We consider the classical nonlinear fourth-order two-point boundary value problem $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=\lambda h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),0<t<1,\hfill \\ u\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}}\left(1\right)=0.\hfill \end{array}\right.$$ In this problem, the nonlinear term $h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right))$ contains the first and second derivatives of the unknown function, and the function $h\left(t\right)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

The problem on the existence of a positive in the interval $]a,b[$ solution of the boundary value problem $$u^{\text{'}\text{'}}=f(t,u)+g(t,u)u^{\text{'}};u(a+)=0,u(b-)=0$$ is considered, where the functions $f$ and $g]a,b[\times ]0,+\infty [\to ℝ$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.

Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)+\lambda u\left(b\right)=c$$ are established, where $\ell :C\left(\right[a,b];R)\to L\left(\right[a,b];R)$ is a linear bounded operator, $q\in L\left(\right[a,b];R)$, $\lambda \in R_{+}$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right),u\left(a\right)+\lambda u\left(b\right)=0$$ is discussed as well.

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation $$x^{\left(n\right)}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right),\cdots ,x^{(n-1)}\left(t\right))+e\left(t\right),0<t<1,(*)$$ and the following multi-point boundary value conditions $$\begin{array}{cc}\hfill 1*-1x^{\left(i\right)}\left(0\right)& =0fori=0,1,\cdots ,n-3,\hfill \\ \hfill x^{(n-1)}\left(0\right)& =\alpha x^{(n-1)}\left(\xi \right),x^{(n-2)}\left(1\right)=\sum _{i=1}^m{\beta }_i{x}^{(n-2)}\left({\eta }_i\right).**\hfill \end{array}$$ Sufficient conditions for the existence of at least one solution of the BVP $(*)$ and $(**)$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002),...