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.

We consider boundary value problems for second order differential equations of the form ${(x^{\text{'}}+g(t,x,x^{\text{'}}))}^{\text{'}}=f(t,x,x^{\text{'}})$ with the boundary conditions $r(x\left(0\right),x^{\text{'}}\left(0\right),x\left(T\right))+\varphi \left(x\right)=0$, $w(x\left(0\right),x\left(T\right),x^{\text{'}}\left(T\right))+\psi \left(x\right)=0$, where $g,r,w$ are continuous functions, $f$ satisfies the local Carathéodory conditions and $\varphi ,\psi$ are continuous and nondecreasing functionals. Existence results are proved by the method of lower and upper functions and applying the degree theory for $\alpha$-condensing operators.

Existence principles for solutions of singular differential systems${\left(\phi \left(u^{\text{'}}\right)\right)}^{\text{'}}=f(t,u,u^{\text{'}})$ satisfying nonlocal boundary conditions are stated. Here $\phi$ is a homeomorphism ${ℝ}^N$ onto ${ℝ}^N$ and the Carathéodory function $f$ may have singularities in its space variables. Applications of the existence principles are given.

Conditions for the existence and uniqueness of a solution of the Cauchy problem $$u^{\text{'}}\left(t\right)=p\left(t\right)u\left(\tau \left(t\right)\right)+q\left(t\right),u\left(a\right)=c,$$ established in [2], are formulated more precisely and refined for the special case, where the function $\tau$ maps the interval $]a,b[$ into some subinterval $[{\tau }_0,{\tau }_1]\subseteq [a,b]$, which can be degenerated to a point.