We establish the existence and uniqueness theorems for a linear and a nonlinear fourth-order boundary value problem. The results obtained generalize the results of Usmani [4] and Yang [5]. The methods used are based, in principle, on [3], [5].

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.

In the paper we prove an Ambrosetti-Prodi type result for solutions $u$ of the third-order nonlinear differential equation, satisfying $u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)=u\left(\eta \right)=0,0\le \eta \le 1$.

Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle$ with the sup norm and $\alpha ,\beta \in X\to R$ be continuous increasing functionals, $\alpha \left(0\right)=\beta \left(0\right)=0$. This paper deals with the functional differential equation (1) $x^{\text{'}\text{'}\text{'}}\left(t\right)=Q[x,x^{\text{'}},x^{\text{'}\text{'}}\left(t\right)]\left(t\right)$, where $Q:X^2\times R\to X$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha \left(x\right)=0=\beta \left(x^{\text{'}}\right)$, $x^{\text{'}\text{'}}\left(1\right)-x^{\text{'}\text{'}}\left(0\right)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional...