Multiple solutions of second-order ordinary differential equation via Morse theory
Qiong Meng X. H. Tang
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.
keywords: Leray-Schauder degree Lazer Solimini. Periodic solution Morse index
Lyapunov-type inequalities for even order differential equations
Xiaofei He X. H. Tang
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.

keywords: Even order differential equation Lyapunov-type inequality.
Solutions of a second-order Hamiltonian system with periodic boundary conditions
Qiong Meng X. H. Tang
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.
keywords: Second-order Hamiltonian system least action principle periodic boundary condition saddle point theorem.

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