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### Open Access Journals

CPAA

In this paper, we consider the the second-order ordinary
differential equation with periodic boundary problem $ -
\ddot{x}(t)=f(t,x(t))$, subject to
$x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0$, where $f:C([0,
2\pi]\times R, R)$. The operator
$K=(-\frac{d^2}{dt^2}+I)^{-1}$ plays an important role. By using
Morse index, Leray-Schauder degree and Morse index theorem of the
type Lazer-Solimini, we obtain that the equation has at least two
or three nontrivial solutions without assuming nondegeneracy of
critical points and has at least four nontrivial solutions
assuming nondegeneracy of critical points.

CPAA

In this paper, we establish several new Lyapunov-type inequalities
for the $2n-$order differential equation

$x^{(2n)}(t)+(-1)^{n-1}q(t)x(t)=0, $

which are sharper than all related existing ones.

CPAA

By using the least action principle and the
minimax methods, some existence theorems are obtained existence of
solutions to a second-order Hamiltonian system with periodic boundary
conditions in the
cases when the gradiant of the nonlinearity is bounded sublinearly and linearly
respectively.

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