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.

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure...

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.

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...

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$-u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon )$$ and $$u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon ),$$ where $\epsilon \in (0,1/2)$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C\left(\right[0,1],(0,+\infty \left)\right)$, $f\in C(ℝ,ℝ)$ with $sf\left(s\right)>0$ for $s\ne 0$. The proof of the main results is based upon bifurcation techniques.