The authors consider the difference equation $${\Delta }^m[y_n-p_n{y}_{n-k}]+\delta q_n{y}_{\sigma (n+m-1)}=0(*)$$ where $m\ge 2$, $\delta =\pm 1$, $k\in N_0=\{0,1,2,\cdots \}$, $\Delta y_n=y_{n+1}-y_n$, $q_n>0$, and $\left\{\sigma \right(n\left)\right\}$ is a sequence of integers with $\sigma \left(n\right)\le n$ and ${lim}_{n\to \infty }\sigma \left(n\right)=\infty$. They obtain results on the classification of the set of nonoscillatory solutions of ($*$) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.

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.

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...