Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Homogeneous initial–boundary value problem of the Rosenau equation posed on a finite interval

This paper considers the IBVP of the Rosenau equation $\{\begin{array}{c}{\partial }_{t}u+{\partial }_t{\partial }_x^{4}u+{\partial }_{x}u+u{\partial }_{x}u=0\text{,}x\in (0,1),t>0\text{,}\\ u(0,x)=u_0\left(x\right)\\ u(0,t)={\partial }_x^{2}u(0,t)=0\text{,}u(1,t)={\partial }_x^{2}u(1,t)=0\text{.}\end{array}$ It is proved that this IBVP has a unique global distributional solution $u\in C([0,T];H^s(0,1))$ as initial data $u_0\in H^s(0,1)$ with $s\in [0,4]$. This is a new global well-posedness result on IBVP of the Rosenau equation with Dirichlet boundary conditions.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$