The higher-order nonlinear ordinary differential equation $$x^{\left(n\right)}+\lambda p\left(t\right)f\left(x\right)=0,t\ge a,$$ is considered and the problem of counting the number of zeros of bounded nonoscillatory solutions $x(t;\lambda )$ satisfying ${lim}_{t\to \infty }x(t;\lambda )=1$ is studied. The results can be applied to a singular eigenvalue problem.

We are concerned with the solvability of the fourth-order four-point boundary value problem ⎧ $u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right),u^{\text{'}\text{'}}\left(t\right))$, t ∈ [0,1], ⎨ u(0) = u(1) = 0, ⎩ au”(ζ₁) - bu”’(ζ₁) = 0, cu”(ζ₂) + du”’(ζ₂) = 0, where 0 ≤ ζ₁ < ζ₂ ≤ 1, f ∈ C([0,1] × [0,∞) × (-∞,0],[0,∞)). By using Guo-Krasnosel’skiĭ’s fixed point theorem on cones, some criteria are established to ensure the existence, nonexistence and multiplicity of positive solutions for this problem.

Let $f:[0,1]\times {ℝ}^2\to ℝ$ be a function satisfying Caratheodory’s conditions and let $e\left(t\right)\in L^1[0,1]$. Let ${\xi }_i,{\tau }_j\in (0,1)$, $c_i,a_j\in ℝ$, all of the $c_i$’s, (respectively, $a_j$’s) having the same sign, $i=1,2,...,m-2$, $j=1,2,...,n-2$, $0<{\xi }_1<{\xi }_2<...<{\xi }_{m-2}<1$, $0<{\tau }_1<{\tau }_2<...<{\tau }_{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems $$\begin{array}{c}\hfill x^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right))+e\left(t\right),t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_i{x}^{\text{'}}\left({\xi }_i\right),x\left(1\right)=\sum _{j=1}^{n-2}{a}_{j}x\left({\tau }_j\right)B{C}_{mn}\end{array}$$ and $$\begin{array}{c}\hfill x^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right))+e\left(t\right),t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_i{x}^{\text{'}}\left({\xi }_i\right),x^{\text{'}}\left(1\right)=\sum _{j=1}^{n-2}{a}_j{x}^{\text{'}}\left({\tau }_j\right),B{C}_{mn}^{\text{'}}\end{array}$$ Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.