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### Open Access Journals

PROC

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].

PROC

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.

PROC

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.

PROC

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.

PROC

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.

PROC

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.

PROC

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.

PROC

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.

CPAA

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

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