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.

We consider a nonlinear evolution inclusion defined in the abstract framework of an evolution triple of spaces and we look for extremal periodic solutions. The nonlinear operator is only pseudomonotone coercive. Our approach is based on techniques of multivalued analysis and on the theory of operators of monotone-type. An example of a parabolic distributed parameter system is also presented.