We study singular boundary value problems with mixed boundary conditions of the form $${\left(p\left(t\right)u^{\text{'}}\right)}^{\text{'}}+p\left(t\right)f(t,u,p\left(t\right)u^{\text{'}})=0,\underset{t\to 0+}{lim}p\left(t\right)u^{\text{'}}\left(t\right)=0,u\left(T\right)=0,$$ where $[0,T]\subset ℝ.$ We assume that ${ℝ}^2,$ $f$ satisfies the Carathéodory conditions on $(0,T)\times$ $p\in C[0,T]$ and $1/p$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$ ...

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 establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f=f(t,x,y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x,y$ may be also singular at time $t=0$. Two examples...

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.