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.

The aim of this paper is to extend the classical linear condition concerning diagonal dominant bloc matrix to fully nonlinear equations. Even if assumptions are strong, we obtain an explicit condition which exactly extend the one known in linear case, and the setting allows also to consider bicontinuous operator instead of the schift and as particular case, we receive periodic or almost periodic solutions for discrete time equations.

In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation $${\Delta }^3(x_n+a_n{x}_{n-l}+b_n{x}_{n+m})+p_n{x}_{n-k}-q_n{x}_{n+r}=0,n\ge n_0$$ via Banach contraction principle. Examples are provided to illustrate the main results. The results obtained in this paper extend and complement some of the existing results.

Based on the fixed-point theorem in a cone and some analysis skill, a new sufficient condition is obtained for the existence of positive periodic solutions for a class of higher-order functional difference equations. An example is used to illustrate the applicability of the main result.