A priori bounds are established for periodic solutions of an nth order Rayleigh equation with delay. From these bounds, existence theorems for periodic solutions are established by means of Mawhin's continuation theorem.

We propose an approach for studying positivity of Green’s operators of a nonlocal boundary value problem for the system of $n$ linear functional differential equations with the boundary conditions $n_i{x}_i-{\sum }_{j=1}^n{m}_{ij}{x}_j={\beta }_i$, $i=1,\cdots ,n$, where $n_i$ and $m_{ij}$ are linear bounded “local” and “nonlocal“ functionals, respectively, from the space of absolutely continuous functions. For instance, $n_i{x}_i=x_i\left(\omega \right)$ or $n_i{x}_i=x_i\left(0\right)-x_i\left(\omega \right)$ and $m_{ij}{x}_j={\int }_0^{\omega }k\left(s\right)x_j\left(s\right)\mathrm{d}s+{\sum }_{r=1}^{n_{ij}}{c}_{ijr}{x}_j\left(t_{ijr}\right)$ can be considered. It is demonstrated that the positivity of Green’s operator of nonlocal problem follows from the positivity of Green’s operator...

The equation to be considered is $$L_{n}y\left(t\right)+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The aim of this paper is to derive sufficient conditions for property (A) of this equation.

We study the existence of positive solutions of the integral equation $$x\left(t\right)=\mu {\int }_0^{1}k(t,s)f(s,x\left(s\right),x^{\text{'}}\left(s\right),...,x^{(n-1)}\left(s\right))ds,n\ge 2$$ in both $C^{n-1}[0,1]$ and $W^{n-1,p}[0,1]$ spaces, where $p\ge 1$ and $\mu >0$. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.

In this paper we present some new existence results for singular positone and semipositone boundary value problems of second order delay differential equations. Throughout our nonlinearity may be singular in its dependent variable.