system $$ \left\{\begin{array}{ll} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\ x\in(0,1),\\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\ u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{array} \right. i=1,\ldots,n. $$ The approach is based on variational methods and critical point theory.
keywords: Multi-point boundary value problem critical point theory.
CPAA
Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems
John R. Graef Shapour Heidarkhani Lingju Kong
Communications on Pure & Applied Analysis 2013, 12(3): 1393-1406 doi: 10.3934/cpaa.2013.12.1393
In this paper, the authors prove the existence of at least three weak solutions for the $(p_{1},\ldots,p_{n})$-- biharmonic system $$\begin{cases} \Delta(|\Delta u_{i}|^{p_i-2}\Delta u_{i}) = \lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}), & \mbox{in} \ \Omega,\\ u_{i}=\Delta u_i=0, & \mbox{on} \ \partial\Omega. \end{cases} $$ The main tool is a recent three critical points theorem of Averna and Bonanno ({\it A three critical points theorem and its applications to the ordinary Dirichlet problem}, Topol. Methods Nonlinear Anal. 22 (2003), 93-104).
keywords: p_{n})$-biharmonic systems critical points Three solutions $(p_{1} variational methods multiplicity results. \ldots

Year of publication

Related Authors

Related Keywords

[Back to Top]