The authors consider the nonlinear difference equation
\[ x_{n+1}=\alpha x_n + x_{n-k}f(x_{n-k}), \quad n=0, 1,\dots .1 \text{where} \alpha \in (0, 1),\hspace{5.0pt}k \in \lbrace 0, 1, \dots \rbrace \hspace{5.0pt}\text{and}\hspace{5.0pt}f\in C^1[[0, \infty),[0, \infty )] \qquad \mathrm{(0)}\] with $f^{\prime }(x)<0$. They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.
We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.
The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition u ′′ + g(t)f(t, u) = 0, t ∈ (0, 1), u(0) = αu(ξ) + λ, u(1) = βu(η) + µ. Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term f(t, x) may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of f(t, x)/x for x near 0 and ±∞, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.