In this note we consider the boundary value problem $y^{\text{'}\text{'}}=f(x,y,y^{\text{'}})$ $(x\in [0,X];X>0)$, $y\left(0\right)=0$, $y\left(X\right)=a>0$; where $f$ is a real function which may be singular at $y=0$. We prove an existence theorem of positive solutions to the previous problem, under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173 (1993), 69–83], that extends and improves Theorem 3.2 of D. O’Regan [J. Differential Equations 84 (1990), 228–251].

The paper deals with the singular nonlinear problem $$u^{\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,u\left(0\right)=0,u^{\text{'}}\left(T\right)=\psi \left(u\left(T\right)\right),$$ where $f\in \mathrm{𝐶𝑎𝑟}\left(\right(0,T)\times D)$, $D=(0,\infty )\times$. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t=0$, $t=T$ and space singularity at $x=0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

We study the singular periodic boundary value problem of the form $${\left(\phi \left(u^{\text{'}}\right)\right)}^{\text{'}}+h\left(u\right)u^{\text{'}}=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),$$ where $\phi ℝ\to ℝ$ is an increasing and odd homeomorphism such that $\phi \left(ℝ\right)=ℝ,$ $h\in C[0,\infty ),$ $e\in L_{1}J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. ${lim\; sup}_{x\to 0+}\left|g\left(x\right)\right|=\infty$ may hold. We prove new existence results both for the case of an attractive singularity, when ${lim\; inf}_{x\to 0+}g\left(x\right)=-\infty ,$ and for the case of a strong repulsive singularity, when ${lim}_{x\to 0+}{\int }_x^{1}g\left(\xi \right)\text{d}\xi =\infty .$ In the latter case we assume that $\phi \left(y\right)={\phi }_p\left(y\right)={\left|y\right|}^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those...