We prove two existence results on abstract differential equations of the type $d\left(Bu\right)/dt+A\left(u\right)=f$ and we give some applications of them to partial differential equations.

Consider the forced higher-order nonlinear neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+C\left(t\right)x(t-\tau )]+\sum _{i=1}^m{Q}_i\left(t\right)f_i\left(x(t-{\sigma }_i)\right)=g\left(t\right),t\ge t_0,$$ where $n,m\ge 1$ are integers, $\tau ,{\sigma }_i\in {ℝ}^{+}=[0,\infty )$, $C,Q_i,g\in C([t_0,\infty ),ℝ)$, $f_i\in C(ℝ,ℝ)$, $(i=1,2,\cdots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i\left(t\right)$$(i=1...$

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.