In this paper we deal with the four-point singular boundary value problem $$\left\{\begin{array}{cc}{\left({\phi }_p\left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+q\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,\hfill & t\in (0,1),\hfill \\ u^{\text{'}}\left(0\right)-\alpha u\left(\xi \right)=0,u^{\text{'}}\left(1\right)+\beta u\left(\eta \right)=0,\hfill \end{array}\right.$$ where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q\left(t\right)>0$, $t\in (0,1)$, and $f\in C\left(\right[0,1]\times (0,+\infty )\times ℝ,(0,+\infty \left)\right)$ may be singular at $u=0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.

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.

The paper deals with the existence of multiple positive solutions for the boundary value problem $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right)u^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),...,u^{(n-2)}\left(t\right))=0,0<t<1,\hfill \\ u^{\left(i\right)}\left(0\right)=0,i=0,1,...,n-3,\hfill \\ u^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha }_i{u}^{(n-2)}\left({\xi }_i\right),u^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.