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 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.