We consider the existence of at least one positive solution to the dynamic boundary value problem $$\begin{array}{cccc}\hfill -y^{\Delta \Delta }\left(t\right)& =\lambda f(t,y\left(t\right))\text{,}t\in {[0,T]}_{𝕋}y\left(0\right)\hfill & \hfill ={\int }_{{\tau }_1}^{{\tau }_2}{F}_1(s,y\left(s\right))\Delta sy\left({\sigma }^2\left(T\right)\right)& ={\int }_{{\tau }_3}^{{\tau }_4}{F}_2(s,y\left(s\right))\Delta s,\hfill \end{array}$$ where $𝕋$ is an arbitrary time scale with $0<{\tau }_1<{\tau }_2<{\sigma }^2\left(T\right)$ and $0<{\tau }_3<{\tau }_4<{\sigma }^2\left(T\right)$ satisfying ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, ${\tau }_4\in 𝕋$, and where the boundary conditions at $t=0$ and $t={\sigma }^2\left(T\right)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.

Let $𝕋$ be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay $x^{▵}\left(t\right)=-a\left(t\right)h\left(x^{\sigma }\left(t\right)\right)+c\left(t\right)x^{\widetilde{▵}}\left(t-r\left(t\right)\right)+G\left(t,x\left(t\right),x\left(t-r\left(t\right)\right)\right)$, $t\in 𝕋$, where $f^{▵}$ is the $▵$-derivative on $𝕋$ and $f^{\widetilde{▵}}$ is the $▵$-derivative on $(id-r)\left(𝕋\right)$. We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show...