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.

This paper discusses the existence of mild solutions for a class of semilinear fractional evolution equations with nonlocal initial conditions in an arbitrary Banach space. We assume that the linear part generates an equicontinuous semigroup, and the nonlinear part satisfies noncompactness measure conditions and appropriate growth conditions. An example to illustrate the applications of the abstract result is also given.

The paper deals with the existence of multiple positive solutions for the boundary value problem $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right)u^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),...,u^{(n-2)}\left(t\right))=0,0<t<1,\hfill \\ u^{\left(i\right)}\left(0\right)=0,i=0,1,...,n-3,\hfill \\ u^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha }_i{u}^{(n-2)}\left({\xi }_i\right),u^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.