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.

L’existence de solutions holomorphes locales d’équations aux dérivées partielles d’ordre infini à coefficients holomorphes de type spécial est étudiée.

In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       $$x\left(t\right)=\underset{a}{\overset{b}{\int }}{K}_1(t,r,x\left(r\right))dr,x\left(t\right)=\underset{a}{\overset{b}{\int }}{K}_2(t,r,x\left(r\right))dr,$$ where a, b...

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.